Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories

FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS F OR ( ∞ , 2) -CA TEGORIES FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI Abstract. In the ﬁrst part of this pap er we study ﬁbrations of ( ∞ , 2) -categories. W e give a simple characterization of suc h ﬁbrations in terms of a certain square being a pullback, a nd apply this to show that in some cases ( ∞ , 2) -categories of functors and partially (op)lax transformations preserve ﬁbrations. W e also describe free ﬁbrations of ( ∞ , 2) -categories, including in the case where we only ask for (co)cartesian lifts of speciﬁed 1- and 2-morphisms in the base, and describe the right adjoin t to pullbac k from ﬁbrations to suc h partial ﬁ- brations along an arbitrary functor. In the second part of the paper we ap- ply these results to study colimits and Kan extensions of ( ∞ , 2) -categories. Most notably , we give a ﬁbrational description of b oth partially (op)lax and weigh ted (co)limits of ( ∞ , 2) -categories and construct partially lax Kan exten- sions. Among other results, w e also include a model-indep endent v ersion of co- ﬁnality for ( ∞ , 2) -categories and brieﬂy consider presen table ( ∞ , 2) -categories, characterizing them as accessible lo calizations of presheav es of ∞ -categories. Contents 1. In tro duction 2 2. Preliminaries 7 2.1. ( ∞ , 2) -categories and Gray tensor pro ducts 7 2.2. Mark ed ( ∞ , 2) -categories 12 2.3. Mark ed Gra y tensors and partially (op)lax transformations 16 2.4. Fibrations and the straightening equiv alence 19 2.5. Lax slices and cones 22 2.6. Decorated ( ∞ , 2) -categories 26 2.7. Decorated Gray tensor pro ducts and (op)lax transformations 31 3. Fibrations of decorated ( ∞ , 2) -categories 34 3.1. P artial ﬁbrations 35 3.2. Decorated partial ﬁbrations 36 3.3. Maps in (op)lax arrows 43 3.4. Characterizing partial ﬁbrations 46 3.5. P artial ﬁbrations on functors 50 3.6. ϵ -equiv alences and ϵ -coﬁbrations 53 4. F ree ﬁbrations and pushforwards for ( ∞ , 2) -categories 59 4.1. Fibrations from (op)lax arrows 59 4.2. F ree partial ﬁbrations 64 4.3. F ree decorated partial ﬁbrations 70 4.4. F ree ﬁbrations on decorated ( ∞ , 2) -categories 73 1 2 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI 4.5. Smo othness for decorated ( ∞ , 2) -categories 76 4.6. Pushforw ard of partial ﬁbrations 80 4.7. Lo calization of ﬁbrations 83 5. (Co)limits and Kan extensions in ( ∞ , 2) -categories 85 5.1. Lax and oplax (co)limits 86 5.2. W eighted (co)limits in ( ∞ , 2) -categories 89 5.3. (Co)limits of ( ∞ , 2) -categories in terms of ﬁbrations 94 5.4. Existence and preserv ation of (co)limits 99 5.5. Coﬁnal functors 102 5.6. Kan extensions 107 5.7. A Bousﬁeld–Kan formula for weigh ted colimits 112 5.8. F ree co completion 114 5.9. Presen table ( ∞ , 2) -categories 118 References 121 1. Intr oduction As our understanding of homotopy-coheren t structures in mathematics has gro wn, it has b ecome increasingly eviden t that in order to adequately capture some of the phenomena encountered in practice, it is necessary to go b eyond the by no w w ell-studied realm of ∞ -categories into ev en higher categorical spheres. The next lev el of complexit y , to which we will restrict our attention in the present pap er, is giv en by ( ∞ , 2) -c ate gories , whic h are the homotopy-coheren t analogue of classical 2-categories or bicategories. One wa y to approach to the theory of ( ∞ , 2) -categories is to regard them as ∞ -categories enriched in ∞ -categories [ GH15 , Hau15 ]. The theory of enriched ∞ - categories has recen tly b een extensiv ely developed by Hinich [ Hin20 ] and Heine [ Hei23 , Hei24 ], so this makes a v ailable to us tools such as the Y oneda lemma, w eighted (co)limits, and Kan extensions. Ho wev er, there are imp ortant asp ects of ( ∞ , 2) -category theory that are not visible from this p ersp ectiv e: most notably , as is well-kno wn from the classical theory of 2-categories we can use 2-morphisms to deﬁne lax versions of many constructions, such as the Gray tensor pro duct and lax transformations [ CM23 , GHL21 ]. The theory of ﬁbr ations and their straighten- ing to functors, whic h has turned out to play an extremely signiﬁcant role in the dev elopment of the theory of ∞ -categories and many of its applications, also do es not make sense for general enrichmen ts, but has an analogue for ( ∞ , 2) -categories [ GHL24 , AS23a , Nui24 ]; straightening of ﬁbrations furthermore gives us access to certain lax phenomena [ HHLN23 , AGH25 ]. Our goal in this pap er is to further develop the theory of ﬁbrations of ( ∞ , 2) - categories, and then apply these ﬁbrational results to obtain a relatively self-contained dev elopment of colimits and Kan extensions of ( ∞ , 2) -categories. In particular, w e giv e a thorough treatmen t of fr e e ﬁbrations and the pushforw ard of ﬁbrations along an arbitrary functor (including c ofr e e ﬁbrations); this allows us to access partially FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 3 lax versions of b oth (co)limits and Kan extensions, which we relate to their ana- logues from the enriched theory . Fibr ations of ( ∞ , 2) -c ate gories. A co cartesian ﬁbration of ( ∞ , 2) -categories, which w e will refer to as a (0 , 1) -ﬁbr ation in order to hav e succinct terminology for all four v ariances of ( ∞ , 2) -ﬁbrations, is a functor p : 𝔼 → 𝔹 where 𝔼 has p -co cartesian lifts of all 1-morphisms in 𝔹 and p -cartesian lifts of all 2-morphisms (see § 2.4 for a more precise deﬁnition). Our ﬁrst main result gives a new 1 c haracterization of such ﬁbrations in terms of a certain square of ( ∞ , 2) -categories b eing a pullback. T o state this, we need to consider an ( ∞ , 2) -category where some 1- and 2-morphisms ha ve b een singled out; to av oid confusion with pre-existing terminology we refer to suc h ob jects as de c or ate d ( ∞ , 2) -categories, reserving the term marke d to refer to the case where we only lab el 1-morphisms. Theorem A. Supp ose p : 𝔼 → 𝔹 is a functor of ( ∞ , 2) -c ate gories and 𝔼 ⋄ is a de c or ation of 𝔼 by b oth 1- and 2-morphisms. Then p is a (0 , 1) -ﬁbr ation, with these de c or ations giving the p -c o c artesian 1-morphisms and p -c artesian 2-morphisms in 𝔼 , if and only if the c ommutative squar e 𝔻𝔸 r oplax ( 𝔼 ⋄ ) 𝔸 r oplax ( 𝔹 ) 𝔼 𝔹 𝔸 r oplax ( p ) ev 1 ev 1 p is a pul lb ack of ( ∞ , 2) -c ate gories. Here 𝔻𝔸 r oplax ( 𝔼 ⋄ ) is the lo cally full sub- ( ∞ , 2) -category of the oplax arro w ( ∞ , 2) -category 𝔸 r oplax ( 𝔼 ) whose ▶ ob jects are the decorated 1-morphisms in 𝔼 ⋄ , ▶ morphisms are the oplax squares • • • • con taining a decorated 2-morphism. W e prov e this theorem in § 3.4 ; in fact, we pro v e more generally such a character- ization of p artial ϵ -ﬁbr ations for all four v ariances ϵ ∈ { 0 , 1 } × 2 , meaning functors p : 𝔼 → 𝔹 where 𝔼 only has (co)cartesian lifts of certain 1- and 2-morphisms sp ec- iﬁed by a decoration of 𝔹 . Let us next note tw o useful results w e prov e as easy consequences of this characterization: Corollary B. Supp ose p : 𝔼 → 𝔹 is a (0 , 1) -ﬁbr ation. Then so is the functor p ∗ : 𝔽 un ( 𝕂 , 𝔼 ) lax → 𝔽 un ( 𝕂 , 𝔹 ) lax , 1 Note, ho w ever, that (as w e discuss in § 3.6 ) this statement can b e rephrased as characterizing partial (0 , 1) -ﬁbrations by an orthogonality condition, which has also previously b een considered by Loubaton in the setting of ( ∞ , ∞ ) -categories [ Lou24 , Theorem 3.2.2.24]. 4 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI given by c omp osition with p , for any ( ∞ , 2) -c ate gory 𝕂 . Its c o c artesian morphisms ar e those lax tr ansformations whose c omp onent at e ach obje ct of 𝕂 is p -c o c artesian and whose lax natur ality squar es al l c ontain a p -c artesian 2-morphism, and its c arte- sian 2-morphisms ar e those whose c omp onent at e ach obje ct is p -c artesian. Corollary C. F or any ( ∞ , 2) -c ate gory 𝔸 , the tar get functor ev 1 : 𝔸 r oplax ( 𝔸 ) → 𝔸 is a (0 , 1) -ﬁbr ation. Its c o c artesian morphisms ar e the c ommutative squar es whose sour c e is an e quivalenc e in 𝔸 , and its c artesian 2-morphisms ar e those whose sour c e is an e quivalenc e. W e prov e the ﬁrst statement, and its partial and decorated generalizations, in § 3.5 , using decorated v ersions of the Gra y tensor pro duct and lax transformations w e set up in § 2.7 . The second statement, whose generalized version is prov ed in § 4.1 , has previously b een prov ed using scaled simplicial sets by Gagna, Harpaz, and Lanari [ GHL24 , Theorem 3.0.7]; for us, it is the starting p oint for the construction of free ﬁbrations: Theorem D. F or any ( ∞ , 2) -c ate gory 𝔹 , the for getful functor Fib (0 , 1) / 𝔹 → Cat ( ∞ , 2) / 𝔹 has a left adjoint, which sends a functor F : 𝔸 → 𝔹 to 𝔸 × 𝔹 𝔸 r oplax ( 𝔹 ) → 𝔹 , wher e the pul lb ack is taken via F and ev 0 and the functor to 𝔹 is given by ev 1 . Here Cat ( ∞ , 2) / 𝔹 is the ∞ -category of ( ∞ , 2) -categories with a map to 𝔹 and Fib (0 , 1) / 𝔹 is its subcategory that con tains the (0 , 1) -ﬁbrations and the maps that preserv e cocartesian 1-morphisms and cartesian 2-morphisms. A version of this result (or more precisely its dual) based on scaled simplicial sets w as previously pro ved b y the ﬁrst author and Stern [ AS23b , Theorem 3.17]. W e prov e Theorem 4.2.1 and its partial v ariants in § 4.2 , and also consider some further v ariations of free ﬁbrations in §§ 4.3 – 4.4 . After this we mov e on to consider- ing right adjoin ts to pullback for ( ∞ , 2) -categories: In § 4.5 w e upgrade the existence of righ t adjoin ts to pullbac ks along ﬁbrations, due to Ab ellán–Stern [ AS23a ], to the decorated setting, and we then apply this in § 4.6 to show that there is a righ t ad- join t to pulling back ﬁbrations to partial ﬁbrations along an arbitrary functor. As a particularly notable sp ecial case, we ha ve: Theorem E. F or any functor F : 𝔸 → 𝔹 and any c ol le ction I of 1-morphisms in 𝔸 , the functor F ∗ : Fib (0 , 1) / 𝔹 → Fib (0 , 1) / ( 𝔸 ,I ) , given by pul lb ack along F , has a right adjoint. Here Fib (0 , 1) / ( 𝔸 ,I ) denotes the subcategory of Cat ( ∞ , 2) / 𝔸 whose ob jects are the (0 , 1) - ﬁbrations and whose morphisms are required to preserve all cartesian 2-morphisms, FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 5 but only those co cartesian morphisms that lie ov er I . Another notable sp ecial case is the existence of c ofr e e ﬁbrations — in other words, the forgetful functor Fib (0 , 1) / 𝔹 → Cat ( ∞ , 2) / 𝔹 has a right adjoint. W e also apply these results in § 4.7 to pro v e that if we hav e a functor to decorated ( ∞ , 2) -categories, then the (0 , 1) -ﬁbration for the functor obtained by in verting the decorations is given by a lo calization of the ﬁbration for the original functor, generalizing a result of Hinich [ Hin16 ] to ( ∞ , 2) -categories. (Co)limits and Kan extensions. In the last part of the pap er we apply our results on ﬁbrations to study (co)limits and Kan extensions for ( ∞ , 2) -categories. Giv en a functor F : 𝔸 → ℂ and a collection I of 1-morphisms in 𝔸 , w e can deﬁne the I -(op)lax limit of F as an ob ject of ℂ that represents the presheaf of ∞ -categories c 7→ Nat I -(op)lax 𝔸 , ℂ ( c, F ) , where the right-hand side denotes the ∞ -category of (op)lax natural transforma- tions from the constan t functor c to F whose naturalit y squares at morphisms in I comm ute. This deﬁnition and its dual gives one notion of ( ∞ , 2) -categorical (co)limits, and in § 5.1 we show that it is equiv alen t to the deﬁnition of partially (op)lax (co)limits previously studied by the ﬁrst author in [ A G22 ] and b y Gagna, Harpaz and Lanari in [ GHL25 ] in the context of scaled simplicial sets. F or a functor F as ab o ve and a copresheaf of ∞ -categories W on 𝔸 , we can also deﬁne the W -weighte d limit of F as an ob ject of ℂ that represen ts the presheaf c 7→ Nat 𝔸 , ℂ at ∞ ( W , ℂ ( c, F )) , where the right-hand side denotes the ∞ -category of natural transformations from W to the ∞ -category of maps from c in to F ( – ) in ℂ . In § 5.2 we show that w eigh ted (co)limits can b e expressed as partially lax and oplax (co)limits o ver the ﬁbrations for the weigh t, and conv ersely that partially (op)lax colimits can b e computed as w eighted colimits. This was also previously prov ed b y Gagna, Harpaz and Lanari [ GHL25 ], but our work on free ﬁbrations allows us to give a more explicit description of the weigh ts that show up. W e then turn to our main new result on (co)limits in § 5.3 , where we compute par- tially (op)lax (co)limits in the ( ∞ , 2) -category ℂ at ( ∞ , 2) of small ( ∞ , 2) -categories in terms of ﬁbrations: Theorem F. Supp ose p : 𝔼 → 𝔸 is the (0 , 1) -ﬁbr ation for a functor F : 𝔸 → ℂ at ( ∞ , 2) . F or a c ol le ction I of 1-morphisms in 𝔸 , we have: (1) The I -lax c olimit of F is obtaine d fr om 𝔼 by inverting al l c artesian 2-morphisms and those c o c artesian 1-morphisms that lie over I . (2) The I -lax limit of F is given by the ( ∞ , 2) -c ate gory of se ctions 𝔸 → 𝔼 of p that send al l 2-morphisms to p -c artesian 2-morphisms and the 1-morphisms in I to p -c o c artesian morphisms. 6 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Other v ariances of ﬁbrations can b e used to get the I -oplax (co)limits of F . Note that a similar description of partially (op)lax (co)limits for ( ∞ , ∞ ) -categories app ears in work of Loubaton [ Lou24 , Examples 4.2.3.12–13]. W e next prov e some results on the existence and preserv ation of (co)limits in § 5.4 , including the existence of certain (co)limits in ( ∞ , 2) -categories of functors and partially lax transformations (Prop osition 5.4.2 ). After this we see in § 5.5 that our framew ork of decorated ( ∞ , 2) -categories and partial ﬁbrations mak es it v ery easy to understand coﬁnalit y for functors of mark ed ( ∞ , 2) -categories, and w e obtain model- indep enden t proofs of most of the results on coﬁnality from [ AS23b , A G22 , GHL25 ] with very little eﬀort. If we com bine Theorem E with the straightening for partially lax transformations from [ A GH25 ], w e get that for a functor of ( ∞ , 2) -categories F : 𝔸 → 𝔹 and a collection I of 1-morphisms in 𝔸 , the restriction functor F ∗ : 𝔽 un ( 𝔹 , ℂ at ( ∞ , 2) ) → 𝔽 un ( 𝔸 , ℂ at ( ∞ , 2) ) I -lax has a righ t adjoint. In § 5.6 we combine this with our work on (co)limits and the Y oneda embedding to show that such adjoints exist more generally: Theorem G. Supp ose F is as ab ove and ℂ is an ( ∞ , 2) -c ate gory that for every b ∈ 𝔹 admits I b -lax limits over 𝔸 b → , then the r estriction functor F ∗ : 𝔽 un ( 𝔹 , ℂ ) → 𝔽 un ( 𝔸 , ℂ ) I -lax has a right adjoint F I -lax ∗ , the I -lax right Kan extension functor along F , given at ϕ : 𝔸 → ℂ and b ∈ 𝔹 by ( F I -lax ∗ ϕ )( b ) ≃ lim I b -lax 𝔸 b → F . Here 𝔸 b → denotes the pullback 𝔸 × 𝔹 𝔹 b → where 𝔹 b → is the ﬁbre of ev 0 : 𝔸 r oplax ( 𝔹 ) → 𝔹 at b , and I b denotes the maps therein that pro ject to I . More generally , we obtain b oth left and right partially (op)lax Kan extensions, extending to a general target results prov ed using scaled simplicial sets in [ Ab e23 ]. T aking the marking I to b e maximal, w e obtain ordinary Kan extensions for ( ∞ , 2) -categories; these can also b e obtained using enriched ∞ -category theory [ Hei24 ], and our (co)limit form ula sp ecializes to the exp ected one in terms of weigh ted (co)limits (Corollary 5.6.6 ). W e apply our results on Kan extensions to pro v e a Bousﬁeld–Kan form ula for w eighted (co)limits in § 5.7 ; this gives a diﬀeren t proof of the ( ∞ , 2) -categorical case of a result of Heine [ Hei24 , Theorem 3.44] for general enrichmen ts. After this w e use our description of pushforward for ﬁbrations to obtain a functorial version of the Y oneda lemma in § 5.8 , whic h w e further apply to pro ve that taking the ( ∞ , 2) -category ℙ𝕊 h ( ℂ ) := 𝔽 un ( ℂ op , ℂ at ∞ ) of preshea v es of ∞ -categories gives the free co completion of a small ( ∞ , 2) -category ℂ . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 7 Finally , we brieﬂy study pr esentable ( ∞ , 2) -categories in § 5.9 . W e deﬁne these rather naïvely as co complete ( ∞ , 2) -categories whose underlying ∞ -category is pre- sen table, and as our main result obtain the following equiv alent c haracterizations of this notion: Theorem H. The fol lowing ar e e quivalent for an ( ∞ , 2) -c ate gory ℂ : (1) ℂ is pr esentable. (2) ℂ is c o c omplete and lo c al ly smal l, and ther e is a r e gular c ar dinal κ and a smal l ful l sub- ( ∞ , 2) -c ate gory ℂ 0 ⊆ ℂ c onsisting of 2- κ -c omp act obje cts, such that every obje ct in ℂ is the c onic al c olimit of a diagr am in ℂ indexe d by a κ -ﬁlter e d ∞ -c ate gory. (3) Ther e is a smal l ( ∞ , 2) -c ate gory 𝕁 and a ful ly faithful functor ℂ  → ℙ𝕊 h ( 𝕁 ) that admits a left adjoint and whose image is close d under c onic al c olimits over κ -ﬁlter e d ∞ -c ate gories for some r e gular c ar dinal κ . (4) Ther e is a smal l ( ∞ , 2) -c ate gory 𝕁 , a smal l set S of 1-morphisms in ℙ𝕊 h ( 𝕁 ) and an e quivalenc e ℂ ≃ Lo c S ( ℙ𝕊 h ( 𝕁 )) with the ful l sub- ( ∞ , 2) -c ate gory of obje cts that ar e 2-lo c al with r esp e ct to S . W e refer the reader to § 5.9 for the precise deﬁnitions of the terms that app ear here. Note that similar results on prese n tabilit y for general enriched ∞ -categories also app ear in the work of Heine, in particular [ Hei24 , Theorem 5.10]. A cknow le dgments. W e thank the Max Planc k Institute for Mathematics in Bonn, where part of this work was carried out, for its hospitality and supp ort. 2. Preliminaries In this section w e ﬁrst recall some terminology and basic results on ( ∞ , 2) - categories in § 2.1 . W e then review marked ( ∞ , 2) -categories in § 2.2 and their mark ed Gra y tensor product in § 2.3 . Next, we review ﬁbrations of ( ∞ , 2) -categories and their straightening in § 2.4 and lax slices and cones in § 2.5 . After this w e are ready to in troduce de c or ate d ( ∞ , 2) -categories, b y whic h w e mean ( ∞ , 2) -categories equipp ed with collections of sp ecial 1- and 2-morphisms, in § 2.6 ; this will provide a v ery conv enien t setting for our w ork on ﬁbrations in the following sections. Finally , w e in tro duce a decorated version of the Gray tensor pro duct in § 2.7 . 2.1. ( ∞ , 2) -categories and Gra y tensor pro ducts. In this section w e introduce our basic terminology and notation and recall some fundamental results on ( ∞ , 2) - categories and their Gra y tensor pro ducts. As we hardly ever need to refer to any particular mo del of ( ∞ , 2) -categories, we will not review any of them here. Notation 2.1.1. ▶ Generic ( ∞ , 2) -categories are denoted 𝔸 , 𝔹 , ℂ , . . . , and generic ∞ -categories A , B , C , . . . . ▶ W e write ∗ = [0] = C 0 for the 0-cell or p oint, [1] = C 1 for the 1-cell or arrow, and C 2 for the 2-cell or free 2-morphism. W e note also the following notation 8 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI for 2-categories built from these: ∂ [1] := { 0 } ⨿ { 1 } , ∂ C 2 := [1] ⨿ ∂ [1] [1] , ∂ C 3 := C 2 ⨿ ∂ C 2 C 2 . ▶ If ℂ is an ( ∞ , 2) -category with ob jects x, y , w e will generally denote the ∞ - category of morphisms from x to y in ℂ by ℂ ( x, y ) , while if C is an ∞ -category w e will write C ( x, y ) or Map( x, y ) for the ∞ -group oid of morphisms. ▶ W e write Cat ∞ and Cat ( ∞ , 2) for the ∞ -categories of (small) ∞ -categories and ( ∞ , 2) -categories, and ℂ at ∞ and ℂ at ( ∞ , 2) for the ( ∞ , 2) -categories thereof, re- sp ectiv ely; similarly , we write Gpd ∞ for the ∞ -category of (small) ∞ -group oids or spaces. ▶ If ℂ and 𝔻 are ( ∞ , 2) -categories, we write 𝔽 un ( ℂ , 𝔻 ) for the ( ∞ , 2) -category of functors b etw een them (the internal Hom in Cat ( ∞ , 2) ), and Fun ( ℂ , 𝔻 ) for its underlying ∞ -category. F or functors F , G : ℂ → 𝔻 we denote the ∞ -category of maps from F to G in 𝔽 un ( ℂ , 𝔻 ) by Nat ℂ , 𝔻 ( F , G ) . ▶ F or the arrow [1] , we write 𝔸 r ( ℂ ) := 𝔽 un ([1] , ℂ ) for the ( ∞ , 2) -category of arro ws in an ( ∞ , 2) -category ℂ and Ar ( ℂ ) for its underlying ∞ -category; the functor to ℂ given by ev aluation at i ∈ [1] is denoted ev i ( i = 0 , 1 ). ▶ W e write ℙ𝕊 h ( ℂ ) := 𝔽 un ( ℂ op , ℂ at ∞ ) for the ( ∞ , 2) -category of preshea ves of ∞ -categories on an ( ∞ , 2) -category ℂ , and h ℂ : ℂ → ℙ𝕊 h ( ℂ ) for the Y oneda em b edding [ Hin20 ]. ▶ W e say an ( ∞ , 2) -category ℂ admits tensors by an ∞ -category K if the co- presheaf Map( K , ℂ ( c, – )) is corepresen table by an ob ject K ⊠ c for every c ∈ ℂ ; dually , ℂ admits c otensors b y K if the presheav es Map( K , ℂ ( – , c )) are repre- sen table b y ob jects c K . If ℂ admits (co)tensors b y all small ∞ -categories, w e sa y that ℂ is (c o)tensor e d . ▶ If ℂ is an ( ∞ , 2) -category, w e write ℂ ≃ or ℂ ≤ 0 for its underlying (or core) ∞ -group oid, and ℂ ≤ 1 for its underlying ∞ -category; these are right adjoint to the inclusions of Gpd ∞ and Cat ∞ in Cat ( ∞ , 2) , resp ectively . ▶ These inclusions also hav e left adjoints, which we denote ∥ – ∥ = τ ≤ 0 : Cat ( ∞ , 2) → Gpd ∞ , τ ≤ 1 : Cat ( ∞ , 2) → Cat ∞ , resp ectiv ely . Deﬁnition 2.1.2. F or n ≥ 0 we deﬁne 𝕆 n to b e the following strict 2-category: ▶ The ob jects are the elements of [ n ] . ▶ F or i, j ∈ [ n ] , the mapping category 𝕆 n ( i, j ) is the p oset of subsets S ⊆ [ n ] suc h that min( S ) = i and max( S ) = j , ordered b y inclusion. ▶ F or a triple i, j, k ∈ [ n ] , the comp osition maps are given by taking unions. The 2-categories 𝕆 n corresp ond to the (2-truncated) orientals or oriented simplices deﬁned by Street [ Str87 ], and 𝕆 n should b e thought of as an n -simplex with com- patibly oriented 2-morphisms inserted in all of its 2-dimensional faces. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 9 Deﬁnition 2.1.3. Supp ose ℂ and 𝔻 are ( ∞ , 2) -categories and that ℂ admits ten- sors by an ∞ -category K . A functor F : ℂ → 𝔻 then induces for c ∈ ℂ a functor ℂ ( K ⊠ c, K ⊠ c ) ≃ ≃ Map( K , ℂ ( c, K ⊠ c )) → Map( K , 𝔻 ( F c, F ( K ⊠ c ))) , so that w e get a canonical functor K → 𝔻 ( F c, F ( K ⊠ c )) from id K ⊠ c . W e sa y that F pr eserves tensors by K if this exhibits the copresheaf Map( K , 𝔻 ( F c, – )) as corepresen ted b y F ( K ⊠ c ) . ( ∞ , 2) -categories can b e deﬁned as ∞ -categories enriche d in Cat ∞ (see [ Hau15 ]). The general theory of enriched ∞ -categories has recen tly been extensively developed b y Hinich [ Hin20 ] and Heine [ Hei23 , Hei24 ]. In particular, it follows from Heine’s w ork that an ( ∞ , 2) -category ℂ is tensored if and only if its enric hmen t arises from a Cat ∞ -mo dule structure on the underlying ∞ -category ℂ ≤ 1 , and this correspondence is part of an equiv alence of ∞ -categories. F rom this we get the following useful pro cedure to construct ( ∞ , 2) -categories and functors among them: Observ ation 2.1.4. Supp ose ϕ : Cat ∞ → C is a pro duct-preserving functor. This mak es C an algebra o ver Cat ∞ , and so a mo dule with the action given b y ( K , c ) 7→ ϕ ( K ) × c . If the functor K 7→ ϕ ( K ) × c has a right adjoint ℂ ( c, – ) for all c ∈ C , then w e can upgrade C to an ( ∞ , 2) -category ℂ , where the mapping ∞ -categories satisfy Map( K , ℂ ( c, c ′ )) ≃ C ( ϕ ( K ) × c, c ′ ) . Moreo ver, if we hav e a commutativ e triangle of ∞ -categories Cat ∞ C D φ ψ F where ϕ and ψ preserve pro ducts and there are righ t adjoin ts as ab ov e, and also the functor F preserves pro ducts, then F upgrades to a functor of ( ∞ , 2) -categories for the Cat ∞ -enric hments of C and D induced by the functors ϕ and ψ . Similarly , if the same prop erties hold with Cat ∞ replaced b y Cat ( ∞ , 2) , we can upgrade (functors of ) ∞ -categories to (functors of ) ( ∞ , 3) -categories. W e next note the following useful criterion for upgrading adjunctions; this is also a sp ecial case of [ Ste20 , Prop osition 4.4.1] (where the presentabilit y assumption is not used in the pro of ) or [ MGS24 , Lemma A.2.14]. Prop osition 2.1.5. Supp ose F : ℂ → 𝔻 is a functor of ( ∞ , 2) -c ate gories such that ℂ admits tensors by [1] and F pr eserves these. If the functor F ≤ 1 is a left adjoint, then so is F . Note that since ( – ) ≤ 1 is a 2-functor (e.g. b y Observ ation 2.1.4 ), it preserves adjunctions, and so in the situtation ab ov e the underlying functor of ∞ -categories of the right adjoint of F is necessarily the right adjoint of F ≤ 1 . Pr o of. By the Y oneda Lemma for ( ∞ , 2) -categories it suﬃces to show that the presheaf of ∞ -categories 𝔻 ( F ( – ) , d ) is representable for all d ∈ 𝔻 . Since F ≤ 1 has a 10 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI righ t adjoin t g , we ha v e a counit map  : F ( g ( d )) → d so that the comp osite ℂ ( c, g ( d )) ≃ → ℂ ( F ( c ) , F ( g ( d ))) ≃ → ℂ ( F ( c ) , d ) ≃ is an equiv alence for every c . W e claim that w e also hav e an equiv alence without taking cores; to see this it is enough to chec k that the comp osite Map([1] , ℂ ( c, g ( d ))) → Map([1] , ℂ ( F ( c ) , F ( g ( d )))) → Map([1] , ℂ ( F ( c ) , d )) is an equiv alence, and since F preserves tensors by [1] w e may identify this as the ﬁrst map with c replaced by [1] ⊠ c . □ Corollary 2.1.6. Supp ose F : C → D is a functor of ∞ -c ate gories as in Observa- tion 2.1.4 . (i) If F has a right adjoint R , then the adjunction up gr ades c anonic al ly to an adjunction F ⊣ R of ( ∞ , 2) -c ate gories. (ii) If F has a left adjoint L , the tensoring of C and D by [1] has a right adjoint, and the c anonic al map L ([1] ⊠ d ) → [1] ⊠ Ld is an e quivalenc e for al l d , then the adjunction up gr ades c anonic al ly to an adjunction L ⊣ F of ( ∞ , 2) -c ate gories. Pr o of. The ﬁrst part is immediate from Prop osition 2.1.5 , while the second part follo ws from this together with the observ ation that F preserves all cotensors b y [1] if and only if its left adjoint preserves all tensors. □ W e also recall some useful terminology for subob jects of ∞ -categories and ( ∞ , 2) - categories: Deﬁnition 2.1.7. A functor F : C → D of ∞ -categories is: ▶ faithful if C ( c, c ′ ) → D ( F c, F c ′ ) is a monomorphism of ∞ -group oids for all c, c ′ ∈ C ; ▶ an inclusion if it is faithful and C ≃ → D ≃ is a monomorphism — w e then also sa y that C is a sub c ate gory of D . Deﬁnition 2.1.8. A functor F : ℂ → 𝔻 of ( ∞ , 2) -categories is called ▶ ful ly faithful if ℂ ( c, c ′ ) → 𝔻 ( F c, F c ′ ) is an equiv alence of ∞ -categories for all ob jects c, c ′ ∈ ℂ — we then also say that ℂ is a ful l sub- ( ∞ , 2) -c ate gory of 𝔻 ; ▶ lo c al ly ful ly faithful if ℂ ( c, c ′ ) → 𝔻 ( F c, F c ′ ) is fully faithful for all c, c ′ ∈ ℂ ; ▶ a lo c al ly ful l inclusion if F is lo cally fully faithful and ℂ ≃ → 𝔻 ≃ is a monomor- phism — we then also say that ℂ is a lo c al ly ful l sub- ( ∞ , 2) -c ate gory of 𝔻 ; ▶ lo c al ly faithful if ℂ ( c, c ′ ) → 𝔻 ( F c, F c ′ ) is faithful for all c, c ′ ∈ ℂ ; ▶ lo c al ly an inclusion if ℂ ( c, c ′ ) → 𝔻 ( F c, F c ′ ) is a sub category inclusion for all c, c ′ ∈ ℂ ; ▶ an inclusion if it is lo cally an inclusion and ℂ ≃ → 𝔻 ≃ is a monomorphism — w e then also say that ℂ is a sub- ( ∞ , 2) -c ate gory of 𝔻 . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 11 W e also sa y that a sub- ( ∞ , 2) -category ℂ of 𝔻 is wide if ℂ ≃ → 𝔻 ≃ is an equiv alence. As discussed in [ AGH25 , §2.5], all of these prop erties can b e characterized by F b eing right orthogonal to some ﬁnite set of maps in Cat ( ∞ , 2) . Notation 2.1.9. The Gr ay tensor pr o duct of ( ∞ , 2) -categories will play an imp or- tan t role in this pap er, but as this is b y now a standard construction w e will not review the deﬁnition here (see for instance [ GHL21 ] for a deﬁnition using scaled simplicial sets or [ CM23 ] for a version based on Θ -spaces). ▶ The Gray tensor pro duct of tw o ( ∞ , 2) -categories ℂ and 𝔻 is denoted ℂ ⊗ 𝔻 . ▶ W e write Fun (op)lax ( – , – ) for the adjoints of the Gray tensor in each of the tw o v ariables, so that we ha ve natural equiv alences Map( ℂ ⊗ 𝔻 , 𝔼 ) ≃ Map( 𝔻 , 𝔽 un ( ℂ , 𝔼 ) oplax ) ≃ Map( ℂ , 𝔽 un ( 𝔻 , 𝔼 ) lax ) . Here F un (op)lax ( ℂ , 𝔻 ) is an ( ∞ , 2) -category whose ob jects are functors from ℂ to 𝔻 , with morphisms given by (op)lax natur al tr ansformations among these. ▶ F or [1] , we write 𝔸 r (op)lax ( ℂ ) := 𝔽 un ([1] , ℂ ) (op)lax . ▶ F or functors F , G : ℂ → 𝔻 , we denote the ∞ -category of maps from F to G in F un (op)lax ( ℂ , 𝔻 ) by Nat (op)lax ℂ , 𝔻 ( F , G ) . Observ ation 2.1.10. There is a pushout square ∂ C 2 C 2 ∂ ([1] ⊗ [1]) [1] ⊗ [1] where ∂ ([1] ⊗ [1]) denotes the b oundary ( ∂ [1] × [1]) ⨿ ∂ [1] × ∂ [1] ([1] × ∂ [1]) ≃ [2] ⨿ { 0 , 2 } [2] , with the left vertical map given by gluing tw o copies of d 1 : [1] → [2] . F or example, using the mo del of scaled simplicial sets it is easy to see that there is a pushout ∂ 𝕆 2 𝕆 2 ∂ ([1] ⊗ [1]) [1] ⊗ [1] , from which the square ab ov e follows using [ AGH25 , Prop osition 2.3.6]. W e emphasize that the Gray tensor pro duct is not compatible with cartesian pro ducts, and so do es not giv e a functor of ( ∞ , 2) -categories. How ev er, it is p ossible to upgrade 𝔽 un (op)lax ( 𝔸 , – ) for a ﬁxed 𝔸 to a 2-functor; we will not prov e this here, but instead w e note the useful consequence that this functor preserves adjunctions: Construction 2.1.11. Supp ose R : Cat ( ∞ , 2) → Cat ( ∞ , 2) is a pro duct-preserving functor and α : id → R is a natural transformation. W e then hav e a natural map 𝔽 un ( 𝔸 , 𝔹 ) → 𝔽 un ( R 𝔸 , R 𝔹 ) 12 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI adjoin t to the comp osite 𝔽 un ( 𝔸 , 𝔹 ) × R 𝔸 α × id − − − → R 𝔽 un ( 𝔸 , 𝔹 ) × R 𝔸 ≃ R ( 𝔽 un ( 𝔸 , 𝔹 ) × 𝔸 ) R (ev) − − − → R 𝔹 . In particular, from a natural transformation ϕ : 𝔸 × [1] → 𝔹 from F to G we get a natural transformation ˜ R ( ϕ ) : R 𝔸 × [1] id × α − − − → R 𝔸 × R [1] ≃ R ( 𝔸 × [1]) Rφ − − → R ( 𝔹 ) . It is easy to chec k that this construction is compatible with b oth horizontal and v ertical comp osition of natural transformations, so that we get: Prop osition 2.1.12. L et R : Cat ( ∞ , 2) → Cat ( ∞ , 2) b e a pr o duct-pr eserving functor and α : id → R a natur al tr ansformation. Supp ose F : ℂ → 𝔻 is a functor of ( ∞ , 2) - c ate gories with right adjoint G , with unit η and c ounit  . Then R ( F ) is left adjoint to R ( G ) with unit ˜ R ( η ) and c ounit ˜ R (  ) . □ Remark 2.1.13. In fact, in the situation abov e we should b e able to upgrade R to a functor of ( ∞ , 2) -categories (or even ( ∞ , 3) -categories) using the transforma- tion α : Since R preserves pro ducts, it is a monoidal functor with resp ect to the cartesian product, and α is automatically a monoidal transformation. W e can in particular regard R as a (lax) linear endofunctor for the canonical action of Cat ( ∞ , 2) on itself. This corresp onds to a Cat ( ∞ , 2) -enric hed functor R ∗ ℂ at ( ∞ , 2) → ℂ at ( ∞ , 2) [ Hin20 , Hei23 ], where we push forw ard the enric hment along R . Moreov er, pushfor- w ard of enrichmen t is a 2-functor (for example b ecause it arises from comp osition in the ( ∞ , 2) -category of ∞ -op erads in the mo del of [ GH15 ]), so the monoidal trans- formation α also gives rise to a Cat ( ∞ , 2) -enric hed functor ℂ at ( ∞ , 2) → R ∗ ℂ at ( ∞ , 2) . The comp osite ℂ at ( ∞ , 2) → R ∗ ℂ at ( ∞ , 2) → ℂ at ( ∞ , 2) should then b e the desired en- ric hed lift of R , but unfortunately we hav e not found any proof in the literature that the underlying functor of ∞ -categories recov ers the original functor R , and we will not go into this here. Example 2.1.14. F or any ( ∞ , 2) -category 𝔸 , the functor 𝔽 un ( 𝔸 , – ) (op)lax : Cat ( ∞ , 2) → Cat ( ∞ , 2) preserv es adjunctions, since it preserves pro ducts (b eing a right adjoint) and we ha ve a natural transformation from the identit y given by the constan t diagrams ℂ → 𝔽 un ( 𝔸 , ℂ ) (op)lax , or equiv alen tly induced by the natural transformation of left adjoints 𝔸 ⊗ ( – ) → [0] ⊗ ( – ) ≃ id . 2.2. Mark ed ( ∞ , 2) -categories. In this subsection we review marke d ( ∞ , 2) - categories, whic h are ( ∞ , 2) -categories equipp ed with a collection of morphisms, and compare tw o descriptions of the ∞ -category thereof. Deﬁnition 2.2.1. A marke d ( ∞ , 2) -c ate gory ( ℂ , E ) consists of an ( ∞ , 2) -category ℂ together with a collection E of morphisms in ℂ . W e ma y assume that E con- tains all equiv alences in ℂ and is moreov er closed under comp osition, so that it FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 13 determines a wide locally full sub- ( ∞ , 2) -category ℂ E  → ℂ . This allo ws us to deﬁne the ∞ -category MCat ( ∞ , 2) of mark ed ( ∞ , 2) -categories as the full sub cate- gory of Ar ( Cat ( ∞ , 2) ) spanned by the wide lo cally full sub category inclusions. W e write u m : MCat ( ∞ , 2) → Cat ( ∞ , 2) for the forgetful functor ( ℂ , E ) 7→ ℂ , given by restricting ev 1 . Prop osition 2.2.2. Given an ∞ -c ate gory C e quipp e d with a c ol le ction S of 1- morphisms, let Ar S ( C ) b e the ful l sub c ate gory of Ar ( C ) sp anne d by the morphisms in S . If al l elements of S ar e monomorphisms, then ev 1 : Ar S ( C ) → C is a faithful functor. Pr o of. It suﬃces to show that for morphisms f : x → y , g : a → b in C , the map Ar ( C )( f , g ) → C ( y , b ) induced by ev aluation at 1 is a monomorphism of ∞ -group oids if g is a monomor- phism in C . Here Ar ( C )( f , g ) ﬁts in a pullbac k square Ar ( C )( f , g ) C ( x, a ) C ( y , b ) C ( x, b ) , ev 0 ev 1 g ∗ f ∗ so this is clear since monomorphisms are closed under pullback. □ Corollary 2.2.3. The functor u m : MCat ( ∞ , 2) → Cat ( ∞ , 2) is faithful. □ Observ ation 2.2.4. By [ AGH25 , Observ ations 2.5.6, 2.5.8], a functor of ( ∞ , 2) - categories is a wide lo cally full sub category inclusion if and only if it is righ t orthog- onal to ∅ → ∗ , ∂ C 2 → C 2 . It follo ws from [ Lur09a , Prop osition 5.5.5.7] that such functors form the right class in a factorization system on Cat ( ∞ , 2) , whic h implies that the inclusion MCat ( ∞ , 2)  → Ar ( Cat ( ∞ , 2) ) has a left adjoint, given by factoring a morphism through this factorization system. As the morphisms w e consider or- thogonalit y for also go b et w een compact ( ∞ , 2) -categories, it is moreov er clear that MCat ( ∞ , 2) is closed under ﬁltered colimits in Ar ( Cat ( ∞ , 2) ) , so that this left adjoint exhibits MCat ( ∞ , 2) as an accessible lo calization. Th us MCat ( ∞ , 2) is a presentable ∞ -category. Observ ation 2.2.5. The functor ev 1 : Ar ( Cat ( ∞ , 2) ) → Cat ( ∞ , 2) has the following prop erties: (1) ev 1 has a left adjoint, which takes an ( ∞ , 2) -category 𝔸 to ∅ → 𝔸 . (2) ev 1 has a right adjoint const , which takes an ( ∞ , 2) -category 𝔸 to 𝔸 = − → 𝔸 . (3) The right adjoin t const is itself left adjoint to ev 0 . F rom this w e can conclude that the restriction u m of ev 0 to has the following corresp onding prop erties: (1’) u m has a left adjoint ( – ) ♭ , which takes an ( ∞ , 2) -category 𝔸 to 𝔸 ♭ := 𝔸 eq → 𝔸 , where 𝔸 eq is the sub- ( ∞ , 2) -category that contains only the in vertible 1-morphisms, but all 2-morphisms among these. 14 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI (2’) u m has a right adjoint ( – ) ♯ , which takes an ( ∞ , 2) -category 𝔸 to 𝔸 ♯ := 𝔸 = − → 𝔸 . (3’) The right adjoin t ( – ) ♯ has a further right adjoint, which takes ( 𝔸 , E ) to 𝔸 E . It is often conv enien t to regard the mark ed 1-morphisms in a marked ( ∞ , 2) - category ( 𝔹 , E ) as b eing speciﬁed b y a wide subcategory of 𝔹 ≤ 1 instead of a sub- ( ∞ , 2) -category of 𝔹 ; this is justiﬁed by the follo wing observ ation, which leads to an alternative description of MCat ( ∞ , 2) : Prop osition 2.2.6. L et Ar lﬀ ( Cat ( ∞ , 2) ) denote the ful l sub c ate gory of Ar ( Cat ( ∞ , 2) ) sp anne d by the lo c al ly ful ly faithful functors, and let Ar fa ( Cat ∞ ) denote the ful l sub c ate gory of Ar ( Cat ∞ ) sp anne d by the faithful functors. Then the c ommutative squar e Ar lﬀ ( Cat ( ∞ , 2) ) Cat ( ∞ , 2) Ar fa ( Cat ∞ ) Cat ∞ ev 1 ( – ) ≤ 1 ( – ) ≤ 1 ev 1 is a pul lb ack. Pr o of. Suppose 𝔹 is an ( ∞ , 2) -category and F : A → 𝔹 ≤ 1 is a faithful functor. Then the comp osite A → 𝔹 admits a unique factorization as A i − → 𝔸 F ′ − → 𝔹 where F ′ is lo cally fully faithful and i is left orthogonal to lo cally fully faithful functors. By [ LMGR + 24 , Theorem 5.3.7] this left orthogonal class consists precisely of the functors that are essentially surjective on ob jects and on all mapping ∞ -categories. It follows that i ≤ 1 : A → 𝔸 ≤ 1 is b oth fully faithful and essentially surjective, i.e. an equiv alence. In particular, F ≃ F ′≤ 1 . Conv ersely , if w e start with a lo cally fully faithful functor G : 𝔸 → 𝔹 , then the comp osite 𝔸 ≤ 1 G ≤ 1 − − − → 𝔹 ≤ 1 → 𝔹 factors as 𝔸 ≤ 1 → 𝔸 G − → 𝔹 where the ﬁrst functor is left orthogonal to lo cally fully faithful functors; by uniqueness, this means that applying the factorization to G ≤ 1 reco vers G . This construction therefore gives an inv erse to the functor Ar lﬀ ( Cat ( ∞ , 2) ) → Ar fa ( Cat ∞ ) × Cat ∞ Cat ( ∞ , 2) from the commutativ e square, as required. □ Corollary 2.2.7. The c ommutative squar e MCat ( ∞ , 2) Cat ( ∞ , 2) MCat ∞ Cat ∞ u m ( – ) ≤ 1 ( – ) ≤ 1 ev 1 is a pul lb ack. □ Lemma 2.2.8. The functor ( – ) ♭ : Cat ( ∞ , 2) → MCat ( ∞ , 2) has a left adjoint τ m : MCat ( ∞ , 2) → Cat ( ∞ , 2) . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 15 This is given by inverting the marke d 1-morphisms, i.e. for a marke d ( ∞ , 2) -c ate gory ( 𝔸 , E ) we have a natur al pushout squar e 𝔸 ≤ 1 E ∥ 𝔸 ≤ 1 E ∥ 𝔸 τ m ( 𝔸 , E ) . Pr o of. It is clear that this pushout deﬁnes a functor τ m . T o prov e that it is a left adjoin t as desired, we observe that we hav e natural equiv alences Map Cat ( ∞ , 2) ( τ m ( 𝔸 , E ) , 𝔹 ) ≃ Map Cat ( ∞ , 2) ( 𝔸 , 𝔹 ) × Map Cat ( ∞ , 2) ( 𝔸 ≤ 1 E , 𝔹 ) Map Cat ( ∞ , 2) ( ∥ 𝔸 ≤ 1 E ∥ , 𝔹 ) ≃ Map Cat ( ∞ , 2) ( 𝔸 , 𝔹 ) × Map Cat ∞ ( 𝔸 ≤ 1 E , 𝔹 ≤ 1 ) Map Cat ∞ ( 𝔸 ≤ 1 E , 𝔹 ≃ ) ≃ Map MCat ( ∞ , 2) (( 𝔸 , E ) , 𝔹 ♭ ) , where the last equiv alence follows from the description of MCat ( ∞ , 2) as a pullback in Corollary 2.2.7 . □ Notation 2.2.9. F or marked ( ∞ , 2) -categories ( 𝔸 , I ) and ( 𝔹 , J ) , we write 𝕄𝔽 un (( 𝔸 , I ) , ( 𝔹 , J )) for the full sub category of 𝔽 un ( 𝔸 , 𝔹 ) spanned by the functors that preserv e the markings. W e give this the marking consisting of the natural transformations 𝔸 × [1] → 𝔹 whose comp onent at every a ∈ 𝔸 is marked in ( 𝔹 , J ) , i .e. this is a mark ed functor ( 𝔸 , I ) × [1] ♯ → ( 𝔹 , J ) . Observ ation 2.2.10. The ∞ -category MCat ( ∞ , 2) is cartesian closed, with in ternal Hom giv en b y 𝕄𝔽 un ( – , – ) . Since u m is faithful, to pro v e this it suﬃces to observe that for mark ed ( ∞ , 2) -categories ( 𝔸 , I ) , ( 𝔹 , J ) , ( ℂ , K ) , a functor 𝔸 × 𝔹 → ℂ is a mark ed functor ( 𝔸 , I ) × ( 𝔹 , J ) → ( ℂ , K ) if and only if its adjoint 𝔸 → 𝔽 un ( 𝔹 , ℂ ) factors through a marked functor ( 𝔸 , I ) → 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) . W e can thus regard MCat ( ∞ , 2) as enric hed in itself; we can also transfer this enric hmen t along u m and make MCat ( ∞ , 2) an ( ∞ , 3) -category with mapping ( ∞ , 2) -categories u m 𝕄𝔽 un ( – , – ) , i.e. the full sub- ( ∞ , 2) -category of 𝔽 un ( – , – ) spanned by the mark ed functors; we write 𝕄ℂ at ( ∞ , 2) for the underlying ( ∞ , 2) -category. Note that this then arises from the product-preserving functor ( – ) ♭ : Cat ∞ → MCat ( ∞ , 2) ; since we hav e commutativ e triangles Cat ∞ Cat ( ∞ , 2) MCat ( ∞ , 2) , ( – ) ♭ ( – ) ♭ Cat ∞ MCat ( ∞ , 2) MCat ( ∞ , 2) , ( – ) ♭ u m the functors ( – ) ♭ , u m , and ( – ) ♯ upgrade to 2-functors with adjunctions of ( ∞ , 2) - categories ( – ) ♭ ⊣ u m ⊣ ( – ) ♯ b y Corollary 2.1.6 . 16 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI 2.3. Mark ed Gray tensors and partially (op)lax transformations. In this subsection we review the mark ed version of the Gray tensor pro duct, where certain 2-morphisms are inv erted. This has previously b een studied in a mo del-categorical setting in [ Ab e23 ], as well as in [ GHL25 , §4.1] (but without the marking of the Gra y tensor). Deﬁnition 2.3.1. Giv en mark ed ( ∞ , 2) -categories ( 𝔸 , I ) and ( 𝔹 , J ) , w e deﬁne 𝔸 ⊗ I ,J 𝔹 as the ( ∞ , 2) -category obtained by inv erting the 2-morphisms in squares [1] ⊗ [1] f ⊗ g − − − → 𝔸 ⊗ 𝔹 where either f lies in I or g lies in J . In other words, we hav e a pushout 𝔸 I ⊗ 𝔹 ⨿ 𝔸 ⊗ 𝔹 J 𝔸 I × 𝔹 ⨿ 𝔸 × 𝔹 J 𝔸 ⊗ 𝔹 𝔸 ⊗ I ,J 𝔹 . This has a marking generated by the image of 𝔸 I × 𝔹 ≃ ⨿ 𝔸 ≃ × 𝔹 J , giving a marked ( ∞ , 2) -category ( 𝔸 , I ) ⊗ m ( 𝔹 , J ) . Observ ation 2.3.2. If either I or J is the maximal marking, then ( 𝔸 , I ) ⊗ m ( 𝔹 , J ) is equiv alen t to the cartesian pro duct ( 𝔸 , I ) × ( 𝔹 , J ) . Deﬁnition 2.3.3. Giv en mark ed ( ∞ , 2) -categories ( 𝔸 , I ) and ( 𝔹 , J ) , w e deﬁne 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) lax to b e the locally full sub- ( ∞ , 2) -category of 𝔽 un ( 𝔹 , ℂ ) lax whose ▶ ob jects are the mark ed functors, i.e. those that take morphisms in 𝔹 J in to ℂ K , ▶ morphisms are the lax transformations [1] ⊗ 𝔹 → ℂ that factor through a mark ed functor [1] ⊗ ♭,I 𝔹 → ℂ , which unpacks to those where the lax naturalit y square asso ciated to a morphism in I commutes, and the restrictions to each ob ject of [1] gives a marked functor. W e equip this with the marking given by the strong natural transformations among mark ed functors (which are precisely the mark ed functors [1] ⊗ ♯,I 𝔹 → ℂ by Obser- v ation 2.3.2 ). Rev ersing the order of the Gray tensor we similarly deﬁne marked ( ∞ , 2) -categories 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) oplax . Lemma 2.3.4. W e have natur al e quivalenc es Map MCat ( ∞ , 2) (( 𝔸 , I ) ⊗ m ( 𝔹 , J ) , ( ℂ , K )) ≃ Map MCat ( ∞ , 2) (( 𝔸 , I ) , 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) lax ) ≃ Map MCat ( ∞ , 2) (( 𝔹 , J ) , 𝕄𝔽 un (( 𝔸 , I ) , ( ℂ , K )) oplax ) , so that ther e ar e adjunctions ( 𝔸 , I ) ⊗ m – ⊣ 𝕄𝔽 un (( 𝔸 , I ) , – ) oplax , – ⊗ m ( 𝔸 , I ) ⊣ 𝕄𝔽 un (( 𝔸 , I ) , – ) lax . Pr o of. W e prov e the ﬁrst equiv alence; the second is prov ed by the same argument. Unpac king the deﬁnition of the mapping space on the left, we see that it can b e iden tiﬁed with the space of functors 𝔸 ⊗ 𝔹 → ℂ such that ▶ the 2-morphism in the lax square asso ciated to a pair of morphisms f from 𝔸 and g from 𝔹 is inv ertible if either f lies in I or g lies in J , FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 17 ▶ the morphism in ℂ asso ciated to a morphism in 𝔸 and an ob ject in 𝔹 lies in K if the morphism from 𝔸 lies in I , and similarly with the roles of 𝔸 and 𝔹 rev ersed. These corresp ond to functors 𝔸 → 𝔽 un ( 𝔹 , ℂ ) lax suc h that ▶ for every ob ject of 𝔸 , the asso ciated functor 𝔹 → ℂ is mark ed, ▶ for every morphism in 𝔸 , the asso ciated lax natural transformation has com- m uting naturalit y squares at all morphisms in J , ▶ for ev ery morphism in I , the asso ciated lax natural transformation has com- m uting naturalit y squares at all morphisms in 𝔹 , i.e. it is strong. These are precisely the marked functors from ( 𝔸 , I ) to 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) lax , as required. □ Prop osition 2.3.5. The marke d Gr ay tensor pr o duct is asso ciative, i.e. we have a natur al e quivalenc e (( 𝔸 , I ) ⊗ m ( 𝔹 , J )) ⊗ m ( ℂ , K ) ≃ ( 𝔸 , I ) ⊗ m (( 𝔹 , J ) ⊗ m ( ℂ , K )) . Pr o of. W e ﬁrst prov e there is an equiv alence on underlying ( ∞ , 2) -categories. F or this, consider the commutativ e square ( 𝔸 I ⊗ 𝔹 ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 J ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ K 𝔸 I × ( 𝔹 ⊗ ℂ ) ⨿ 𝔹 J × ( 𝔸 ⊗ ℂ ) ⨿ ( 𝔸 ⊗ 𝔹 ) × ℂ K ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ ( 𝔸 ⊗ I ,J 𝔹 ) ⊗ L,K ℂ , where L denotes the marking of 𝔸 ⊗ I ,J 𝔹 ; we claim this is a pushout. Indeed, we can factor this horizontally through the square ( 𝔸 I ⊗ 𝔹 ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 J ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ K ( 𝔸 I × 𝔹 ) ⊗ ℂ ⨿ ( 𝔸 × 𝔹 J ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ K ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ ( 𝔸 ⊗ I ,J 𝔹 ) ⊗ ℂ , whic h is a pushout since ⊗ preserv es colimits in eac h v ariable, follo w ed b y the square ( 𝔸 I × 𝔹 ) ⊗ ℂ ⨿ ( 𝔸 × 𝔹 J ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ K 𝔸 I × ( 𝔹 ⊗ ℂ ) ⨿ 𝔹 J × ( 𝔸 ⊗ ℂ ) ⨿ ( 𝔸 ⊗ 𝔹 ) × ℂ K ( 𝔸 ⊗ I ,J 𝔹 ) ⊗ ℂ ( 𝔸 ⊗ I ,J 𝔹 ) ⊗ L,K ℂ . It suﬃces to show that this is a pushout, for which we consider the diagram ( 𝔸 I × 𝔹 ≃ ) ⊗ ℂ ⨿ ( 𝔸 ≃ × 𝔹 J ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ K ( 𝔸 I × 𝔹 ≃ ) × ℂ ⨿ ( 𝔸 ≃ × 𝔹 J ) × ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) × ℂ K ( 𝔸 I × 𝔹 ) ⊗ ℂ ⨿ ( 𝔸 × 𝔹 J ) ⊗ ℂ ⨿ ( 𝔸 ⊗ 𝔹 ) ⊗ ℂ K 𝔸 I × ( 𝔹 ⊗ ℂ ) ⨿ 𝔹 J × ( 𝔸 ⊗ ℂ ) ⨿ ( 𝔸 ⊗ 𝔹 ) × ℂ K ( 𝔸 ⊗ I ,J 𝔹 ) ⊗ ℂ ( 𝔸 ⊗ I ,J 𝔹 ) ⊗ L,K ℂ . 18 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Here the top square is a pushout b y [ A GH25 , Prop osition 2.8.1], while the composite square is a pushout by the deﬁnition of the mark ed Gray tensor. Hence the b ottom square is also a pushout. The same argument applied to 𝔸 ⊗ I ,M ( 𝔹 ⊗ J,K ℂ ) , where M denotes the marking of 𝔹 ⊗ J,K ℂ , pro duces a pushout square 𝔸 I ⊗ ( 𝔹 ⊗ ℂ ) ⨿ 𝔸 ⊗ ( 𝔹 J ⊗ ℂ ) ⨿ 𝔸 ⊗ ( 𝔹 ⊗ ℂ K ) 𝔸 I × ( 𝔹 ⊗ ℂ ) ⨿ 𝔹 J × ( 𝔸 ⊗ ℂ ) ⨿ ( 𝔸 ⊗ 𝔹 ) × ℂ K 𝔸 ⊗ ( 𝔹 ⊗ ℂ ) 𝔸 ⊗ I ,M ( 𝔹 ⊗ L,K ℂ ) , whic h is clearly equiv alent to the ﬁrst pushout via the associativity of the Gray tensor. Moreov er, the marking is in b oth cases giv en b y the image of 𝔸 I × 𝔹 ≃ × ℂ ≃ ⨿ 𝔸 ≃ × 𝔹 J × ℂ ≃ ⨿ 𝔸 ≃ × 𝔹 ≃ × ℂ K , so we get the required natural equiv alence of marked ( ∞ , 2) -categories. □ Corollary 2.3.6. F or marke d ( ∞ , 2) -c ate gories ( 𝔸 , I ) , ( 𝔹 , J ) and ( ℂ , K ) , we have natur al e quivalenc es of marke d ( ∞ , 2) -c ate gories 𝕄𝔽 un (( 𝔸 , I ) , 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) lax ) oplax ≃ 𝕄𝔽 un (( 𝔹 , J ) , 𝕄𝔽 un (( 𝔸 , I ) , ( ℂ , K )) oplax ) lax , 𝕄𝔽 un (( 𝔸 , I ) , 𝕄𝔽 un (( 𝔹 , J ) , ( ℂ , K )) lax ) lax ≃ 𝕄𝔽 un (( 𝔸 , I ) ⊗ m ( 𝔹 , J ) , ( ℂ , K )) lax , 𝕄𝔽 un (( 𝔹 , J ) , 𝕄𝔽 un (( 𝔸 , I ) , ( ℂ , K )) oplax ) oplax ≃ 𝕄𝔽 un (( 𝔸 , I ) ⊗ m ( 𝔹 , J ) , ( ℂ , K )) oplax . Pr o of. Apply the Y oneda lemma and the asso ciativit y of the marked Gray tensor. (Cf. [ AGH25 , Lemma 2.2.10] for the unmarked case.) □ Deﬁnition 2.3.7. Given a mark ed ( ∞ , 2) -category ( 𝔸 , I ) and an (unmark ed) ( ∞ , 2) - category 𝔹 , we deﬁne 𝔽 un ( 𝔸 , 𝔹 ) I -lax := u m 𝕄𝔽 un (( 𝔸 , I ) , 𝔹 ♯ ) lax , 𝔽 un ( 𝔸 , 𝔹 ) I -oplax := u m 𝕄𝔽 un (( 𝔸 , I ) , 𝔹 ♯ ) oplax , Unpac king the v arious adjunctions, w e see that functors 𝕂 → 𝔽 un ( 𝔸 , 𝔹 ) I -(op)lax corresp ond to functors 𝕂 ⊗ ♭,I 𝔸 → 𝔹 , 𝔸 ⊗ I ,♭ 𝕂 → 𝔹 in th e lax and oplax cases, resp ectiv ely . Here 𝔽 un ( 𝔸 , 𝔹 ) I -(op)lax can b e identiﬁed as a wide and lo cally full sub- ( ∞ , 2) -category of 𝔽 un ( 𝔸 , 𝔹 ) (op)lax whose morphisms are the I -(op)lax tr ansformations , meaning those (op)lax transformations [1] ⊗ 𝔸 → 𝔹 , 𝔸 ⊗ [1] → 𝔹 that factor through [1] ⊗ ♭,I 𝔸 and 𝔸 ⊗ I ,♭ [1] , resp ectiv ely , meaning that the (op)lax naturalit y squares asso ciated to the morphisms in I actually commute. Notation 2.3.8. Giv en functors of ( ∞ , 2) -categories F , G : 𝔸 → 𝔹 and a marking ( 𝔸 , I ) , we write Nat I -(op)lax 𝔸 , 𝔹 ( F , G ) for the ∞ -category of morphisms from F to G in 𝔽 un ( 𝔸 , 𝔹 ) I -(op)lax . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 19 Observ ation 2.3.9. Applying Corollary 2.3.6 with ( 𝔸 , I ) = 𝔸 ♭ and ( ℂ , K ) = ℂ ♯ , w e see that there are natural equiv alences 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ ) J -lax ) lax ≃ 𝔽 un ( 𝔸 ⊗ ♭,J 𝔹 , ℂ ) J ′ -lax , 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ ) J -oplax ) oplax ≃ 𝔽 un ( 𝔹 ⊗ J,♭ 𝔸 , ℂ ) J ′′ -oplax , where J ′ and J ′′ are b oth generated by the image of 𝔹 J × 𝔸 ≃ . This means we ha v e a pullback square 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ ) J -lax ) lax 𝔽 un ( 𝔸 ⊗ ♭,J 𝔹 , ℂ ) lax lim 𝔸 ≃ 𝔽 un ( 𝔹 , ℂ ) J -lax lim 𝔸 ≃ 𝔽 un ( 𝔹 , ℂ ) lax , and similarly in the oplax case. W e also see that 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ ) J -lax ) oplax is equiv alent to the full sub- ( ∞ , 2) -category of 𝔽 un ( 𝔹 , 𝔽 un ( 𝔸 , ℂ ) oplax ) J -lax spanned b y the functors that take morphisms in J to strong natural transformations. 2.4. Fibrations and the straigh tening equiv alence. In this section we re- view the ( ∞ , 2) -categorical analogues of (co)cartesian ﬁbrations, ﬁrst introduced in [ GHL24 ], and their straightening equiv alence with functors ℂ at ( ∞ , 2) , due to Ab ellán–Stern [ AS23a ] and Nuiten [ Nui24 ]. W e also recall the extension to straigh t- ening for partially (op)lax transformations prov ed in [ AGH25 ]. Deﬁnition 2.4.1. Let p : 𝔼 → 𝔹 b e a functor of ( ∞ , 2) -categories. ▶ A morphism f : x → y in 𝔼 is p -c o c artesian (or p - 0 -c artesian ) if for all ob jects z ∈ 𝔼 the comm utative square of ∞ -categories 𝔼 ( y , z ) 𝔼 ( x, z ) 𝔹 ( p ( y ) , p ( z )) 𝔹 ( p ( x ) , p ( z )) f ∗ p ( f ) ∗ is a pullbac k. Dually , w e sa y f is p -c artesian (or p - 1 -c artesian ) if it is p op - co cartesian when viewed as a morphism in 𝔼 op . ▶ A 2-morphism α in 𝔼 b et ween 1-morphisms from x to y is we akly p -(c o)c artesian (or we akly p - i -c artesian ) if it is a (co)cartesian morphism for the functor p x,y : 𝔼 ( x, y ) → 𝔹 ( p ( x ) , p ( y )) . ▶ A 2-morphism α in 𝔼 b etw een 1-morphisms from x to y is p -(c o)c artesian (or p - i -c artesian ) if for any morphisms f : x ′ → x and g : y → y ′ the whisk ering g ◦ α ◦ f is weakly p -(co)cartesian. Lemma 2.4.2. Supp ose f : x → y is a p -c o c artesian morphism for a functor of ( ∞ , 2) -c ate gories p : 𝔼 → 𝔹 . L et u : h → h ′ b e a 2-morphism b etwe en morphisms y → z . If u ◦ f is we akly p -(c o)c artesian then so is u . 20 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Pr o of. Since f is p -co cartesian, we ha ve a pullback square of ∞ -categories 𝔼 ( y , z ) 𝔼 ( x, z ) 𝔹 ( py , pz ) 𝔹 ( px, pz ) f ∗ p y,z p x,z p ( f ) ∗ and b y assumption f ∗ u is p x,z -(co)cartesian. This implies that u is p y ,z -(co)cartesian b y [ Lur09a , 2.4.1.3(2)]. □ Observ ation 2.4.3. Let p : 𝔼 → 𝔹 b e a functor of ( ∞ , 2) -categories. Then p - (co)cartesian lifts are unique when they exist. More precisely , if w e let Map i -cart ([1] , 𝔼 ) denote the subspace of Map([1] , 𝔼 ) spanned by the i -cartesian morphis ms, then the functor Map i -cart ([1] , 𝔼 ) → Map([1] , 𝔹 ) × 𝔹 ≃ 𝔼 ≃ , induced by the commutativ e square Map([1] , 𝔼 ) 𝔼 ≃ Map([1] , 𝔹 ) 𝔹 ≃ , ev i Map([1] ,p ) p ≃ ev i is a monomorphism, i.e. its ﬁbres are either empt y or con tractible. T o see this, w e can observe that if there exists a cartesian lift ¯ f : x → y of some morphism f : a → b in 𝔹 , then comp osition with ¯ f identiﬁes the ∞ -group oid of cartesian lifts of f with target y with that of co cartesian lifts of id b with target y ; the latter is the ∞ -group oid of equiv alences with target y , which is alwa ys contractible. Deﬁnition 2.4.4. Giv en a functor of ( ∞ , 2) -categories p : 𝔼 → 𝔹 , w e sa y that 𝔼 has p -(c o)c artesian lifts of a morphism f : b → b ′ in 𝔹 if for all x ∈ 𝔼 , p ( x ) ≃ b , there exists a p -co cartesian morphism ¯ f : x → y with an equiv alence p ( ¯ f ) ≃ f extending that for the source. Similarly , we say that 𝔼 has p -(c o)c artesian lifts of a 2-morphism if the induced functor on mapping ∞ -categories has p -(co)cartesian lifts in the previous sense, and these are preserv ed under pre- and p ostcomp osition with 1-morphisms. Deﬁnition 2.4.5. A functor π : 𝔼 → 𝔹 of ( ∞ , 2) -categories is a (0 , 1) -ﬁbr ation if: (1) 𝔼 has p -co cartesian lifts of all morphisms in 𝔹 . (2) 𝔼 has p -cartesian lifts of all 2-morphisms in 𝔹 . If π : 𝔼 → 𝔹 and π ′ : 𝔼 ′ → 𝔹 ′ are (0 , 1) -ﬁbrations, we say that a commutativ e square of ( ∞ , 2) -categories 𝔼 𝔼 ′ 𝔹 𝔹 ′ ψ π π ′ φ is a morphism of (0 , 1) -ﬁbr ations if ψ preserv es π -co cartesian 1-morphisms and π -cartesian 2-morphisms. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 21 Similarly , for all c hoices of i, j ∈ { 0 , 1 } we hav e the notion of ( i, j ) -ﬁbrations (whic h hav e i -cartesian 1-morphisms and j -cartesian 2-morphisms) and their mor- phisms. W e write 𝔽 ib ( i,j ) / 𝔹 for the lo cally full sub- ( ∞ , 2) -category (cf. Deﬁnition 2.1.8 ) of ℂ at ( ∞ , 2) / 𝔹 con taining the ( i, j ) -ﬁbrations, the morphisms of ( i, j ) -ﬁbrations, and all 2-morphisms among these. Deﬁnition 2.4.6. An ϵ -ﬁbration p : 𝔼 → 𝔹 is 1-ﬁbr e d if its ﬁbres are ∞ -categories. (See [ AGH25 , Prop osition 3.2.14] for an alternativ e characterization of these.) Observ ation 2.4.7. F or a functor p : 𝔼 → 𝔹 , the following are equiv alent: ▶ p is a (1 , 0) -ﬁbration. ▶ p op is a (0 , 0) -ﬁbration. ▶ p co is a (1 , 1) -ﬁbration. ▶ p coop is a (0 , 1) -ﬁbration. Notation 2.4.8. T o lighten the notation, we will usually write ϵ -ﬁbration for ϵ ∈ { 0 , 1 } × 2 , and we will similarly denote 𝔽 ib ( i,j ) / 𝔹 as 𝔽 ib ϵ / 𝔹 . Giv en ϵ = ( i, j ) we will also write ϵ = (1 − i, 1 − j ) for the conjugate v ariance. This allows us to use short-hand notation for v arious structures asso ciated with ﬁbrations; in particular, w e deﬁne 𝔹 ϵ -op :=              𝔹 , ϵ = (0 , 1) , 𝔹 op , ϵ = (1 , 0) , 𝔹 co , ϵ = (0 , 0) , 𝔹 coop , ϵ = (1 , 1) . Theorem 2.4.9 (Nuiten [ Nui24 ], Ab ellán–Stern [ AS23a ]) . Ther e ar e e quivalenc es of ( ∞ , 2) -c ate gories 𝔽 ib ϵ / 𝔹 ≃ 𝔽 un ( 𝔹 ϵ -op , ℂ at ( ∞ , 2) ) , i.e. 𝔽 ib (0 , 1) / 𝔹 ≃ − → 𝔽 un ( 𝔹 , ℂ at ( ∞ , 2) ) , 𝔽 ib (1 , 0) / 𝔹 ≃ − → 𝔽 un ( 𝔹 op , ℂ at ( ∞ , 2) ) , 𝔽 ib (0 , 0) / 𝔹 ≃ − → 𝔽 un ( 𝔹 co , ℂ at ( ∞ , 2) ) , 𝔽 ib (1 , 1) / 𝔹 ≃ − → 𝔽 un ( 𝔹 coop , ℂ at ( ∞ , 2) ) , which ar e c ontr avariantly natur al in 𝔹 with r esp e ct to pul lb ack on the left and c om- p osition on the right. □ Theorem 2.4.10 (Ab ellán–Stern [ AS23a ]) . L et p : 𝕏 → 𝕊 b e an ( i, j ) -ﬁbr ation. Then the pul lb ack functor p ∗ : ℂ at ( ∞ , 2) / 𝕊 → ℂ at ( ∞ , 2) / 𝕏 admits a right adjoint. Pr o of. By [ AS23a , Theorem 3.90], the functor p ∗ admits a right adjoint at the level of underlying ∞ -categories. The ( ∞ , 2) -category structure on ℂ at ( ∞ , 2) / 𝕊 arises from the pro duct functor ( – ) × 𝕊 : Cat ∞ → Cat ( ∞ , 2) / 𝕊 , 22 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI whic h is preserved by p ∗ as p ∗ ( K × 𝕊 ) ≃ K × 𝕏 . It therefore follo ws from Corol- lary 2.1.6 that the adjunction p ∗ ⊣ p ∗ upgrades to an adjunction of ( ∞ , 2) -categories. □ Deﬁnition 2.4.11. Let ( ℂ , E ) b e a marked ( ∞ , 2) -category. F or ϵ = ( i, j ) we write 𝔽 ib ϵ / ( ℂ ,E ) for the lo cally full sub- ( ∞ , 2) -category of ℂ at ( ∞ , 2) / ℂ whose ob jects are ϵ -ﬁbrations ov er ℂ and whose morphisms preserve i -cartesian morphisms that lie ov er E as well as all j -cartesian 2-morphisms. Theorem 2.4.12 (Ab ellán–Gagna–Haugseng, [ AGH25 , Prop osition 3.5.7]) . F or ( ℂ , E ) a marke d ( ∞ , 2) -c ate gory, we have e quivalenc es of ( ∞ , 2) -c ate gories 𝔽 ib (0 , 1) / ( ℂ ,E ) ≃ 𝔽 un ( ℂ , ℂ at ( ∞ , 2) ) E - lax , 𝔽 ib (0 , 0) / ( ℂ ,E ) ≃ 𝔽 un ( ℂ co , ℂ at ( ∞ , 2) ) E - lax 𝔽 ib (1 , 1) / ( ℂ ,E ) ≃ 𝔽 un ( ℂ coop , ℂ at ( ∞ , 2) ) E - oplax , 𝔽 ib (1 , 0) / ( ℂ ,E ) ≃ 𝔽 un ( ℂ op , ℂ at ( ∞ , 2) ) E - oplax , given on obje cts by str aightening. These e quivalenc es ar e al l c ontr avariantly natur al in ( ℂ , E ) with r esp e ct to pul lb ack on the left and c omp osition on the right. □ 2.5. Lax slices and cones. In this subsection we review the lax versions of slices and cones, and observe that they are related in the exp ected wa y . The analogous constructions in the setting of scaled simplicial sets w ere previously studied by Gagna–Harpaz–Lanari in [ GHL25 , §5.2]. Deﬁnition 2.5.1. Let ℂ b e a ( ∞ , 2) -category and let c ∈ ℂ . W e deﬁne lax versions of the slice construction by means of the following pullbac k squares ℂ c → 𝔸 r (op)lax ( ℂ ) [0] ℂ , ev 0 c ℂ c → 𝔸 r lax ( ℂ ) [0] ℂ , ev 0 c ℂ → c 𝔸 r (op)lax ( ℂ ) [0] ℂ , ev 1 c ℂ → c 𝔸 r lax ( ℂ ) [0] ℂ . ev 1 c The lax slices come equipp ed with: ▶ a functor ℂ c → → ℂ induced by ev 1 whic h is a (0 , 1) -ﬁbration, which classiﬁes the corepresentable functor ℂ ( c, − ) : ℂ → ℂ at ∞ ; ▶ a functor ℂ c → → ℂ induced by ev 1 whic h is a (0 , 0) -ﬁbration, whic h classiﬁes the corepresentable functor ℂ ( c, − ) op : ℂ co → ℂ at ∞ ; ▶ a functor ℂ → c → ℂ induced by ev 0 whic h is a (1 , 0) -ﬁbration, which classiﬁes the representable functor ℂ ( − , c ) : ℂ op → ℂ at ∞ ; ▶ a functor ℂ → c → ℂ induced by ev 0 whic h is a (1 , 1) -ﬁbration, whic h classiﬁes the representable functor ℂ ( − , c ) op : ℂ coop → ℂ at ∞ . The ﬁrst claim is [ Lur09b , Prop osition 4.1.8], and the others follow by reversing 1- and 2-morphisms. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 23 Remark 2.5.2. The notation chosen for the lax slices is designed to giv e a de- scription of the 1-morphisms. More precisely , an ob ject in ℂ → c is given by a map u : x → c and a morphism from u to v : y → c can b e represented b y triangle x y c. u v whic h comm utes up to a (non-inv ertible) 2-morphism. Similarly , a 1-morphism in ℂ → c is given b y a laxly commuting triangle where the asso ciated 2-cell p oints in the other direction. Deﬁnition 2.5.3. Giv en a mark ed ( ∞ , 2) -category ( 𝕁 , E ) and a functor F : 𝕁 → ℂ , w e deﬁne the ( ∞ , 2) -category ℂ E - lax → F of E -lax c ones on F as the pullback ℂ E - lax → F 𝔽 un ( 𝕁 , ℂ ) E - lax → F ℂ 𝔽 un ( 𝕁 , ℂ ) E - lax ; const an E -lax cone on F is th us by deﬁnition an E -lax natural transformation c → F from a constant functor. Similarly , we deﬁne ( ∞ , 2) -categories ℂ E - oplax → F of E -oplax c ones , ℂ E - lax → F of E -lax c o c ones , and ℂ E - oplax → F of E -oplax c o c ones . Observ ation 2.5.4. Since pullback of ﬁbrations corresp onds to comp osition under straigh tening, the (1 , 1) -ﬁbration ℂ E - lax → F → ℂ corresp onds to the functor Nat E - lax 𝕁 , ℂ ( – , F ) op : ℂ coop → ℂ at ∞ . Similarly , ▶ ℂ E - oplax → F corresp onds to Nat E - oplax 𝕁 , ℂ ( – , F ) , ▶ ℂ E - lax → F corresp onds to Nat E - lax 𝕁 , ℂ ( F , – ) op , ▶ ℂ E - oplax → F corresp onds to Nat E - oplax 𝕁 , ℂ ( F , – ) . W e can also describ e the ( ∞ , 2) -categories of partially lax (co)cones in terms of joins: Deﬁnition 2.5.5. Let ( 𝕁 , E ) b e a marked ( ∞ , 2) -category. W e deﬁne lax versions of the cone construction by means of the following pushout diagrams: { 0 } × 𝕁 [0] [1] ⊗ ♭,E 𝕁 𝕁 ◁ E -lax , { 1 } × 𝕁 [0] [1] ⊗ ♭,E 𝕁 𝕁 ▷ E -lax . Similarly , we deﬁne 𝕁 ◁ E - oplax and 𝕁 ▷ E -(op)lax b y reversing the order of the Gra y tensor pro duct ab ov e. 24 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Lemma 2.5.6. F or ( 𝕁 , E ) a marke d ( ∞ , 2) -c ate gory and a functor F : 𝕁 → ℂ , ther e is a pul lb ack squar e ℂ E - lax → F 𝔽 un ( 𝕁 ◁ E -lax , ℂ ) lax { F } 𝔽 un ( 𝕁 , ℂ ) lax , and similarly for the other thr e e variants. Pr o of. By deﬁnition of the lax slice, we hav e a pullback ℂ E - lax → F 𝔸 r lax ( 𝔽 un ( 𝕁 , ℂ ) E - lax ) ℂ × [0] ( 𝔽 un ( 𝕁 , ℂ ) E - lax ) × 2 . const × F By Observ ation 2.3.9 , we also hav e a pullback 𝔸 r lax ( 𝔽 un ( 𝕁 , ℂ ) E - lax ) 𝔽 un ([1] ⊗ ♭,E 𝕁 , ℂ ) lax ( 𝔽 un ( 𝕁 , ℂ ) E - lax ) × 2 ( 𝔽 un ( 𝕁 , ℂ ) lax ) × 2 . Com bining the tw o, we get a pullback square ℂ E - lax → F 𝔽 un ([1] ⊗ ♭,E 𝕁 , ℂ ) lax ℂ × [0] ( 𝔽 un ( 𝕁 , ℂ ) lax ) × 2 . const × F W e can factor the b ottom horizontal map as ℂ × [0] id × F − − − − → ℂ × 𝔽 un ( 𝕁 , ℂ ) lax const × id − − − − − − → ( 𝔽 un ( 𝕁 , ℂ ) lax ) × 2 . By deﬁnition of 𝕁 ◁ E - lax , this gives a horizontal factorization of our square through the pullback 𝔽 un ( 𝕁 ◁ E - lax , ℂ ) E - lax 𝔽 un ([1] ⊗ ♭,E 𝕁 , ℂ ) lax ℂ × [0] ( 𝔽 un ( 𝕁 , ℂ ) lax ) × 2 , const × F from which the pullback we wan t to prov e is clear. □ Lemma 2.5.7. L et p : 𝔸 → [1] and c onsider the functor 𝔸 → 𝔸 ′ = 𝔸 ` 𝔸 1 [0] over [1] , wher e 𝔸 1 denotes the ﬁbr e over 1 . Then we have an e quivalenc e of ( ∞ , 2) - c ate gories, 𝔸 0 ≃ − → 𝔸 ′ 0 . Pr o of. The map 𝔸 0 → 𝔸 ′ 0 is clearly essentially surjective, so we only need to show that it is fully faithful. T o see this w e will compute 𝔸 ′ in a model of ( ∞ , 2) - categories. W e start by taking A → ∆ 1 to b e a mo del of 𝔸 → [1] as a ﬁbration in scaled simplicial sets; then the ﬁbres A i will mo del 𝔸 i ( i = 0 , 1 ) as these are giv en by homotop y pullbac ks. Next, we tak e A := C sc ( A ) to b e the coﬁbrant mo del FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 25 for 𝔸 as a category enriched in marked simplicial sets (or Set + ∆ -enric hed categories) obtained from the left Quillen equiv alence C sc (see [ Lur09b ]). W e deﬁne A i for i = 0 , 1 similarly; then the resulting functors A i → A are coﬁbrations (since C sc is a left Quillen functor), and also full sub category inclusions by the deﬁnition of this functor. W e now deﬁne a Set + ∆ -enric hed category b A as follows: ▶ The set of ob jects is given by those of A 0 in addition to a “cone p oint”, which w e denote by v . ▶ The marked simplicial sets of maps are giv en by b A ( a, b ) = A 0 ( a, b ) , b A ( v , v ) = ∗ , b A ( v , b ) = ∅ , and ﬁnally b A ( a, v ) is deﬁned to b e the (conical) Set + ∆ -enric hed colimit of the functor A ( a, ι ( − )) : A 1 → Set + ∆ where ι : A 1 → A denotes the ob vious inclusion. This datum assembles naturally into a Set + ∆ -enric hed category that ﬁts into a com- m utative diagram A 1 [0] A b A . p T o ﬁnish the proof we will show that this is a pushout square; as the left v ertical morphism is a coﬁbration b et w een coﬁbrant ob jects and [0] is also coﬁbrant, this implies it is a homotop y pushout and so mo dels a pushout in the ∞ -category of ( ∞ , 2) -categories. In order to prov e this, supp ose that we hav e a functor f : A → X suc h that its restriction to A 1 is constant on an ob ject x . W e deﬁne a functor ˆ f : b A → X suc h that f = ˆ f ◦ p . T o deﬁne ˆ f it will b e enough to deﬁne for every a ∈ A 0 a map ϕ a : b A ( p ( a ) , v ) → X ( f ( a ) , x ) and show that the all of these choices assemble into a functor. Note that by con- struction w e ha ve a natural transformation of functors A ( a, ι ( − )) → X ( p ( a ) , x ) where we view the latter functor as b eing constan t. The universal prop ert y of the colimit provides us with the desired functor ϕ a . It is immediate to verify that the c hoices ab ov e assem ble to yield a functor ˆ f as desired. It is clear that ˆ f is the unique such extension, so this concludes the pro of. □ Prop osition 2.5.8. F or any marke d ( ∞ , 2) -c ate gory ( 𝕀 , E ) , the c anonic al functors 𝕀 → 𝕀 ▷ E - (op)lax , 𝕀 → 𝕀 ◁ E - (op)lax induc e d by the inclusions { 0 } , { 1 } → [1] , r esp e ctively, ar e al l ful ly faithful. Pr o of. W e will only pro v e the case { 0 } → [1] ; the remaining case is completely analogous. W e ﬁrst show that the map 𝕀 → [1] ⊗ ♭,E 𝕀 is fully faithful. T o see this w e observ e that the comp osite 𝕀 → [1] ⊗ ♭,E 𝕀 π − → [1] × 𝕀 is fully faithful, so it will suﬃce to show that π induces an equiv alence of mapping ∞ -categories at the ob jects in the essential image of the ﬁrst functor. 26 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI T o see this, we can implement this map using scaled simplicial sets ([ Lur09b ]) and v erify the previous claim after applying the rigidiﬁcation functor C sc , cf. [ Lur09b , Deﬁnition 3.1.10, Theorem 4.2.2.]. Note that this is immediate, as the resulting functor b etw een Set + ∆ -enric hed categories sets is an isomorphism on the underlying simplicially enriched categories and the decorations on the corresp onding mapping simplicial sets agree. T o ﬁnish the pro of, we apply Lemma 2.5.7 to the map [1] ⊗ ♭,E 𝕀 → [1] . □ Observ ation 2.5.9. Let ∗ ∈ 𝕀 ▷ E - (op)lax denote the cone point and consider the follo wing pullbac k square 𝕀 ( ∗ ) =  𝕀 ▷ E - (op)lax  → j × 𝕀 ▷ E - (op)lax 𝕀  𝕀 ▷ E - (op)lax  → ∗ 𝕀 𝕀 ▷ E - (op)lax W e can use the description of the straightening functor [ Lur09b , Deﬁnition 3.5.1] to see that 𝕀 ( ∗ ) → 𝕀 is the 1-ﬁbred (1 , 0) -ﬁbration corresp onding to straightening the iden tity functor on 𝕀 , where w e equip the source with the marking given by the edges corresponding to E . Putting this all together it follows that for every 1-ﬁbred (1 , 0) -ﬁbration p : 𝕏 → 𝕀 we can iden tify 𝔽 ib (1 , 0) / 𝕀 ( 𝕀 ( ∗ ) , 𝕏 ) with the full sub- ( ∞ , 2) - category of ℂ at ( ∞ , 2) / 𝕀 ( 𝕀 , 𝕏 ) on those functors o ver 𝕀 which send the 1-morphisms in E to cartesian morphisms in 𝕏 . 2.6. Decorated ( ∞ , 2) -categories. In this subsection we in troduce the notion of de c or ate d ( ∞ , 2) -c ate gories , which will provide a conv enient framew ork for our work on ﬁbrations in the subsequent sections. By a decorated ( ∞ , 2) -category we mean an ( ∞ , 2) -category equipp ed with collections of (“decorated”) 1- and 2-morphisms (whic h are c hosen indep enden tly , i.e. the decorated 2-morphisms do not ha v e to go b etw een decorated 1-morphisms). W e can further assume that these collections are closed under comp osition and include all equiv alences, so that w e can formally deﬁne these ob jects as follows: Deﬁnition 2.6.1. A de c or ate d ( ∞ , 2) -c ate gory 𝔸 ⋄ is a span of ( ∞ , 2) -categories 𝔸 ⋄ (1) i 1 − → 𝔸 i 2 ← − 𝔸 ⋄ (2) suc h that (see Deﬁnition 2.1.8 ) ▶ i 1 is a wide lo cally full sub- ( ∞ , 2) -category, ▶ i 2 is a wide and lo cally wide sub- ( ∞ , 2) -category inclusion. W e deﬁne the ∞ -category DCat ( ∞ , 2) as the full sub category of Fun (Λ 2 , op 0 , Cat ( ∞ , 2) ) spanned by the decorated ( ∞ , 2) -categories, where Λ 2 , op 0 denotes the category 1 → 0 ← 2 . Observ ation 2.6.2. It follows easily from [ AGH25 , Observ ations 2.5.6, 2.5.8] that a functor of ( ∞ , 2) -categories is ▶ a wide lo cally full sub category inclusion if and only if it is righ t orthogonal to ∅ → ∗ , ∂ C 2 → C 2 , FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 27 ▶ a wide and lo cally wide sub category inclusion if and only if it is right orthogonal to ∅ → ∗ , ∂ [1] → [1] , ∂ C 3 → C 2 . In particular, by [ Lur09a , Prop osition 5.5.5.7] b oth of these types of functors form the righ t class in a factorization system on Cat ( ∞ , 2) . F rom this it is easy to see that the inclusion DCat ( ∞ , 2)  → F un (Λ 2 , op 0 , Cat ( ∞ , 2) ) has a left adjoint, given by factoring a span of ( ∞ , 2) -categories using these factorization systems. Moreov er, as the morphisms w e consider orthogonality for all go b etw een compact ( ∞ , 2) -categories, it is clear that DCat ( ∞ , 2) is closed under ﬁltered colimits in Fun (Λ 2 , op 0 , Cat ( ∞ , 2) ) , so that this left adjoin t exhibits DCat ( ∞ , 2) as an accessible lo calization. In particular, DCat ( ∞ , 2) is a presen table ∞ -category. Moreo v er, w e can describ e (co)limits of decorated ( ∞ , 2) -categories as follows: ▶ the limit of a diagram in DCat ( ∞ , 2) is computed in Fun (Λ 2 , op 0 , Cat ( ∞ , 2) ) , and so is giv en by the limit of the underlying ( ∞ , 2) -categories equipp ed with the limits of the sub categories of decorations; ▶ the colimit of a diagram in DCat ( ∞ , 2) is giv en b y factoring the colimit in F un (Λ 2 , op 0 , Cat ( ∞ , 2) ) , and so is given b y the colimit of the underlying ( ∞ , 2) - categories equipp ed with the decorations generated by the images of those in the diagram. Observ ation 2.6.3. It is easy to see that the functor ev 0 : F un (Λ 2 , op 0 , Cat ( ∞ , 2) ) → Cat ( ∞ , 2) has the following prop erties: (1) ev 0 has a left adjoint, which takes an ( ∞ , 2) -category 𝔸 to the span ∅ → 𝔸 ← ∅ . (2) ev 0 has a right adjoint const , which takes an ( ∞ , 2) -category 𝔸 to the span 𝔸 = − → 𝔸 = ← − 𝔸 . (3) The right adjoin t const has a further right adjoint, which tak es a span 𝔸 → 𝔹 ← ℂ to its limit 𝔸 × 𝔹 ℂ . (4) ev 0 is a co cartesian ﬁbration; the co cartesian transp ort of a span 𝔸 F − → 𝔹 G ← − ℂ along ϕ : 𝔹 → 𝔹 ′ is given by comp osition with ϕ , giving 𝔸 φF − − → 𝔹 ′ φG ← − − ℂ . (5) ev 0 is a cartesian ﬁbration; the cartesian transp ort of a span 𝔸 → 𝔹 ← ℂ along ψ : 𝔹 ′ → 𝔹 is given by pullback along ψ , giving 𝔸 × 𝔹 𝔹 ′ → 𝔹 ′ ← ℂ × 𝔹 𝔹 ′ . 28 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI F rom this we can conclude that the restriction of ev 0 to u d : DCat ( ∞ , 2) → Cat ( ∞ , 2) has the following corresp onding prop erties: (1’) u d has a left adjoint ( – ) ♭♭ , which takes an ( ∞ , 2) -category 𝔸 to the span 𝔸 ♭♭ := 𝔸 eq → 𝔸 ← 𝔸 ≤ 1 , where 𝔸 eq is the sub category that contains only the inv ertible 1-morphisms, but all 2-morphisms among these. (2’) u d has a right adjoint ( – ) ♯♯ , which takes an ( ∞ , 2) -category 𝔸 to the span 𝔸 ♯♯ := 𝔸 = − → 𝔸 = ← − 𝔸 . (3’) The right adjoint ( – ) ♯♯ has a further right adjoint D , which takes a decorated ( ∞ , 2) -category 𝔸 ⋄ to the pullback D ( 𝔸 ⋄ ) := 𝔸 ⋄ (1) × 𝔸 𝔸 ⋄ (2) , i.e. the sub- ( ∞ , 2) -category of 𝔸 that contains the decorated 1-morphisms and the decorated 2-morphisms among these. (4’) u d is a co cartesian ﬁbration; the co cartesian transp ort of a decorated ( ∞ , 2) - category 𝔸 ⋄ along ϕ : 𝔸 → 𝔸 ′ is given b y comp osing with ϕ and then applying the lo calization to DCat ( ∞ , 2) (i.e. factoring the span 𝔸 ⋄ (1) → 𝔸 ′ ← 𝔸 ⋄ (2) using the appropriate factorization systems). (5’) u d is a cartesian ﬁbration; the cartesian transp ort of a decorated ( ∞ , 2) - category 𝔸 ⋄ along a functor ψ : 𝔸 ′ → 𝔸 is given b y pullback along ψ (since this preserves the right classes in our factorization systems). Observ ation 2.6.4. Let 2DCat ( ∞ , 2) ⊆ Ar ( Cat ( ∞ , 2) ) denote the full sub category spanned by functors that are wide and lo cally wide sub- ( ∞ , 2) -category inclusions. Then the equiv alence F un (Λ 2 , op 0 , Cat ( ∞ , 2) ) ≃ Ar ( Cat ( ∞ , 2) ) × Cat ( ∞ , 2) Ar ( Cat ( ∞ , 2) ) , arising from the ob vious pushout decomp osition of Λ 2 , op 0 , restricts to an equiv alence DCat ( ∞ , 2) ≃ MCat ( ∞ , 2) × Cat ( ∞ , 2) 2DCat ( ∞ , 2) . F rom Corollary 2.2.7 we then get an equiv alence, (2.1) DCat ( ∞ , 2) ≃ MCat ∞ × Cat ∞ 2DCat ( ∞ , 2) , so that we can regard a decorated ( ∞ , 2) -category ℂ ⋄ as a diagram ℂ ⋄ , ≤ 1 (1) → ℂ ← ℂ ⋄ (2) where ℂ ⋄ , ≤ 1 (1) is a wide subcategory of ℂ ≤ 1 and ℂ ⋄ (2) is a wide and lo cally wide sub- ( ∞ , 2) -category of ℂ . W e note that the forgetful functor u m d : DCat ( ∞ , 2) → MCat ( ∞ , 2) has FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 29 ▶ a right adjoint ( – ) ♯ , which takes ( ℂ , E ) ∈ MCat ( ∞ , 2) to the decorated ( ∞ , 2) - category ( ℂ , E ) ♯ := ℂ E → ℂ = ← − ℂ , ▶ a left adjoint ( – ) ♭ , which takes ( ℂ , E ) ∈ MCat ( ∞ , 2) to the decorated ( ∞ , 2) - category ( ℂ , E ) ♯ := ℂ E → ℂ ← ℂ ≤ 1 , Moreo ver, u m d ( – ) ♯ ≃ id ≃ u m d ( – ) ♭ , so b oth adjoints are fully faithful. Notation 2.6.5. If K is an ∞ -category, then it has a unique class of decorated 2-morphisms, and it is sometimes conv enien t to write K ♯ = K ♯♯ = K ♯♭ , K ♭ = K ♭♭ = K ♭♯ . Similarly , an ∞ -group oid X has a unique decoration and w e may write X = X ♭♭ = X ♯♯ . Prop osition 2.6.6. u d : DCat ( ∞ , 2) → Cat ( ∞ , 2) is a faithful functor. Pr o of. This is immediate from Prop osition 2.2.2 . □ Lemma 2.6.7. The functor ( – ) ♭♭ has a left adjoint τ d : DCat ( ∞ , 2) → Cat ( ∞ , 2) , given by inverting the de c or ate d 1-morphisms and 2-morphisms. Pr o of. W e use the description of DCat ( ∞ , 2) from ( 2.1 ), and view a decorated ( ∞ , 2) - category 𝔸 ⋄ as 𝔸 ⋄ , ≤ 1 (1) → 𝔸 ← 𝔸 ⋄ (2) ; then we deﬁne τ d ( 𝔸 ⋄ ) as the iterated pushout τ d ( 𝔸 ⋄ ) := ∥ 𝔸 ⋄ , ≤ 1 (1) ∥ ⨿ 𝔸 ⋄ , ≤ 1 (1) 𝔸 ⨿ 𝔸 ⋄ (2) τ ( ∞ , 1) ( 𝔸 ⋄ (2) ) . It is clear that this deﬁnes a functor τ d : DCat ( ∞ , 2) → Cat ( ∞ , 2) . F or an ( ∞ , 2) - category 𝔹 , we then hav e a natural equiv alence Map Cat ( ∞ , 2) ( τ d ( 𝔸 ⋄ ) , 𝔹 ) ≃ Map( ∥ 𝔸 ⋄ , ≤ 1 (1) ∥ , 𝔹 ) × Map( 𝔸 ⋄ , ≤ 1 (1) , 𝔹 ) Map( 𝔸 , 𝔹 ) × Map( 𝔸 ⋄ (2) , 𝔹 ) Map( τ ( ∞ , 1) 𝔸 ⋄ (2) , 𝔹 ) ≃ Map( 𝔸 ⋄ , ≤ 1 (1) , 𝔹 ≃ ) × Map( 𝔸 ⋄ , ≤ 1 (1) , 𝔹 ) Map( 𝔸 , 𝔹 ) × Map( 𝔸 ⋄ (2) , 𝔹 ) Map( 𝔸 ⋄ (2) , 𝔹 ≤ 1 ) ≃ Map DCat ( ∞ , 2) ( 𝔸 ⋄ , 𝔹 ♭♭ ) , since the decorated ( ∞ , 2) -category 𝔹 ♭♭ is given by the diagram 𝔹 ≃ → 𝔹 ← 𝔹 ≤ 1 in terms of the description from ( 2.1 ). This shows that the functor τ d is left adjoint to ( – ) ♭♭ , as required. □ Deﬁnition 2.6.8. F or decorated ( ∞ , 2) -categories 𝔸 ⋄ and 𝔹 ⋄ , w e write 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) for the full sub category of 𝔽 un ( 𝔸 , 𝔹 ) spanned by the decorated functors. W e equip this with the decoration where ▶ a natural transformation 𝔸 × [1] → 𝔹 is decorated if its component at ev ery a ∈ 𝔸 is a decorated 1-morphism in 𝔹 , i.e. if it is a decorated functor 𝔸 ⋄ × [1] ♯ → 𝔹 ⋄ ; 30 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI ▶ a 2-morphism 𝔸 × C 2 → 𝔹 is decorated if its comp onent at ev ery a ∈ 𝔸 is a decorated 2-morphism in 𝔹 , i.e. if it is a decorated functor 𝔸 ⋄ × C ♭♯ 2 → 𝔹 ⋄ . Prop osition 2.6.9. The ∞ -c ate gory DCat ( ∞ , 2) is c artesian close d, with internal Hom given by 𝔻𝔽 un ( – , – ) . Pr o of. Since u d is faithful, it suﬃces to chec k that for decorated ( ∞ , 2) -categories 𝔸 ⋄ , 𝔹 ⋄ , ℂ ⋄ , a functor 𝔸 × 𝔹 → ℂ is decorated if and only if its adjoin t 𝔸 → 𝔽 un ( 𝔹 , ℂ ) factors through a decorated functor 𝔸 ⋄ → 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) , whic h is clear from un- pac king the tw o conditions. □ W e can thus regard DCat ( ∞ , 2) as enriched in itself. More importantly , we can upgrade it to an ( ∞ , 2) - (or ( ∞ , 3) -)category: Construction 2.6.10. The functor ( – ) ♭♭ : Cat ( ∞ , 2) → DCat ( ∞ , 2) preserv es carte- sian pro ducts, and for 𝔸 ⋄ ∈ Cat ( ∞ , 2) the functor ( – ) ♭♭ × 𝔸 ⋄ has a right adjoint, namely u d 𝔻𝔽 un ( 𝔸 ⋄ , – ) . It follo ws from Observ ation 2.1.4 that we can upgrade DCat ( ∞ , 2) to an ( ∞ , 3) -category; we write 𝔻ℂ at ( ∞ , 2) for its underlying ( ∞ , 2) - category. Unpac king the deﬁnition, we see that the mapping ∞ -category b et w een t wo ob jects 𝔸 ⋄ and 𝔹 ⋄ is ( u d 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ )) ≤ 1 , whic h is the full sub category of F un ( 𝔸 , 𝔹 ) spanned by the decorated functors: its morphisms are the decorated functors 𝔸 ⋄ × [1] ♭ → 𝔹 ⋄ , whic h are al l the natural transformations among decorated functors. W e also note that the functors u d and ( – ) ♭♭ preserv e pro ducts (b eing right adjoints) and that there is an equiv alence u d ◦ ( – ) ♭♭ , so using Corollary 2.1.6 we get adjunctions of ( ∞ , 2) -categories ( – ) ♭♭ ⊣ u d ⊣ ( – ) ♯♯ . The functor u d is given on mapping ∞ -categories by the fully faithful inclusions ( u d 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ )) ≤ 1 → F un ( 𝔸 , 𝔹 ) ≤ 1 , so that this is a lo cally fully faithful functor. Moreov er, the functor ( – ) ♭♭ preserv es cotensors by [1] , as in general the canonical map 𝔽 un ( ℂ , 𝔻 ) ♭♭ → 𝔻𝔽 un ( ℂ ♭♭ , 𝔻 ♭♭ ) is an equiv alence, so its left adjoint τ d also upgrades to an adjunction of ( ∞ , 2) - categories τ d ⊣ ( – ) ♭♭ b y the dual of Prop osition 2.1.5 . W arning 2.6.11. The ( ∞ , 2) -category 𝔻ℂ at ( ∞ , 2) is not a sub- ( ∞ , 2) -category of 𝔽 un (Λ 2 , op 0 , ℂ at ( ∞ , 2) ) . This is because w e take al l 2-morphisms among decorated functors, i.e. the decorated functors of the form 𝔸 ⋄ × [1] ♭ → 𝔹 ⋄ , as 2-morphisms in 𝔻ℂ at ( ∞ , 2) , while only those of the form 𝔸 ⋄ × [1] ♯ → 𝔹 ⋄ come from 2-morphisms in 𝔽 un (Λ 2 , op 0 , ℂ at ( ∞ , 2) ) . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 31 Observ ation 2.6.12. The forgetful functor u m d : DCat ( ∞ , 2) → MCat ( ∞ , 2) ﬁts in a comm utative triangle Cat ∞ DCat ( ∞ , 2) MCat ( ∞ , 2) ; ( – ) ♭♭ ( – ) ♭ u m d b oth this functor and its right adjoint ( – ) ♯ therefore upgrade to 2-functors (and an adjunction of ( ∞ , 2) -categories) by Corollary 2.1.6 ; the same go es for the left adjoin t ( – ) ♯ , so we get an adjoint triple ( – ) ♭ ⊣ u m d ⊣ ( – ) ♯ also on the level of ( ∞ , 2) - categories. Moreov er, we hav e a natural equiv alence of decorated ( ∞ , 2) -categories 𝕄𝔽 un ( – , – ) ♯ ≃ 𝔻𝔽 un (( – ) ♯ , ( – ♯ )) , whic h in particular sho ws that ( – ) ♯ enhances to a fully faithful functor of ( ∞ , 2) - categories 𝕄ℂ at ( ∞ , 2)  → 𝔻ℂ at ( ∞ , 2) . 2.7. Decorated Gray tensor pro ducts and (op)lax transformations. In this subsection we introduce a decorated version of the Gray tensor pro duct and show that it has righ t adjoin ts in each v ariable. W e w arn the reader that this is given b y the ordinary Gray tensor pro duct (as reviewed in § 2.1 ) equipp ed with certain decorations, and should not b e confused with the marke d Gray tensor pro duct of § 2.3 , where certain 2-morphisms are inv erted. In fact, as we will brieﬂy discuss, we can also deﬁne a decorated version of marked Gra y tensors. Deﬁnition 2.7.1. Given tw o decorated ( ∞ , 2) -categories 𝔸 ⋄ and 𝔹 ⋄ , w e deﬁne an (undecorated) ( ∞ , 2) -category 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) (op)lax as the lo cally full sub- ( ∞ , 2) - category of 𝔽 un ( 𝔸 , 𝔹 ) (op)lax where ▶ the ob jects are decorated functors 𝔸 ⋄ → 𝔹 ⋄ , ▶ the morphisms are (op)lax natural transformations such that for every deco- rated 1-morphism in 𝔸 , the 2-morphism in the corresp onding (op)lax naturalit y square in 𝔹 is decorated. W e enchance this to a decorated ( ∞ , 2) -category by declaring that ▶ an (op)lax transformation is a decorated 1-morphism if all of its comp onents are decorated 1-morphisms in 𝔹 and all of the 2-morphisms in its (op)lax naturalit y squares are decorated, ▶ a 2-morphism is decorated if its comp onent 2-morphism for every ob ject in 𝔸 is decorated. Deﬁnition 2.7.2. Given decorated ( ∞ , 2) -categories 𝔸 ⋄ and 𝔹 ⋄ , we deﬁne a deco- rated ( ∞ , 2) -category 𝔸 ⋄ ⊗ d 𝔹 ⋄ to b e given by the decoration of 𝔸 ⊗ 𝔹 so that ▶ the decorated 1-morphisms are generated by those in 𝔸 ⋄ (1) × 𝔹 ≃ and 𝔸 ≃ × 𝔹 ⋄ (1) ▶ the decorated 2-morphisms are generated by those in 𝔸 ⋄ (2) × 𝔹 ≃ , 𝔸 ≃ × 𝔹 ⋄ (2) , 𝔸 ⋄ , ≤ 1 (1) ⊗ 𝔹 ≤ 1 and 𝔸 ≤ 1 ⊗ 𝔹 ⋄ , ≤ 1 (1) . 32 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI In other w ords, in addition to decorating the (2-)morphisms associated to an ob ject of 𝔸 and a decorated (2-)morphism in 𝔹 and vice versa, we also decorate the 2- morphism in the lax square asso ciated to a pair of morphisms from 𝔸 and 𝔹 if one of them is marked in 𝔸 ⋄ or 𝔹 ⋄ . Observ ation 2.7.3. Let 𝔸 ⋄ , 𝔹 ⋄ , ℂ ⋄ b e decorated ( ∞ , 2) -categories. A functor ℂ → 𝔽 un ( 𝔸 , 𝔹 ) oplax factors through 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) oplax if and only if the adjoin t functor 𝔸 ⊗ ℂ → 𝔹 satisﬁes: ▶ for every ob ject c ∈ ℂ , the restriction to { c } × 𝔸 → 𝔹 is decorated, ▶ for ev ery morphism c → c ′ in ℂ , the restriction to 𝔸 ⊗ [1] → 𝔹 takes any decorated 1-morphism in 𝔸 to an oplax square with a decorated 2-morphism. Moreo ver, this functor is decorated if in addition ▶ for every decorated morphism c → c ′ ∈ ℂ , the restriction to 𝔸 ⊗ [1] → 𝔹 takes ev ery ob ject of 𝔸 to a decorated 1-morphism in 𝔹 and every morphism in 𝔸 to an oplax square with a decorated 2-morphism, ▶ for ev ery decorated 2-morphism in ℂ , the restriction to 𝔸 ⊗ C 2 → 𝔹 takes every ob ject of 𝔸 to a decorated 2-morphism in 𝔹 . In total, these conditions say that we ha v e a decorated functor 𝔸 ⋄ ⊗ d ℂ ⋄ → 𝔹 ⋄ , and this is moreo v er equiv alent to the other adjoint functor 𝔸 → 𝔽 un ( ℂ , 𝔹 ) lax factoring through a decorated functor 𝔸 ⋄ → 𝔻𝔽 un ( ℂ ⋄ , 𝔹 ⋄ ) lax . Thus the functor – ⊗ d – : DCat ( ∞ , 2) × DCat ( ∞ , 2) → DCat ( ∞ , 2) has a right adjoint in each v ariable, given by Map DCat ( ∞ , 2) ( ℂ ⋄ , 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) oplax ) ≃ Map DCat ( ∞ , 2) ( 𝔸 ⋄ ⊗ d ℂ ⋄ , 𝔹 ⋄ ) ≃ Map DCat ( ∞ , 2) ( 𝔸 ⋄ , 𝔻𝔽 un ( ℂ ⋄ , 𝔹 ⋄ ) lax ) . Notation 2.7.4. T o make our notation more compact, it will be conv enient to write 𝔸 ⋄ ⊗ d , ϵ 𝔹 ⋄ :=    𝔸 ⋄ ⊗ d 𝔹 ⋄ , ϵ = (0 , 1) , (1 , 0) , 𝔹 ⋄ ⊗ d 𝔸 ⋄ , ϵ = (0 , 0) , (1 , 1) , 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) ϵ -lax :=    𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) oplax , ϵ = (0 , 1) , (1 , 0) , 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) lax , ϵ = (0 , 0) , (1 , 1) . W e then hav e natural equiv alences Map( 𝔸 ⋄ , 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) ϵ -lax ) ≃ Map( 𝔹 ⋄ ⊗ d , ϵ 𝔸 ⋄ , ℂ ⋄ ) ≃ Map( 𝔹 ⋄ , 𝔻𝔽 un ( 𝔸 ⋄ , ℂ ⋄ ) ϵ -lax ) . W arning 2.7.5. The con v ention in Notation 2.7.4 is motiv ated by the character- ization of partial ϵ -ﬁbrations in Theorem 3.4.1 and related results, and do es not matc h the v ariance of straightening to (op)lax transformations in Theorem 2.4.12 . In fact, a conv ention that ﬁts b oth situations is imp ossible, as ab ov e w e pair (0 , 1) with (1 , 0) , while in 2.4.12 we instead pair (0 , 1) with (0 , 0) . Prop osition 2.7.6. F or de c or ate d ( ∞ , 2) -c ate gories 𝔸 ⋄ , 𝔹 ⋄ , ℂ ⋄ we have a natur al e quivalenc e 𝔸 ⋄ ⊗ d ( 𝔹 ⋄ ⊗ d ℂ ⋄ ) ≃ ( 𝔸 ⋄ ⊗ d 𝔹 ⋄ ) ⊗ d ℂ ⋄ . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 33 Pr o of. W e need to see that the decorations matc h up under the asso ciativit y equiv- alence for the Gray tensor pro duct, whic h is immediate from the deﬁnition. □ Remark 2.7.7. W e expect that the decorated Gra y tensors can in fact be en- hanced to make DCat ( ∞ , 2) a monoidal ∞ -category. As we do not need to know an ything beyond the asso ciativit y statemen t ab ov e, we will not attempt to prov e this, how ev er. Corollary 2.7.8. Ther e ar e natur al e quivalenc es of de c or ate d ( ∞ , 2) -c ate gories 𝔻𝔽 un ( 𝔸 ⋄ , 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) lax ) oplax ≃ 𝔻𝔽 un ( 𝔹 ⋄ , 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) oplax ) lax , 𝔻𝔽 un ( 𝔸 ⋄ ⊗ d 𝔹 ⋄ , ℂ ⋄ ) oplax ≃ 𝔻𝔽 un ( 𝔹 ⋄ , 𝔻𝔽 un ( 𝔸 ⋄ , ℂ ⋄ ) oplax ) oplax , 𝔻𝔽 un ( 𝔸 ⋄ ⊗ d 𝔹 ⋄ , ℂ ⋄ ) lax ≃ 𝔻𝔽 un ( 𝔸 ⋄ , 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) lax ) lax , for 𝔸 ⋄ , 𝔹 ⋄ , ℂ ⋄ ∈ DCat ( ∞ , 2) . □ W arning 2.7.9. Our de c or ate d Gray tensor pro duct consists of the normal Gray tensor of ( ∞ , 2) -categories equipp ed with decorated 1- and 2-morphisms. It should not b e confused with the marke d Gray tensor pro duct of marked ( ∞ , 2) -categories, as reviewed ab o ve in § 2.2 , where certain 2-morphisms in the Gray tensor pro duct determined b y the marking are inverte d . In fact, we can com bine the t w o b y considering decorated ( ∞ , 2) -categories equipp ed with an additional collection of marke d morphisms: V arian t 2.7.10. Supp ose we hav e a decorated ( ∞ , 2) -category 𝔸 ⋄ and in addi- tion a marking ( 𝔸 , I ) on its underlying ( ∞ , 2) -category. F or 𝔹 ⋄ another deco- rated ( ∞ , 2) -category, we then deﬁne 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) I -(op)lax to b e the intersection of 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) (op)lax and 𝔽 un ( 𝔸 , 𝔹 ) I -(op)lax in 𝔽 un ( 𝔸 , 𝔹 ) (op)lax , with decorations inherited from the former. W e can then also deﬁne decorated versions – ⊗ d ♭,I 𝔸 ⋄ and 𝔸 ⋄ ⊗ d I ,♭ – of these marked Gray tensor pro ducts, which participate in adjunctions – ⊗ d ♭,I 𝔸 ⋄ ⊣ 𝔻𝔽 un ( 𝔸 ⋄ , – ) I -lax , 𝔸 ⋄ ⊗ d I ,♭ – ⊣ 𝔻𝔽 un ( 𝔸 ⋄ , – ) I -oplax . Lemma 2.7.11. Supp ose 𝔸 ⋄ , 𝔹 ⋄ , ℂ ⋄ ar e de c or ate d ( ∞ , 2) -c ate gories and we also have a marke d ( ∞ , 2) -c ate gory ( 𝔹 , I ) . Then as an ( ∞ , 2) -c ate gory, 𝔻𝔽 un ( 𝔸 ⋄ , 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) I -lax ) oplax is e quivalent to the ful l sub- ( ∞ , 2) -c ate gory of 𝔻𝔽 un ( 𝔹 ⋄ , 𝔻𝔽 un ( 𝔸 ⋄ , ℂ ⋄ ) oplax ) I -lax sp anne d by the functors that take the morphisms in I to str ong tr ansformations. Pr o of. By deﬁnition, 𝔻𝔽 un ( 𝔸 ⋄ , 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) I -lax ) oplax is a lo cally full sub- ( ∞ , 2) - category of 𝔽 un ( 𝔸 , 𝔻𝔽 un ( 𝔹 ⋄ , ℂ ⋄ ) I -lax ) oplax , whic h in turn is a lo cally full sub- ( ∞ , 2) - category of 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ ) I -lax ) oplax b y [ AGH25 , Theorem 2.6.3]. F rom Obser- v ation 2.3.9 we kno w that w e can identify this with the full sub- ( ∞ , 2) -category of 34 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI 𝔽 un ( 𝔹 , 𝔽 un ( 𝔸 , ℂ ) oplax ) I -lax spanned by the functors that take the morphisms in I to strong transformations. This ( ∞ , 2) -category in turn contains 𝔻𝔽 un ( 𝔹 ⋄ , 𝔻𝔽 un ( 𝔸 ⋄ , ℂ ⋄ ) oplax ) I -lax as a lo cally full sub- ( ∞ , 2) -category, so it suﬃces to observe that these sub- ( ∞ , 2) - categories match up. □ Remark 2.7.12. More generally , one can also consider decorated upgrades of the mark ed Gray tensor 𝔸 ⊗ I ,J 𝔹 of tw o marked ( ∞ , 2) -categories ( 𝔸 , I ) and ( 𝔹 , J ) . This gives a decorated marked Gray tensor on pairs of decorated marked ( ∞ , 2) - categories with corresp onding adjunctions, but w e shall refrain from sp elling this out. Observ ation 2.7.13. Consider the sp ecial case of V ariant 2.7.10 where 𝔸 ⋄ is mark ed by the collection I consisting of all of its 1-morphisms. Then for any deco- rated ( ∞ , 2) -category 𝔹 ⋄ b oth 𝔸 ⋄ ⊗ d I ,♭ 𝔹 ⋄ and 𝔹 ⋄ ⊗ d ♭,I 𝔸 ⋄ coincide with the cartesian pro duct 𝔸 ⋄ × 𝔹 ⋄ , and the right adjoints 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) I -(op)lax b oth coincide with the internal Hom 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) from Deﬁnition 2.6.8 . Observ ation 2.7.14. Supp ose 𝔸 ⋄ and 𝔹 ⋄ are decorated ( ∞ , 2) -categories, and 𝔸 is equipp ed with a marking I such that all decorated 1-morphisms in 𝔸 are also mark ed. Then 𝔻𝔽 un ( 𝔸 ⋄ , 𝔹 ⋄ ) I -(op)lax is a full sub- ( ∞ , 2) -category of 𝔽 un ( 𝔸 , 𝔹 ) I -(op)lax , since for any I -(op)lax transformation the (op)lax naturalit y square for an y deco- rated 1-morphism in 𝔸 commutes, and so in particular con tains a decorated 2- morphism. 3. Fibra tions of decora ted ( ∞ , 2) -ca tegories In this section w e will use the formalism of decorated ( ∞ , 2) -categories to study p artial ﬁbr ations , by which w e mean functors of ( ∞ , 2) -categories that admit carte- sian or co cartesian lifts of certain sp eciﬁed 1- and 2-morphisms in the base. W e in tro duce this notion in § 3.1 and then consider the v ariant notion of de c or ate d partial ﬁbrations in § 3.2 , where there is also a well-behav ed decoration that con- tains these (co)cartesian lifts; when the base is maximally decorated, w e prov e that decorated ﬁbrations classify functors to 𝔻ℂ at ( ∞ , 2) . In § 3.3 we study mapping ∞ - categories in (op)lax arrow ( ∞ , 2) -categories, as a preliminary to pro ving a v ery useful c haracterization of partial ﬁbrations in § 3.4 , in terms of a certain commuta- tiv e square b eing a pullback. W e apply this criterion in § 3.5 to sho w that under certain conditions the functor 𝔽 un ( ℂ , – ) (op)lax , as well as its decorated and par- tially (op)lax v ersions, preserves ﬁbrations. Our pullbac k criterion also implies a c haracterization of partial ϵ -ﬁbrations as b eing right orthogonal to certain maps of decorated ( ∞ , 2) -categories, and in § 3.6 we study the corresp onding left orthogonal class, the ϵ -c oﬁbr ations , as well as the morphisms that are lo cal with resp ect to partial ϵ -ﬁbrations on a ﬁxed base, the ϵ -e quivalenc es . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 35 3.1. P artial ﬁbrations. In this subsection we in tro duce the notion of p artial ϵ - ﬁbr ations of ( ∞ , 2) -categories. After giving the deﬁnition, we relate them to deco- rated ( ∞ , 2) -categories and mention some examples. Deﬁnition 3.1.1. Let 𝔹 ⋄ b e a decorated ( ∞ , 2) -category. F or ϵ = ( i, j ) , we say that a morphism of ( ∞ , 2) -categories p : 𝔼 ⋄ → 𝔹 ⋄ is a p artial ϵ -ﬁbr ation with resp ect to the decoration 𝔹 ⋄ if ▶ 𝔼 has p - i -cartesian lifts of 1-morphisms in 𝔹 ⋄ (1) , ▶ 𝔼 has p - j -cartesian lifts of 2-morphisms in 𝔹 ⋄ (2) . W e also say that a commutativ e triangle 𝔼 𝔽 𝔹 f p q is a morphism of p artial ϵ -ﬁbr ations if f preserves these i -cartesian 1-morphisms and j -cartesian 2-morphisms; we write PFib ϵ / 𝔹 ⋄ for the subcategory of Cat ( ∞ , 2) / 𝔹 con taining the partial ϵ -ﬁbrations ov er 𝔹 and the morphisms thereof. Similarly , w e deﬁne ℙ𝔽 ib ϵ / 𝔹 ⋄ to b e the lo cally full sub- ( ∞ , 2) -category of ℂ at ( ∞ , 2) / 𝔹 on these ob jects and morphisms. V arian t 3.1.2. More generally , for a decorated functor F : 𝔹 ⋄ → ℂ ⋄ , w e say that a commutativ e square 𝔼 𝔽 𝔹 ℂ G p q F is a morphism of p artial ϵ -ﬁbr ations if p and q are partial ϵ -ﬁbrations with resp ect to 𝔹 ⋄ and ℂ ⋄ , resp ectively , and G preserves i -cartesian morphisms and j -cartesian 2-morphisms ov er the decorations in 𝔹 ⋄ . Examples 3.1.3. Let p : 𝔼 → 𝔹 b e a functor of ( ∞ , 2) -categories and ϵ = ( i, j ) . ▶ p is alwa ys a partial ϵ -ﬁbration with resp ect to 𝔹 ♭♭ . ▶ p is a partial ϵ -ﬁbration with resp ect to 𝔹 ♯♭ if and only if 𝔼 has all p - i -cartesian lifts of morphisms in 𝔹 . ▶ p is a partial ϵ -ﬁbration with resp ect to 𝔹 ♭♯ if and only if 𝔼 ( x, y ) → 𝔹 ( p ( x ) , p ( y )) is a j -cartesian ﬁbration of ∞ -categories for all x, y ∈ 𝔼 , and the j -cartesian morphisms are closed under whisk erings. This is equiv alen t to the composition functors 𝔼 ( x, y ) × 𝔼 ( y , z ) → 𝔼 ( x, z ) preserving j -cartesian morphisms for all x, y , z ∈ 𝔼 , so this condition on p corresp onds to b eing j -c artesian-enriche d in the terminology of [ AGH25 ]. ▶ Com bining the last tw o examples, we see that p is a partial ϵ -ﬁbration with resp ect to 𝔹 ♯♯ if and only if p is an ϵ -ﬁbration as deﬁned ab o v e in § 2.4 . Observ ation 3.1.4. F or p : 𝔼 → 𝔹 a functor of ( ∞ , 2) -categories, it follows from the comp osability of pullbac k squares that p -(co)cartesian morphisms are closed 36 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI under comp osition. Similarly , weakly p -(co)cartesian 2-morphisms are closed under v ertical comp osition. Since horizontal comp osition can b e decomp osed into whisker- ing and vertical comp osition, it follows that p -(co)cartesian 2-morphisms are closed under both vertical and horizontal comp osition. If p is a partial ( i, j ) -ﬁbration with resp ect to 𝔹 ⋄ , this means that ▶ the p - i -cartesian morphisms ov er 𝔹 ⋄ (1) determine a lo cally full sub- ( ∞ , 2) -category of 𝔼 , ▶ the p - j -cartesian 2-morphisms o ver 𝔹 ⋄ (2) determine a wide and locally wide sub- ( ∞ , 2) -category of 𝔼 . Th us when p is a partial ﬁbration we obtain a canonical lift of the decoration of 𝔹 ⋄ to a decoration 𝔼 ♮ of 𝔼 , which leads us to the following deﬁnition: Deﬁnition 3.1.5. Let 𝔹 ⋄ b e a decorated ( ∞ , 2) -category. F or ϵ = ( i, j ) , we say that a morphism of ( ∞ , 2) -categories p : 𝔼 ⋄ → 𝔹 ⋄ is a p artial ϵ -ﬁbr ation if ▶ 𝔼 has p - i -cartesian lifts of 1-morphisms in 𝔹 ⋄ (1) , and 𝔼 ⋄ (1) is precisely the wide lo cally full sub- ( ∞ , 2) -category of these, ▶ 𝔼 has p - j -cartesian lifts of 2-morphisms in 𝔹 ⋄ (2) , and 𝔼 ⋄ (2) is precisely the wide lo cally wide sub- ( ∞ , 2) -category of these. W e can then identify PFib ϵ / 𝔹 ⋄ with the full sub category of DCat ( ∞ , 2) / 𝔹 ⋄ spanned by the partial ϵ -ﬁbrations in this sense. Example 3.1.6. F or any decorated ( ∞ , 2) -category 𝔹 ⋄ and an y ( ∞ , 2) -category 𝕂 , the pro jection 𝕂 ♭♭ × 𝔹 ⋄ → 𝔹 ⋄ is a partial ϵ -ﬁbration for any ϵ . (Note, how ever, that the pro jection 𝕂 ⋄ × 𝔹 ⋄ → 𝔹 ⋄ for some non-minimal decoration of 𝕂 will not b e a partial ﬁbration, since 𝕂 ⋄ × 𝔹 ⋄ then has decorated 1 - or 2 -morphisms that are not (co)cartesian.) Example 3.1.7. Let ( 𝔼 , I ) b e a marked ( ∞ , 2) -category with a functor p : 𝔼 → 𝔹 . Then ( 𝔼 , I ) ♯ → 𝔹 ♯♯ is a partial ϵ -ﬁbration for ϵ = ( i, j ) if and only if p is an ϵ - ﬁbration, I is the collection of p - i -cartesian 1-morphisms, and all 2-morphisms in 𝔼 are j -cartesian. Equiv alen tly , p is a 1-ﬁbr e d ϵ -ﬁbration with I as p - i -cartesian 1-morphisms. 3.2. Decorated partial ﬁbrations. In this section w e introduce a v ariant of the deﬁnition of partial ﬁbrations where we allow additional decorations b eyond the (co)cartesian 1- and 2-morphisms; these will b e useful at several p oin ts in the next section. W e will show that in the case where the base 𝔹 ♯♯ is maximally decorated, these can b e straightened to functors to 𝔻ℂ at ( ∞ , 2) . Deﬁnition 3.2.1. A morphism of decorated ( ∞ , 2) -categories p : 𝔼 ⋄ → 𝔹 ⋄ is a de c or ate d p artial ϵ -ﬁbr ation for ϵ = ( i, j ) if in the commutativ e diagram (where we FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 37 use the description from ( 2.1 )) 𝔼 ⋄ , ≤ 1 (1) 𝔼 𝔼 ⋄ (2) 𝔹 ⋄ , ≤ 1 (1) 𝔹 𝔹 ⋄ (2) , p ≤ 1 (1) p p (2) ▶ the functor p : 𝔼 → 𝔹 is a partial ϵ -ﬁbration with resp ect to 𝔹 ⋄ , ▶ the functor p ≤ 1 (1) : 𝔼 ⋄ , ≤ 1 (1) → 𝔹 ⋄ , ≤ 1 (1) is an i -ﬁbration of ∞ -categories, ▶ the functor p (2) : 𝔼 ⋄ (2) → 𝔹 ⋄ (2) is a partial ϵ -ﬁbration with resp ect to the restric- tion of the decorations on 𝔹 ⋄ , ▶ b oth commutativ e squares are morphisms of partial ϵ -ﬁbrations, i.e. they pre- serv e the i -cartesian 1-morphisms and j -cartesian 2-morphisms. Observ ation 3.2.2. Let us sp ell this out more concretely in the case ϵ = (0 , 1) : a functor p : 𝔼 ⋄ → 𝔹 ⋄ is a decorated partial (0 , 1) -ﬁbration if ▶ the underlying functor of ( ∞ , 2) -categories p : 𝔼 → 𝔹 is a partial (0 , 1) -ﬁbration, ▶ the co cartesian morphisms ov er 𝔹 ⋄ (1) are among the decorated morphisms of 𝔼 ⋄ , and given a commutativ e triangle x y z f h g in 𝔼 whose image in 𝔹 consists of decorated morphisms, suc h that h is decorated and f is co cartesian, then g is also decorated. ▶ the cartesian 2-morphisms ov er 𝔹 ⋄ (2) are among the decorated 2-morphisms of 𝔼 ⋄ , and – if f : x → y in 𝔼 is a co cartesian morphism ov er a decorated morphism in 𝔹 , then for all z ∈ 𝔼 a 2-morphism in 𝔼 ( y , z ) whose image is decorated in 𝔹 ⋄ is decorated in 𝔼 if and only if its comp osition with f in 𝔼 ( x, z ) is decorated; – giv en a comm utative triangle f g h α γ β in 𝔼 ( x, y ) whose image in 𝔹 ( px, py ) consists of decorated 2-morphisms, suc h that γ is decorated and β is cartesian, then α is also decorated. W arning 3.2.3. Note that w e do not ask for the func tor p (1) : 𝔼 ⋄ (1) → 𝔹 ⋄ (1) to b e a partial ϵ -ﬁbration of ( ∞ , 2) -categories. This would require decorated 1-morphisms to b e preserved under j -cartesian transp ort along decorated 2-morphisms. Since this is generally false for i -cartesian 1-morphisms in a partial ϵ -ﬁbration, this is far to o restrictive a condition to imp ose. 38 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Deﬁnition 3.2.4. W e say that a commutativ e triangle 𝔼 ⋄ 𝔽 ⋄ 𝔹 ⋄ , F p q in DCat ( ∞ , 2) , where p and q are decorated partial ϵ -ﬁbrations, is a morphism of de c or ate d p artial ϵ -ﬁbr ations if the decorated functor F also preserves i -cartesian 1-morphisms and j -cartesian 2-morphisms ov er the decorations in 𝔹 ⋄ . The ( ∞ , 2) - category 𝔻ℙ𝔽 ib ϵ / 𝔹 ⋄ of decorated partial ϵ -ﬁbrations ov er 𝔹 ⋄ is then deﬁned to b e the lo cally full sub- ( ∞ , 2) -category of 𝔻ℂ at ( ∞ , 2) / 𝔹 ⋄ whose ob jects are the decorated partial ϵ -ﬁbrations and whose 1-morphisms are the morphisms thereof. Observ ation 3.2.5. The forgetful functor 𝔻ℂ at ( ∞ , 2) / 𝔹 ⋄ → ℂ at ( ∞ , 2) / 𝔹 restricts to a lo cally fully faithful functor 𝔻ℙ𝔽 ib ϵ / 𝔹 ⋄ → ℙ𝔽 ib ϵ / 𝔹 ⋄ . V arian t 3.2.6. In the sp ecial case where the base is maximally decorated, w e refer to decorated partial ϵ -ﬁbrations ov er 𝔹 ♯♯ as de c or ate d ϵ -ﬁbr ations and write 𝔻𝔽 ib ϵ / 𝔹 := 𝔻ℙ𝔽 ib ϵ / 𝔹 ♯♯ . Examples 3.2.7. ▶ Let 𝔹 ⋄ b e a decorated ( ∞ , 2) -category and suppose p : 𝔼 → 𝔹 is a partial ϵ -ﬁbration with resp ect to 𝔹 ⋄ . Then p : 𝔼 ♮ → 𝔹 ⋄ is a decorated partial ϵ - ﬁbration using the canonical decoration of 𝔼 (Observ ation 3.1.4 ); this follows from Lemma 2.4.2 . ▶ F or any decorated ( ∞ , 2) -categories 𝕂 ⋄ and 𝔹 ⋄ , the pro jection 𝕂 ⋄ × 𝔹 ⋄ → 𝔹 ⋄ is a decorated partial ϵ -ﬁbration for any ϵ . ▶ Supp ose p : 𝔼 → 𝔹 is an ϵ -ﬁbration. Then 𝔼 ♯♯ → 𝔹 ♯♯ is a decorated ϵ -ﬁbration. W e now turn to straigh tening for decorated ϵ -ﬁbrations: Theorem 3.2.8. L et 𝔹 b e an ( ∞ , 2) -c ate gory. The str aightening e quivalenc e for ϵ -ﬁbr ations extends to de c or ate d ﬁbr ations to give a natur al c ommutative squar e 𝔻𝔽 ib ϵ / 𝔹 𝔽 un ( 𝔹 ϵ -op , 𝔻ℂ at ( ∞ , 2) ) 𝔽 ib ϵ / 𝔹 𝔽 un ( 𝔹 ϵ -op , ℂ at ( ∞ , 2) ) . ∼ ∼ W e will prov e this in the case ϵ = (0 , 1) , lea ving the 3 other v arian ts implicit. Since the vertical functors here are locally fully faithful (b y [ AGH25 , Theorem 2.6.3] in the right-hand case), the following observ ation shows it suﬃces to prov e that we get an equiv alence on underlying ∞ -categories: Observ ation 3.2.9. Suppose we ha ve lo cally fully faithful functors 𝔸 , 𝔹 → ℂ and an equiv alence 𝔸 ≤ 1 ≃ 𝔹 ≤ 1 o ver ℂ ≤ 1 . Then we can lift this to an equiv alence 𝔸 ≃ 𝔹 FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 39 o ver ℂ . Indeed, we hav e a commutativ e square 𝔸 ≤ 1 𝔹 𝔸 ℂ where the left vertical map is left orthogonal to the righ t vertical map, so that there is a unique dotted lift 𝔸 → 𝔹 . Similarly , the top horizontal map is left orthogonal to the b ottom horizontal map, and by uniqueness the comp osites of these lifts are iden tities. W e ﬁrst consider the case where the base is an ∞ -category: Prop osition 3.2.10. F or B an ∞ -c ate gory ther e is a natur al c ommutative squar e of ∞ -c ate gories DFib (0 , 1) / B F un ( B , DCat ( ∞ , 2) ) Fib (0 , 1) / B F un ( B , Cat ( ∞ , 2) ) . ∼ ∼ Pr o of. Since B is an ∞ -category, we can think of a decorated (0 , 1) -ﬁbration o ver B as a diagram 𝔼 ⋄ (1) 𝔼 𝔼 ⋄ (2) B B B , p (1) p p (2) = = where p , p (1) , and p (2) are all 0 -ﬁbrations, and the top horizontal maps are mor- phisms of 0 -ﬁbrations. Similarly , morphisms in DFib (0 , 1) / B corresp ond to maps of suc h cospans ov er B where all comp onen ts preserve co cartesian morphisms. W e can thus identify DFib (0 , 1) / B with a full sub category of Fun (Λ 2 , op 0 , Fib (0 , 1) / B ) . Under straigh tening, the latter is equiv alent to Fun ( B , Fun (Λ 2 , op 0 , F un ( B , Cat ( ∞ , 2) )) . As it is easy to see that a morphism of ﬁbrations ov er B is a wide lo cally full inclusion or a wide and lo cally wide inclusion if and only if it is so ﬁbrewise, it follows that the full sub category corresp onding to DFib (0 , 1) / B can b e identiﬁed as F un ( B , DCat ( ∞ , 2) ) , as required. □ Prop osition 3.2.11. F or 𝔹 an ( ∞ , 2) -c ate gory, the natur al c ommutative squar e of ∞ -c ate gories DFib (0 , 1) / 𝔹 Fib (0 , 1) / 𝔹 DFib (0 , 1) / 𝔹 ≤ 1 Fib (0 , 1) / 𝔹 ≤ 1 is a pul lb ack. F or the pro of, we need the following observ ation: Lemma 3.2.12. Supp ose p : 𝔼 → 𝔹 is a (0 , 1) -ﬁbr ation and set 𝔼 ′ := 𝔼 × 𝔹 𝔹 ≤ 1 . If 𝔼 ′⋄ → 𝔹 ≤ 1 ,♯ is a de c or ate d (0 , 1) -ﬁbr ation, then 40 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI (i) the 2-morphisms in 𝔼 that factor as a vertic al c omp osite of a de c or ate d 2- morphism in 𝔼 ′⋄ (2) fol lowe d by a c artesian 2-morphism ar e close d under b oth horizontal and vertic al c omp osition, and so form a wide and lo c al ly wide sub c ate gory 𝔼 ⋄ (2) of 𝔼 , which is the minimal such c ontaining b oth 𝔼 ′⋄ (2) and 𝔼 ♮ (2) ; (ii) the r estriction of p to 𝔼 ⋄ (2) → 𝔹 is a (0 , 1) -ﬁbr ation, and the inclusion 𝔼 ⋄ (2) → 𝔼 is a morphism of (0 , 1) -ﬁbr ations; (iii) if 𝔼 ⋄ is the de c or ation of 𝔼 given by the 1-morphisms in 𝔼 ′⋄ (1) and the 2- morphisms in 𝔼 ⋄ (2) , then 𝔼 ⋄ → 𝔹 ⋄ is a de c or ate d (0 , 1) -ﬁbr ation. Pr o of. In (i), it is clear that the sp eciﬁed 2-morphisms are closed under horizontal comp osition, since this is true for b oth the decorated 2-morphisms in 𝔼 ′⋄ and the p -cartesian 2-morphisms in 𝔼 . It therefore suﬃces to consider a vertical comp osition x y f g h α β with α cartesian and β decorated ov er an equiv alence (i.e. in the “wrong” order). Then β factors as x x ′ y γ g ′ h ′ β ′ with γ co cartesian ov er p ( g ) ≃ p ( h ) , while α factors as x x ′ y f ′ γ g ′ α ′ with α ′ cartesian. Th us our vertical comp osition is equiv alent to the horizontal comp osition x x ′ y , f ′ γ g ′ h ′ β ′ α ′ whic h w e can in turn rewrite as a v ertical comp osition in the “right” order. This pro ves (i). F or (ii), we note ﬁrstly that co cartesian morphisms in 𝔼 are also cocartesian in 𝔼 ⋄ (2) , since b oth the morphisms in 𝔼 ′⋄ (2) and 𝔼 ♮ (2) are detected after whiskering with a co cartesian morphism. Moreov er, it is clear from the deﬁnition of 𝔼 ⋄ (2) that for x, y ∈ 𝔼 , the functor 𝔼 ⋄ (2) ( x, y ) → 𝔹 ( px, py ) is a cartesian ﬁbration with the cartesian morphisms inherited from 𝔼 ⋄ ( x, y ) , as required. Finally , (iii) is immediate from (ii) and the deﬁnition of a decorated (0 , 1) - ﬁbration. □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 41 Pr o of of Pr op osition 3.2.11 . Using Lemma 3.2.12 , we can deﬁne a functor from Fib (0 , 1) / 𝔹 × Fib (0 , 1) / 𝔹 ≤ 1 DFib (0 , 1) / 𝔹 ≤ 1 → DFib (0 , 1) / 𝔹 b y taking a pair ( p : 𝔼 → 𝔹 , p ′ : 𝔼 ′⋄ → 𝔹 ≤ 1 ) (where 𝔼 ′ := 𝔼 × 𝔹 𝔹 ≤ 1 ) to the lo caliza- tion to DCat ( ∞ , 2) of the diagram 𝔼 ′≤ 1 (1) → 𝔼 ← 𝔼 ♮ (2) ⨿ 𝔼 ′⋄ (2) o ver 𝔹 ♯♯ , which is clearly functorial in ( p, p ′ ) . In this case, 𝔼 ⋄ (2) × 𝔹 𝔹 ≤ 1 reco vers 𝔼 ′⋄ (2) ; on the other hand, for 𝔼 ⋄ → 𝔹 ♯♯ in DFib (0 , 1) / 𝔹 w e see that 𝔼 ⋄ (2) consists precisely of those 2-morphisms that factor as a decorated 2-morphism ov er an equiv alence follo wed by a cartesian 2-morphism. Th us the comp osites in b oth directions are equiv alences, which completes the pro of. □ Pr o of of The or em 3.2.8 . F rom Observ ation 3.2.9 it suﬃces to show that w e hav e an equiv alence on underlying ∞ -categories. By orthogonality we also ha v e a natural pullbac k square of ∞ -categories F un ( 𝔹 , 𝔻ℂ at ( ∞ , 2) ) F un ( 𝔹 , ℂ at ( ∞ , 2) ) F un ( 𝔹 ≤ 1 , 𝔻ℂ at ( ∞ , 2) ) F un ( 𝔹 ≤ 1 , ℂ at ( ∞ , 2) ) . Com bined with the pullbac k square from Proposition 3.2.11 this means that it suﬃces to construct a natural cospan of equiv alences F un ( 𝔹 ≤ 1 , DCat ( ∞ , 2) ) F un ( 𝔹 ≤ 1 , Cat ( ∞ , 2) ) F un ( 𝔹 , ℂ at ( ∞ , 2) ) , DFib (0 , 1) / 𝔹 ≤ 1 Fib (0 , 1) / 𝔹 ≤ 1 Fib (0 , 1) / 𝔹 , ∼ ∼ ∼ whic h w e get from Prop osition 3.2.10 and the naturality of straigh tening. □ P assing to left adjoints, we also get: Corollary 3.2.13. Ther e is a natur al c ommutative squar e of ( ∞ , 2) -c ate gories 𝔽 ib ϵ / 𝔹 𝔽 un ( 𝔹 ϵ -op , ℂ at ( ∞ , 2) ) 𝔻𝔽 ib ϵ / 𝔹 𝔽 un ( 𝔹 ϵ -op , 𝔻ℂ at ( ∞ , 2) ) .. ∼ ( – ) ♮ ( – ) ♭♭ ∗ ∼ F urthermor e, the left adjoint to ( – ) ♮ c orr esp onds under str aightening to c omp osition with the lo c alization functor τ d : 𝔻ℂ at ( ∞ , 2) → ℂ at ( ∞ , 2) . □ W e exp ect that it is also p ossible to extend Theorem 2.4.12 to describ e the ( ∞ , 2) - categories 𝔽 un ( 𝔹 , 𝔻ℂ at ( ∞ , 2) ) E -(op)lax in terms of decorated ﬁbrations. Ho wev er, as this seems somewhat annoying to pro v e and is not required in this pap er, w e conten t ourselv es with the following statement in this case: 42 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Prop osition 3.2.14. L et ( 𝔹 , E ) b e a marke d ( ∞ , 2) -c ate gory. F or functors F , G : 𝔹 → 𝔻ℂ at ( ∞ , 2) with c orr esp onding de c or ate d ϵ -ﬁbr ations ℙ ⋄ ϵ , ℚ ⋄ ϵ → 𝔹 ♯♯ , ther e is a natur al e quivalenc e Nat E - lax 𝔹 , 𝔻ℂ at ( ∞ , 2) ( F , G ) ≃ DF un E -co c / ( 𝔹 ,E ) ♯ ( ℙ ⋄ (0 , 1) | E , ℚ ⋄ (0 , 1) | E ) , wher e ℙ ⋄ (0 , 1) | E denotes the pul lb ack ℙ ⋄ (0 , 1) × 𝔹 ♯♯ ( 𝔹 , E ) ♯ (given by r estricting the de c- or ate d 1-morphisms of ℙ ⋄ (0 , 1) to those that lie over E ) and DF un E -co c / ( 𝔹 ,E ) ♯ ( ℙ ⋄ (0 , 1) | E , ℚ ⋄ (0 , 1) | E ) ⊆ DF un / ( 𝔹 ,E ) ♯ ( ℙ ⋄ (0 , 1) | E , ℚ ⋄ (0 , 1) | E ) is the ful l sub c ate gory of functors that pr eserve c o c artesian morphisms over E and al l c artesian 2-morphisms. Similarly, we have natur al e quivalenc es Nat E - oplax 𝔹 , 𝔻ℂ at ( ∞ , 2) ( F , G ) ≃ DF un E -cart / ( 𝔹 op ,E ) ♯ ( ℙ ⋄ (1 , 0) | E , ℚ ⋄ (1 , 0) | E ) ≃ DF un E -cart / ( 𝔹 coop ,E ) ♯ ( ℙ ⋄ (1 , 1) | E , ℚ ⋄ (1 , 1) | E ) , Nat E - lax 𝔹 , 𝔻ℂ at ( ∞ , 2) ( F , G ) ≃ DF un E -co c / ( 𝔹 co ,E ) ♯ ( ℙ ⋄ (0 , 0) | E , ℚ ⋄ (0 , 0) | E ) . Lemma 3.2.15. Supp ose F : ℂ → 𝔻 is a lo c al ly ful ly faithful functor. Then so is F ∗ : 𝔽 un ( 𝔸 , ℂ ) E -(op)lax → 𝔽 un ( 𝔸 , 𝔻 ) E -(op)lax for any marke d ( ∞ , 2) -c ate gory ( 𝔸 , E ) , and we c an identify Nat E -(op)lax 𝔸 , ℂ ( α, β ) ⊆ Nat E -(op)lax 𝔸 , 𝔻 ( F α , F β ) as the ful l sub c ate gory of those E -(op)lax tr ansformations η : F α → F β whose c om- p onent η x ∈ 𝔻 ( F α ( x ) , F β ( x )) lies in ℂ ( α ( x ) , α ( y )) for every obje ct x ∈ 𝔸 . Pr o of. It follows from [ AGH25 , Corollary 2.7.13] that F ∗ is lo cally fully faithful, so w e are left with identifying its image on mapping ∞ -categories. The stated condition is clearly necessary , so we need to prov e that it suﬃces. W e consider the oplax case; using the adjunction b etw een 𝔽 un ( 𝔸 , – ) E -(op)lax and 𝔸 ⊗ E ,♮ – w e can rephrase this as: F is right orthogonal to the map 𝔸 ≃ × [1] ⨿ 𝔸 ≃ × ∂ [1] 𝔸 ≤ 1 × ∂ [1] → 𝔸 ⊗ E ,♮ [1] . Since F is lo cally fully faithful if and only if it is right orthogonal to the map ∂ C 2 → C 2 , it suﬃces to show that this map is in the saturated class generated b y the latter map, which we know from [ LMGR + 24 , Theorem 5.3.7] consists of the functors that are essentially surjective on ob jects and 1-morphisms. The canonical description of 𝔸 as a colimit of ob jects in Θ 2 is preserv ed by ( – ) ≤ 1 and ( – ) ≃ , so arguing as in the pro of of [ AGH25 , Corollary 2.5.14] we can reduce to the case where ( 𝔸 , E ) is [0] , [1] ♭ , [1] ♯ and C ♭ 2 . In these cases it is clear that the given map is surjectiv e on ob jects and 1-morphisms, which completes the pro of. □ Pr o of of Pr op osition 3.2.14 . W e pro v e the lax case. By Lemma 3.2.15 the lo cally fully faithful forgetful functor u d : 𝔻ℂ at ( ∞ , 2) → ℂ at ( ∞ , 2) iden tiﬁes Nat E -lax 𝔹 , 𝔻ℂ at ( ∞ , 2) ( F , G ) ⊆ Nat E -lax 𝔹 , ℂ at ( ∞ , 2) ( u d F , u d G ) as the full sub category of E -lax transformations ϕ : u d F → u d G such that each comp onen t ϕ b : u d F ( b ) → u d G ( b ) is a decorated functor F ( b ) → G ( b ) for all b ∈ 𝔹 . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 43 No w Theorem 2.4.12 gives a natural equiv alence Nat E -lax 𝔹 , ℂ at ( ∞ , 2) ( u d F , u d G ) ≃ 𝔽 ib (0 , 1) / ( 𝔹 ,E ) ( ℙ , ℚ ) , where the right-hand side is the full sub category of Fun / 𝔹 ( ℙ , ℚ ) on functors that pre- serv e co cartesian morphisms o v er E and all cartesian 2-morphisms. W e can there- fore iden tify Nat E -lax 𝔹 , 𝔻ℂ at ( ∞ , 2) ( F , G ) with the full sub category of the latter spanned by the functors r : ℙ → ℚ o v er 𝔹 for which the map on ﬁbres is a decorated functor r b : ℙ ⋄ b → ℚ ⋄ b for all b ∈ 𝔹 . A decorated morphism in ℙ ⋄ factors as a co cartesian mor- phism follow ed by a decorated morphism in a ﬁbre, so this condition on 1-morphisms is equiv alen t to r preserving all decorated morphisms ov er E ; similarly , the condi- tion on 2-morphisms is equiv alent to r preserving al l decorated 2-morphisms. The ∞ -category in question is therefore DF un E -co c / ( 𝔹 ,E ) ♯ ( ℙ ⋄ (0 , 1) | E , ℚ ⋄ (0 , 1) | E ) , as required. □ 3.3. Maps in (op)lax arro ws. Our goal in this subsection is to identify the map- ping ∞ -categories in 𝔸 r (op)lax ( ℂ ) , which is the key input needed for the characteri- zation of partial ﬁbrations in the following subsection. Prop osition 3.3.1. F or an ( ∞ , 2) -c ate gory ℂ and morphisms f : x → y and g : a → b in ℂ , ther e ar e pul lb ack squar es of ∞ -c ate gories 𝔸 r lax ( ℂ )( f , g ) Ar ( ℂ ( x, b )) ℂ ( x, a ) × ℂ ( y , b ) ℂ ( x, b ) × ℂ ( x, b ) , ev 0 , ev 1 g ∗ × f ∗ 𝔸 r oplax ( ℂ )( f , g ) Ar ( ℂ ( x, b )) ℂ ( x, a ) × ℂ ( y , b ) ℂ ( x, b ) × ℂ ( x, b ) . ev 1 , ev 0 g ∗ × f ∗ Remark 3.3.2. In the notation of [ AM24 ], Prop osition 3.3.1 says that the ∞ - categories 𝔸 r (op)lax ( ℂ )( f , g ) are oriente d pul lb acks or c omma obje cts : 𝔸 r lax ( ℂ )( f , g ) ≃ ℂ ( x, a ) ← − × ℂ ( x,b ) ℂ ( y , b ) , 𝔸 r oplax ( ℂ )( f , g ) ≃ ℂ ( x, a ) − → × ℂ ( x,b ) ℂ ( y , b ) . Pr o of of Pr op osition 3.3.1 . W e prov e the oplax case; the lax case is prov ed similarly . W e can identify the mapping ∞ -category 𝔸 r oplax ( ℂ )( f , g ) as the pullback 𝔸 r oplax ( ℂ )( f , g ) 𝔸 r oplax ( 𝔸 r oplax ( ℂ )) ∗ 𝔸 r oplax ( ℂ ) × 𝔸 r oplax ( ℂ ) . ev 0 , ev 1 ( f ,g ) Here 𝔸 r oplax ( 𝔸 r oplax ( ℂ )) is equiv alen t to 𝔽 un ([1] ⊗ [1] , ℂ ) oplax . The pushout square from Observ ation 2.1.10 gives a commutativ e cub e 𝔽 un ([1] ⊗ [1] , ℂ ) oplax 𝔽 un ( C 2 , ℂ ) oplax 𝔽 un ( ∂ ([1] ⊗ [1]) , ℂ ) 𝔽 un ( ∂ C 2 , ℂ ) oplax 𝔸 r oplax ( ℂ ) × 2 ℂ × 2 𝔸 r oplax ( ℂ ) × 2 ℂ × 2 = = 44 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI where b oth the top and b ottom squares are pullbac ks, and we recognize 𝔸 r oplax ( ℂ )( f , g ) as the ﬁbre at ( f , g ) in the top left corner. Our desired pullback square will b e the induced pullback on ﬁbres from this cub e. T o identify the ﬁbre at the b ottom left we consider the pushout decomp osition [0] ⨿ 4 [1] ⨿ [1] [1] ⨿ [1] ∂ ([1] ⊗ [1]) , whic h induces a pullback square 𝔽 un ( ∂ ([1] ⊗ [1]) , ℂ ) oplax 𝔸 r oplax ( ℂ ) × 2 𝔸 r oplax ( ℂ ) × 2 ℂ × 4 ; from this we see that the ﬁbre at ( f : x → y , g : a → b ) on the left is equiv alent to ℂ ( x, a ) × ℂ ( y , b ) . T o describ e the ﬁb res on the right side of the cub e we consider the pushouts [1] ⨿ [1] ∂ ([1] ⊗ [1]) [1] ⊗ [1] [0] ⨿ [0] ∂ C 2 C 2 , whic h giv e pullbacks 𝔽 un ( C 2 , ℂ ) oplax 𝔽 un ([1] ⊗ [1] , ℂ ) oplax 𝔽 un ( ∂ C 2 , ℂ ) oplax 𝔽 un ( ∂ ([1] ⊗ [1]) , ℂ ) oplax ℂ × 2 𝔸 r oplax ( ℂ ) × 2 . Here the ﬁbre at ( x, b ) at the top right is identiﬁed with Ar ℂ ( x, b ) b y applying 𝔸 r oplax ( – ) to the pullback ℂ ( x, b ) 𝔸 r oplax ( ℂ ) { ( x, b ) } ℂ × 2 , while the pushout for ∂ ([1] ⊗ [1]) ab ov e again identiﬁes the ﬁbre at the middle right as ℂ ( x, b ) × 2 , with the map from Ar ℂ ( x, b ) giv en by taking source and target. W e thus hav e the desired pullback square 𝔸 r oplax ( ℂ )( f , g ) Ar ℂ ( x, b ) ℂ ( x, a ) × ℂ ( y , b ) ℂ ( x, b ) × 2 FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 45 where the b ottom horizon tal map can b e iden tiﬁed as given by comp osition with f and g since this ultimately came from the map ∂ C 2 → ∂ ([1] ⊗ [1]) given by tw o copies of d 1 . □ Observ ation 3.3.3. Let j : ∂ C 2 → C 2 b e the b oundary inclusion. F or later use, w e note that in the course of this pro of w e saw that in the commutativ e triangle 𝔽 un ( C 2 , ℂ ) oplax 𝔽 un ( ∂ C 2 , ℂ ) oplax ℂ × 2 j ∗ the map on ﬁbres ov er x, y was the ev aluation map Ar ℂ ( x, y ) → ℂ ( x, y ) × 2 . Hence the ﬁbre of j ∗ at the ob ject corresp onding to f , g : x → y is the ∞ -group oid ℂ ( x, y )( f , g ) . This shows in particular that j ∗ is conserv ativ e on 1-morphisms. W e can extend this description to decorated ( ∞ , 2) -categories, for which w e in tro duce the following notation: Deﬁnition 3.3.4. F or a decorated ( ∞ , 2) -category ℂ ⋄ , let 𝔻𝔸 r (op)lax ( ℂ ⋄ ) := 𝔻𝔽 un ([1] ♯ , ℂ ⋄ ) (op)lax . In other words, this is the lo cally full sub- ( ∞ , 2) -category of 𝔸 r (op)lax ( ℂ ) where ▶ the ob jects are the morphisms in ℂ ⋄ (1) , ▶ the morphisms are the (op)lax squares where the 2-morphisms lie in ℂ ⋄ (2) . It is conv enien t to write 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) :=    𝔻𝔸 r oplax ( ℂ ⋄ ) , ϵ = (0 , 1) , (1 , 0) , 𝔻𝔸 r lax ( ℂ ⋄ ) , ϵ = (0 , 0) , (1 , 1) . Viewing 𝔻𝔸 r (op)lax ( ℂ ⋄ ) for a decorated ( ∞ , 2) -category ℂ ⋄ as a lo cally full sub- category of 𝔸 r (op)lax ( ℂ ) , we get: Corollary 3.3.5. F or a de c or ate d ( ∞ , 2) -c ate gory ℂ ⋄ and de c or ate d morphisms f : x → y and g : a → b in ℂ , ther e ar e pul lb ack squar es of ∞ -c ate gories 𝔻𝔸 r lax ( ℂ ⋄ )( f , g ) D Ar ( ℂ ( x, b )) ℂ ( x, a ) × ℂ ( y , b ) ℂ ( x, b ) × ℂ ( x, b ) , s,t g ∗ × f ∗ 𝔻𝔸 r oplax ( ℂ ⋄ )( f , g ) D Ar ( ℂ ( x, b )) ℂ ( x, a ) × ℂ ( y , b ) ℂ ( x, b ) × ℂ ( x, b ) , t,s g ∗ × f ∗ wher e DAr ( ℂ ( x, b )) denotes the ful l sub c ate gory of Ar ( ℂ ( x, b )) sp anne d by the de c o- r ate d 2-morphisms. □ Observ ation 3.3.6. F or a marked ( ∞ , 2) -category ( ℂ , I ) , Corollary 3.3.5 says that w e get a pullback 𝔻𝔸 r (( ℂ , I ) ♭ )( f , g ) ℂ ( x, a ) ℂ ( y , b ) ℂ ( x, b ) . g ∗ f ∗ 46 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI This applies in particular to 𝔸 r ( ℂ ) = 𝔻𝔸 r ( ℂ ♯♭ ) . Observ ation 3.3.7. Specializing Corollary 3.3.5 to iden tity morphisms, we get pullbac k squares 𝔻𝔸 r lax ( ℂ ⋄ )(id x , id y ) D Ar ( ℂ ( x, y )) ℂ ( x, y ) × ℂ ( x, y ) ℂ ( x, y ) × ℂ ( x, y ) , s,t = 𝔻𝔸 r oplax ( ℂ ⋄ )(id x , id y ) D Ar ( ℂ ( x, y )) ℂ ( x, y ) × ℂ ( x, y ) ℂ ( x, y ) × ℂ ( x, y ) , t,s = so that we ha v e equiv alences 𝔻𝔸 r (op)lax ( ℂ ⋄ )(id x , id y ) ≃ D Ar ( ℂ ( x, y )) , appropriately compatible with the ev aluation maps to ℂ ( x, y ) . Com bining the last tw o observ ations, w e see that the functor s ∗ 0 : ℂ → 𝔸 r ( ℂ ) , giv en by restriction along the degeneracy s 0 : [1] → [0] , is fully faithful. This has the following useful consequence: Lemma 3.3.8. Supp ose f : ℂ ′ → ℂ is essential ly surje ctive on obje cts. Then the functor f ∗ : 𝔽 un ( ℂ , 𝔻 ) → 𝔽 un ( ℂ ′ , 𝔻 ) is c onservative on 1-morphisms for any ( ∞ , 2) - c ate gory 𝔻 . Pr o of. A functor is conserv ative on morphisms if it is right orthogonal to s 0 : [1] → [0] , so we wan t to show that the commutativ e square Map([0] , 𝔽 un ( ℂ , 𝔻 )) Map([0] , 𝔽 un ( ℂ ′ , 𝔻 )) Map([1] , 𝔽 un ( ℂ , 𝔻 )) Map([1] 𝔽 un ( ℂ ′ , 𝔻 )) ( f ∗ ) ∗ s ∗ 0 s ∗ 0 ( f ∗ ) ∗ is a pullback. This is equiv alent to the square Map( ℂ , 𝔻 ) Map( ℂ ′ , 𝔻 ) Map( ℂ , 𝔸 r ( 𝔻 )) Map( ℂ ′ , 𝔸 r ( 𝔻 )) , f ∗ s ∗ 0 ( s ∗ 0 ) ∗ f ∗ whic h is a pullback since s ∗ 0 : 𝔻 → 𝔸 r ( 𝔻 ) is fully faithful, f is essentially surjective, and the essentially surjectiv e functors are precisely those that are left orthogonal to the fully faithful ones (e.g. by [ LMGR + 24 , Theorem 5.3.7]). □ 3.4. Characterizing partial ﬁbrations. Our goal in this subsection is to prov e the following useful characterization of partial ﬁbrations 2 : Theorem 3.4.1. A morphism of de c or ate d ( ∞ , 2) -c ate gories p : 𝔼 ⋄ → 𝔹 ⋄ is a p ar- tial ϵ -ﬁbr ation for ϵ = ( i, j ) if and only if the c ommutative squar e (3.1) 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) 𝔼 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) 𝔹 ev i 𝔻𝔸 r ϵ -lax ( p ) p ev i is a pul lb ack of ( ∞ , 2) -c ate gories. 2 A version of this result can also be extracted from Loubaton’s work on ( ∞ , ∞ ) -categories (see Remark 3.6.3 ). FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 47 W arning 3.4.2. If p : 𝔼 ⋄ → 𝔹 ⋄ is a partial ﬁbration, the square ( 3.1 ) is gener- ally not a pullback of de c or ate d ( ∞ , 2) -categories when 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) is equipp ed with its standard decoration from Deﬁnition 2.7.1 : F or ϵ = (0 , 1) , a morphism in 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) is a diagram • • • • in 𝔼 , where the v ertical morphisms and diagonal 2-morphism are decorated. In the standard decoration this is decorated if also the horizontal morphisms are deco- rated. On the other hand, for the pullback of the decorations in ( 3.1 ), a morphism is decorated whenever the top horizontal morphism is dec orated in 𝔼 ⋄ and the im- age of the bottom horizontal morphism is decorated in 𝔹 ⋄ . F or these decorations to agree w e would need the cartesian transp ort of a co cartesian 1-morphism to again b e co cartesian, which is usually not the case. How ev er, we will see b elo w in Prop o- sition 3.5.4 that we do get a pullbac k of decorated ( ∞ , 2) -categories if we mo dify the decorations on 𝔻𝔸 r ϵ -lax ( – ) appropriately . Remark 3.4.3. In Theorem 3.4.1 the assumption that we start with a decoration on 𝔼 can b e w eak ened slightly: since (co)cartesian 1-morphisms are automatically closed under comp osition, we do not strictly need the assumption that the decorated morphisms in 𝔼 are closed under comp osition. Similarly , it is enough to assume the decorated 2-morphisms are closed under whiskering. The starting p oin t for the pro of of Theorem 3.4.1 is the following description of (co)cartesian morphisms in terms of arrow ( ∞ , 2) -categories: Prop osition 3.4.4. Given a functor p : 𝔼 → 𝔹 of ( ∞ , 2) -c ate gories, the fol lowing ar e e quivalent for a morphism ¯ f : ¯ x → ¯ y in 𝔼 over f : x → y in 𝔹 : (i) ¯ f is p - i -c artesian. (ii) F or every morphism ¯ g : ¯ a → ¯ b in 𝔼 over g : a → b in 𝔹 , we have ▶ for i = 0 , the c ommutative squar e of ∞ -c ate gories 𝔸 r ( 𝔼 )( ¯ f , ¯ g ) 𝔼 ( ¯ x, ¯ a ) 𝔸 r ( 𝔹 )( f , g ) 𝔹 ( x, a ) is a pul lb ack. ▶ for i = 1 , the c ommutative squar e of ∞ -c ate gories 𝔸 r ( 𝔼 )( ¯ g , ¯ f ) 𝔼 (¯ a, ¯ x ) 𝔸 r ( 𝔹 )( g , f ) 𝔹 ( a, x ) is a pul lb ack. (iii) The pr evious c ondition holds when ¯ g = id ¯ a for al l ¯ a in 𝔼 . 48 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Pr o of. W e prov e the co cartesian case. Then the commutativ e square in (ii) ﬁts in a commutativ e cub e 𝔸 r ( 𝔼 )( ¯ f , ¯ g ) 𝔼 ( ¯ x, ¯ a ) 𝔸 r ( 𝔹 )( f , g ) 𝔹 ( x, a ) 𝔼 ( ¯ y , ¯ b ) 𝔼 ( ¯ x, ¯ b ) 𝔹 ( y , b ) 𝔹 ( x, b ) where the back and front faces are pullbac ks b y Observ ation 3.3.6 . If ¯ f is p - co cartesian then the b ottom square is also a pullback, hence so is the top square; th us (i) implies (ii). On the other hand, if ¯ g is an equiv alence, then the vertical maps in the right square are inv ertible, and hence so are the tw o other vertical maps, since they are pullbacks of these. The top square is therefore a pullback if and only if the b ottom square is one. Thus if the top square is a pullback for ¯ g = id ¯ a for all ¯ a , then ¯ f is co cartesian, so that (iii) implies (i). □ F rom this we immediately obtain the sp ecial case of Theorem 3.4.1 where only 1-morphisms are decorated: Corollary 3.4.5. Supp ose p : ( 𝔼 , I ) → ( 𝔹 , J ) is a functor of marke d ( ∞ , 2) -c ate gories. Then p ♭ is a p artial ϵ -ﬁbr ation for ϵ = ( i, j ) , i.e. 𝔼 has p - i -c artesian lifts of marke d 1-morphisms in 𝔹 , if and only if the c ommutative squar e (3.2) 𝔻𝔸 r (( 𝔼 , I ) ♭ ) 𝔼 𝔻𝔸 r (( 𝔹 , J ) ♭ ) 𝔹 ev i ev i is a pul lb ack, wher e 𝔻𝔸 r (( 𝔼 , I ) ♭ ) denotes the ful l sub c ate gory of 𝔸 r ( 𝔼 ) on the de c o- r ate d 1-morphisms. Pr o of. W e consider the case ϵ = (0 , 1) and ﬁrst supp ose that p is a partial (0 , 1) - ﬁbration. F rom Observ ation 3.3.6 and Prop osition 3.4.4 we then know that in ( 3.2 ) w e hav e pullbac ks on all mapping ∞ -categories. It th us only remains to show we ha ve a pullback of ∞ -group oids Map ⋄ ([1] , 𝔼 ) 𝔼 ≃ Map ⋄ ([1] , 𝔹 ) 𝔹 ≃ on cores, where Map ⋄ ([1] , – ) denotes the sub- ∞ -group oid of Map([1] , – ) on deco- rated morphisms. This follo ws from the uniqueness of co cartesian lifts as in Obser- v ation 2.4.3 , since Map ⋄ ([1] , 𝔼 ) is precisely the ∞ -group oid of co cartesian lifts of the morphisms in 𝔹 ⋄ (1) , and these all exist by assumption. Con versely , if the square is a pullbac k, then Prop osition 3.4.4 implies that the decorated morphisms in 𝔼 are all p -co cartesian, while the pullbac k on cores sho ws that there is a (unique) decorated lift of every decorated morphism in 𝔹 with an y FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 49 giv en source. Th us 𝔼 has all p -co cartesian lifts of the decorated morphisms in 𝔹 , and these are precisely its decorated morphisms, as required. □ Pr o of of The or em 3.4.1 . W e consider the case of partial (0 , 1) -ﬁbrations; the other 3 cases are prov ed in the same w ay , or follo w by reversing 1- and/or 2-morphisms. Let us ﬁrst assume that p is a partial (0 , 1) -ﬁbration. F or decorated 1-morphisms ¯ f : ¯ x → ¯ y and ¯ g : ¯ a → ¯ b in 𝔼 ov er f : x → y and g : a → b , w e then ha ve a comm utative cub e 𝔻𝔸 r oplax ( 𝔼 ⋄ )( ¯ f , ¯ g ) 𝔻𝔸 r oplax ( 𝔹 ⋄ )( f , g ) D Ar ( 𝔼 ( ¯ x, ¯ b )) D Ar ( 𝔹 ( x, b )) 𝔼 ( ¯ x, ¯ a ) × 𝔼 ( ¯ y , ¯ b ) 𝔹 ( x, a ) × 𝔹 ( y , b ) 𝔼 ( ¯ x, ¯ b ) × 2 𝔹 ( x, b ) × 2 where the left and right faces are pullbacks by Corollary 3.3.5 . Here we can factor the b ottom square as 𝔼 ( ¯ x, ¯ a ) × 𝔼 ( ¯ y , ¯ b ) 𝔼 ( ¯ x, ¯ a ) × 𝔹 ( y , b ) 𝔹 ( x, a ) × 𝔹 ( y , b ) 𝔼 ( ¯ x, ¯ b ) × 𝔼 ( ¯ x, ¯ b ) 𝔼 ( ¯ x, ¯ b ) × 𝔹 ( x, b ) 𝔹 ( x, b ) × 𝔹 ( x, b ) , where the left square is a pullback since ¯ f is a co cartesian 1-morphism. Using this w e can reorganize our cub e as 𝔻𝔸 r oplax ( 𝔼 ⋄ )( ¯ f , ¯ g ) 𝔻𝔸 r oplax ( 𝔹 ⋄ )( f , g ) D Ar ( 𝔼 ( ¯ x, ¯ b )) D Ar ( 𝔹 ( x, b )) 𝔼 ( ¯ x, ¯ a ) × 𝔹 ( y , b ) 𝔹 ( x, a ) × 𝔹 ( y , b ) 𝔼 ( ¯ x, ¯ b ) × 𝔹 ( x, b ) 𝔹 ( x, b ) × 2 , where we know that the left and righ t faces are pullbacks. Moreov er, the front face here is a pullbac k by Corollary 3.4.5 , since 𝔼 ( ¯ x, ¯ b ) → 𝔹 ( x, b ) has cartesian lifts of the decorated morphisms in 𝔹 ( x, b ) . It follows that the back face is also a pullback, whic h implies that the commutativ e square of mapping ∞ -categories 𝔻𝔸 r oplax ( 𝔼 ⋄ )( ¯ f , ¯ g ) 𝔻𝔸 r oplax ( 𝔹 ⋄ )( f , g ) 𝔼 ( ¯ x, ¯ a ) 𝔹 ( x, a ) is a pullback. T o see that ( 3.1 ) is a pullback, it then only remains to prov e that it giv es a pullback square on cores. This again follows from the uniqueness of co cartesian lifts (Observ ation 2.4.3 ), just as in the pro of of Corollary 3.4.5 . 50 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI W e no w prov e the conv erse: Since equiv alences are alwa ys decorated, for ob jects ¯ x and ¯ y in 𝔼 ov er x, y ∈ 𝔹 we get a pullback square 𝔻𝔸 r oplax ( 𝔼 ⋄ )(id ¯ x , id ¯ y ) 𝔻𝔸 r oplax ( 𝔹 ⋄ )(id x , id y ) 𝔼 ( ¯ x, ¯ y ) 𝔹 ( x, y ) . Using Observ ation 3.3.7 we can identify this with the square D Ar ( 𝔼 ( ¯ x, ¯ y )) DAr ( 𝔹 ( x, y )) 𝔼 ( ¯ x, ¯ y ) 𝔹 ( x, y ) . Since this is a pullback, Corollary 3.4.5 implies that 𝔼 ( ¯ x, ¯ y ) has cartesian lifts of the decorated 2-morphisms in 𝔹 ( x, y ) , and these are precisely the decorated 2- morphisms in 𝔼 ( x, y ) . It follows that a decorated 2-morphism in 𝔼 ⋄ is inv ertible precisely if its image in 𝔹 is inv ertible, which means that we hav e a pullback square 𝔻𝔸 r (( u m d 𝔼 ⋄ ) ♭ ) 𝔻𝔸 r oplax ( 𝔼 ⋄ ) 𝔻𝔸 r (( u m d 𝔹 ⋄ ) ♭ ) 𝔻𝔸 r oplax ( 𝔹 ⋄ ) . Applying Corollary 3.4.5 to the combination of this with the pullback square ( 3.1 ), w e can conclude that 𝔼 has co cartesian lifts of the decorated 1-morphisms in 𝔹 , and these are precisely the decorated 1-morphisms in 𝔼 , as required. □ 3.5. P artial ﬁbrations on functors. Supp ose p : E → B is a (co)cartesian ﬁbra- tion of ∞ -categories; then so is p ∗ : Fun ( K , E ) → Fun ( K , B ) for any ∞ -category K , and the p ∗ -(co)cartesian m orphisms are precisely the natural transformations that are comp onen t wise (co)cartesian. Our goal in this subsection is to prov e an ( ∞ , 2) - categorical v ersion of this statement and its generalization to partially (op)lax trans- formations. W e will do this in the framew ork of decorated ( ∞ , 2) -categories, and so w e will more precisely giv e conditions under whic h the functors 𝔻𝔽 un ( 𝕂 ⋄ , – ) I -(op)lax preserv e partial ﬁbrations. Com bining our discussion of decorated Gray tensors with the criterion of Theo- rem 3.4.1 , we immediately get the following useful case: Corollary 3.5.1. Supp ose p : 𝔼 ⋄ → 𝔹 ⋄ is a p artial ϵ -ﬁbr ation. Then for any ( ∞ , 2) - c ate gory 𝕂 , the induc e d functor of de c or ate d ( ∞ , 2) -c ate gories p ∗ : 𝔻𝔽 un ( 𝕂 ♭♭ , 𝔼 ⋄ ) ϵ -lax → 𝔻𝔽 un ( 𝕂 ♭♭ , 𝔹 ⋄ ) ϵ -lax is again a p artial ϵ -ﬁbr ation. Remark 3.5.2. More explicitly , restricting to the case ϵ = (0 , 1) for concreteness, Corollary 3.5.1 says that the functor p ∗ : 𝔽 un ( 𝕂 , 𝔼 ) lax → 𝔽 un ( 𝕂 , 𝔹 ) lax is a partial (0 , 1) -ﬁbration with resp ect to the decoration of the target where FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 51 ▶ the decorated 1-morphisms are the lax transformations whose comp onent at ev ery ob ject of 𝕂 is a decorated 1-morphism in 𝔹 and whose lax naturalit y squares contain decorated 2-morphisms, ▶ the decorated 2-morphisms are those whose comp onent at every ob ject of 𝕂 is a decorated 2-morphism in 𝔹 . Moreo ver, a 1-morphism in the source o v er such a decorated 1-morphism is co carte- sian precisely when its comp onent at every ob ject of 𝕂 is p -co cartesian and its lax naturalit y squares all contain p -cartesian 2-morphisms. A 2-morphism is similarly cartesian if its comp onen t at ev ery ob ject of 𝕂 is a p -cartesian 2-morphism. In particular, if p is a (0 , 1) -ﬁbration, then so is p ∗ , with these as its (co)cartesian 1- and 2-morphisms. Pr o of of Cor ol lary 3.5.1 . W e prov e the case of partial (0 , 1) -ﬁbrations. Applying Theorem 3.4.1 , we need to show that the comm utative square 𝔻𝔸 r oplax ( 𝔻𝔽 un ( 𝕂 ♭♭ , 𝔼 ⋄ ) lax ) 𝔻𝔸 r oplax ( 𝔻𝔽 un ( 𝕂 ♭♭ , 𝔹 ⋄ ) lax ) 𝔻𝔽 un ( 𝕂 ♭♭ , 𝔼 ) lax 𝔻𝔽 un ( 𝕂 ♭♭ , 𝔹 ) lax is a pullback of ( ∞ , 2) -categories. But using Corollary 2.7.8 we can rewrite this as the square 𝔽 un ( 𝕂 , 𝔻𝔸 r oplax ( 𝔼 ⋄ )) lax 𝔽 un ( 𝕂 , 𝔻𝔸 r oplax ( 𝔹 ⋄ )) lax 𝔽 un ( 𝕂 , 𝔼 ) lax 𝔽 un ( 𝕂 , 𝔹 ) lax , whic h is a pullback since 𝔽 un ( 𝕂 , – ) lax preserv es limits, b eing a right adjoint. □ W e can generalize this result using a decorated upgrade of Theorem 3.4.1 (which w e pro ve in the more general case of decorated partial ﬁbrations for later use). Notation 3.5.3. Supp ose 𝔹 ⋄ is a decorated ( ∞ , 2) -category. W e write 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb for the decoration of 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) where ▶ an (op)lax square is a decorated 1-morphism if and only if it commutes and its image under b oth ev 0 and ev 1 are decorated in 𝔹 ⋄ , ▶ a 2-morphism is decorated if and only if its image under ev 0 and ev 1 are b oth decorated in 𝔹 ⋄ . (Note that this diﬀers from the decoration considered in Deﬁnition 2.7.1 , whic h is adjoint to the decorated Gray tensor pro duct: the diﬀerence is that we ask for decorated 1-morphisms to b e c ommutative squares rather than contain a decorated 2-morphism.) If 𝔼 ⋄ → 𝔹 ⋄ is a decorated partial ﬁbration, and 𝔼 ♮ as usual denotes the (co)cartesian decoration of 𝔼 , we also slightly abusively write 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) ⋄ -ﬁb for the decoration of 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) induced by viewing it as a full sub- ( ∞ , 2) -category of 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) ⋄ -ﬁb . 52 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Prop osition 3.5.4. Supp ose p : 𝔼 ⋄ → 𝔹 ⋄ is a de c or ate d p artial ϵ -ﬁbr ation. Then the c ommutative squar e 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) ⋄ -ﬁb 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb 𝔼 ⋄ 𝔹 ⋄ is a pul lb ack of de c or ate d ( ∞ , 2) -c ate gories. Pr o of. Since the underlying functor 𝔼 → 𝔹 is a partial ϵ -ﬁbration, w e know that the underlying square of ( ∞ , 2) -categories is a pullback by Theorem 3.4.1 . It therefore suﬃces to show that the decorations match; we chec k this in the case ϵ = (0 , 1) . A 1-morphism in 𝔻𝔸 r oplax ( 𝔼 ♮ ) is then an oplax square • • • • ♮ ♮ ♮ in 𝔼 , whose v ertical morphisms and diagonal 2-morphism are decorated in 𝔼 ♮ , whic h means they are co cartesian and cartesian, resp ectiv ely . This corresp onds to a dec- orated 1-morphism in the pullback 𝔻𝔸 r oplax ( 𝔹 ⋄ ) ⋄ -ﬁb × 𝔹 ⋄ 𝔼 ⋄ if and only if ▶ the top horizontal morphism is decorated in 𝔼 ⋄ , ▶ and the image in 𝔻𝔸 r oplax ( 𝔹 ⋄ ) ⋄ -ﬁb is decorated, i.e. it commutes and its hori- zon tal morphisms are decorated. It follo ws that the 2-morphism in our oplax square in 𝔼 is cartesian o ver an equiv- alence, so it m ust b e inv ertible and w e get a commutativ e square; the b ottom horizon tal morphism is then also decorated since p was a decorated (0 , 1) -ﬁbration, as noted in Observ ation 3.2.2 . The case of 2-morphisms follows similarly from the t wo prop erties of decorated 2-morphisms in a decorated partial ﬁbrations stated in 3.2.2 . □ Using this, we can strengthen Corollary 3.5.1 as follows: Corollary 3.5.5. Supp ose p : 𝔼 ⋄ → 𝔹 ⋄ is a p artial ϵ -ﬁbr ation and 𝕂 ⋄ is a de c or ate d ( ∞ , 2) -c ate gory e quipp e d with a marking E such that al l de c or ate d 1-morphisms ar e marke d. Then the induc e d functor of de c or ate d ( ∞ , 2) -c ate gories p ∗ : 𝔻𝔽 un ( 𝕂 ⋄ , 𝔼 ⋄ ) E − ϵ -lax → 𝔻𝔽 un ( 𝕂 ⋄ , 𝔹 ⋄ ) E − ϵ -lax is again a p artial ϵ -ﬁbr ation. Pr o of. Applying 𝔻𝔽 un ( 𝕂 ⋄ , – ) E − ϵ -lax to the pullbac k square of decorated ( ∞ , 2) - categories from Prop osition 3.5.4 , we get the pullbac k square 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) ⋄ -ﬁb ) E − ϵ -lax 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb ) E − ϵ -lax 𝔻𝔽 un ( 𝕂 ⋄ , 𝔼 ⋄ ) E − ϵ -lax 𝔻𝔽 un ( 𝕂 ⋄ , 𝔹 ⋄ ) E − ϵ -lax . W e also know from Lemma 2.7.11 that we can identify 𝔻𝔸 r ϵ -lax ( 𝔻𝔽 un ( 𝕂 ⋄ , 𝔹 ⋄ ) E − ϵ -lax ) FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 53 with the full sub- ( ∞ , 2) -category of 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ )) E − ϵ -lax ) on functors that take morphisms in E to commuting squares (where 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) has the stan- dard decoration). On the other hand, 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb ) E − ϵ -lax is the full sub- ( ∞ , 2) -category on functors that take the decorated 1-morphisms to com- m uting squares. Since the decorated morphisms are all marked, this means that 𝔻𝔸 r ϵ -lax ( 𝔻𝔽 un ( 𝕂 ⋄ , 𝔹 ⋄ ) E − ϵ -lax ) is a full sub- ( ∞ , 2) -category of 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb ) E − ϵ -lax . Moreo ver, w e ha ve a pullback square 𝔻𝔸 r ϵ -lax ( 𝔻𝔽 un ( 𝕂 ⋄ , 𝔼 ⋄ ) E − ϵ -lax ) 𝔻𝔸 r ϵ -lax ( 𝔻𝔽 un ( 𝕂 ⋄ , 𝔹 ⋄ ) E − ϵ -lax ) 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) ⋄ -ﬁb ) E − ϵ -lax 𝔻𝔽 un ( 𝕂 ⋄ , 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb ) E − ϵ -lax , since the ob jects of 𝔻𝔸 r ϵ -lax ( 𝔼 ⋄ ) are squares that con tain a j -cartesian 2-morphism, whic h therefore commute if and only if their images in 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) also commute. Putting these squares together w e conclude that p ∗ is a partial ϵ -ﬁbration b y the criterion of Theorem 3.4.1 . □ In particular, we hav e the extreme case where all morphisms are marked: Corollary 3.5.6. Supp ose p : 𝔼 ⋄ → 𝔹 ⋄ is a p artial ϵ -ﬁbr ation and 𝕂 ⋄ is a de c or ate d ( ∞ , 2) -c ate gory. Then the induc e d functor of de c or ate d ( ∞ , 2) -c ate gories p ∗ : 𝔻𝔽 un ( 𝕂 ⋄ , 𝔼 ⋄ ) → 𝔻𝔽 un ( 𝕂 ⋄ , 𝔹 ⋄ ) is again a p artial ϵ -ﬁbr ation. □ 3.6. ϵ -equiv alences and ϵ -coﬁbrations. W e can interpret Theorem 3.4.1 as char- acterizing partial ϵ -ﬁbrations by a right orthogonality prop erty . In this section w e will sp ell this out and then study the corresp onding left orthogonal class, the ϵ - c oﬁbr ations , as well as the morphisms that are lo cal equiv alences with resp ect to partial ϵ -ﬁbrations ov er a ﬁxed base, which we call ϵ -e quivalenc es . Using Notation 2.7.4 , we hav e the following reinterpretation of Theorem 3.4.1 : Corollary 3.6.1. A morphism of de c or ate d ( ∞ , 2) -c ate gories 𝔼 ⋄ → 𝔹 ⋄ is a p artial ϵ -ﬁbr ation for ϵ = ( i, j ) if and only if it is right ortho gonal to { i } ⊗ d , ϵ C ♭♭ k → [1] ♯ ⊗ d , ϵ C ♭♭ k for k = 0 , 1 , 2 . □ Deﬁnition 3.6.2. A morphism in DCat ( ∞ , 2) is an ϵ -c oﬁbr ation for ϵ = ( i, j ) if it lies in the saturated class generated by { i } ⊗ d , ϵ C ♭♭ k → [1] ♯ ⊗ d , ϵ C ♭♭ k for k = 0 , 1 , 2 . Remark 3.6.3. Loubaton [ Lou24 ] deﬁne s (tw o cases of ) ﬁbrations of decorated ( ∞ , ∞ ) -categories as morphisms right orthogonal to generalizations of our ϵ -coﬁbrations. His Theorem 3.2.2.24 then c haracterizes these ﬁbrations in terms of the existence of (co)cartesian lifts of decorated i -morphisms for all i , and so gives by adjunction a version of our criterion from Theorem 3.4.1 for ( ∞ , ∞ ) -categories. 54 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Observ ation 3.6.4. W e saw in Observ ation 2.6.2 that DCat ( ∞ , 2) is a presentable ∞ -category. It therefore follows from [ Lur09a , Prop osition 5.5.5.7] that partial ϵ - ﬁbrations form the right class in a factorization system on DCat ( ∞ , 2) whose left class consists of the ϵ -coﬁbrations. In particular, partial ϵ -ﬁbrations are closed under base c hange, retracts, and limits in Ar ( DCat ( ∞ , 2) ) , and satisfy a cancellation prop ert y: if we hav e a commutativ e triangle of decorated ( ∞ , 2) -categories 𝔼 ⋄ ℙ ⋄ 𝔹 ⋄ f p q suc h that q is a partial ϵ -ﬁbration, then p is a partial ϵ -ﬁbration if and only if f is so. It follows, for example, that in this situation a morphism in 𝔼 is p -(co)cartesian ov er a decorated morphism in 𝔹 ⋄ if and only if it is f -(co)cartesian ov er a q -(co)cartesian morphism in ℙ . Observ ation 3.6.5. It also follows from Corollary 3.6.1 that the full sub category PFib ϵ / 𝔹 ⋄ ⊆ DCat ( ∞ , 2) / 𝔹 ⋄ consists precisely of the ob jects that are lo cal with resp ect to all morphisms of the form (3.3) { i } ⊗ d , ϵ C ♭♭ k [1] ♯ ⊗ d , ϵ C ♭♭ k 𝔹 ⋄ . Since there is a set of equiv alence classes of suc h morphisms and DCat ( ∞ , 2) is presen table, this implies that there exists a lo calization functor L ϵ 𝔹 ⋄ : DCat ( ∞ , 2) / 𝔹 ⋄ → PFib ϵ / 𝔹 ⋄ , left adjoint to the inclusion. Deﬁnition 3.6.6. W e say that a morphism in DCat ( ∞ , 2) / 𝔹 ⋄ is an ϵ -e quivalenc e o ver 𝔹 ⋄ if it is taken to an equiv alence by L ϵ 𝔹 ⋄ , or equiv alen tly if it is in the strongly saturated class generated by the morphisms ( 3.3 ). Observ ation 3.6.7. W e make some elemen tary observ ations on ϵ -coﬁbrations and ϵ -equiv alences: ▶ A morphism f : 𝔸 ⋄ → 𝔹 ⋄ is an ϵ -coﬁbration if and only if it is an ϵ -equiv alence when viewed as a map f → id 𝔹 ⋄ in DCat ( ∞ , 2) / 𝔹 ⋄ . ▶ F or any decorated functor p : 𝔸 ⋄ → 𝔹 ⋄ , the functor p ! : DCat ( ∞ , 2) / 𝔸 ⋄ → DCat ( ∞ , 2) / 𝔹 ⋄ giv en by comp osition with p preserves ϵ -equiv alences, since its right adjoint p ∗ preserv es partial ϵ -ﬁbrations. ▶ If p is itself a partial ϵ -ﬁbration, then p ! furthermore reﬂects ϵ -equiv alences: In this case p ! restricts to a functor PFib ϵ / 𝔸 ⋄ → PFib ϵ / 𝔹 ⋄ left adjoin t to pull- bac k, so that we get an equiv alence of left adjoin ts L ϵ 𝔹 ⋄ p ! ≃ p ! L ϵ 𝔸 ⋄ . Since equiv alences in PFib ϵ / 𝔹 ⋄ are detected in Cat ( ∞ , 2) , it follo ws that for a mor- phism f in DCat ( ∞ , 2) / 𝔸 ⋄ w e hav e that L ϵ 𝔸 ⋄ ( f ) is an equiv alence if and only if FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 55 p ! L ϵ 𝔸 ⋄ ( f ) ≃ L ϵ 𝔹 ⋄ ( p ! f ) is an equiv alence. Note that if p factors as 𝔸 ⋄ q − → 𝕏 ⋄ r − → 𝔹 ⋄ , then this implies that q ! also detects ϵ -equiv alences. ▶ As a sp ecial case, this means that if a morphism 𝔸 ⋄ 𝔹 ⋄ ℂ ⋄ f p q in DCat ( ∞ , 2) / ℂ ⋄ is an ϵ -equiv alence and q is a partial ϵ -ﬁbration, then f is an ϵ -coﬁbration. Prop osition 3.6.8. [1] ♯ → [0] and C ♭♯ 2 → [1] ♭ ar e ϵ -c oﬁbr ations for al l ϵ . Pr o of. Let ϵ = ( i, j ) . F or the ﬁrst map, we observe that the comp osition { i } → [1] ♯ → [0] , is the identit y , from which it follows by cancellation that [1] ♯ → [0] must b e an ϵ -coﬁbration. Similarly , from the comp osition { i } × [1] ♭ → [1] ♯ ⊗ d , ϵ [1] ♭ → [1] ♭ w e see that [1] ♯ ⊗ d , ϵ [1] ♭ → [1] ♭ is an ϵ -coﬁbration. Since we also hav e a pushout [1] ♯ ⨿ [1] ♯ [0] ⨿ [0] [1] ♯ ⊗ d , ϵ [1] ♭ C ♭♯ 2 , it again follows by cancellation that C ♭♯ 2 → [1] ♭ is an ϵ -coﬁbration. □ Since ϵ -coﬁbrations are closed under cobase change and colimits, w e get: Corollary 3.6.9. F or any de c or ate d ( ∞ , 2) -c ate gory ℂ ⋄ , the morphism ℂ ⋄ → τ d ( ℂ ⋄ ) ♭♭ is an ϵ -e quivalenc e for any ϵ , as is any pushout ther e of, i.e. any morphism obtaine d by inverting a c ol le ction of de c or ate d 1- and 2-morphisms. □ Observ ation 3.6.10. The pro jection [1] ♯ ⊗ d 𝔻 ⋄ → 𝔻 ⋄ is a localization at decorated 1- and 2-morphisms, and so is an ϵ -coﬁbration by Corollary 3.6.9 . Hence the triangle [1] ♯ ⊗ d 𝔻 ⋄ 𝔻 ⋄ 𝔹 ⋄ pro j p ◦ pro j p is an ϵ -equiv alence ov er 𝔹 ⋄ . By the 2-out-of-3 prop erty , the tw o inclusions 𝔻 ⋄ → [1] ♯ ⊗ d 𝔻 ⋄ are also ϵ -equiv alences, and moreo ver b ecome equiv alent after applying L ϵ 𝔹 ⋄ , since they b oth give inv erses of the equiv alence L ϵ 𝔹 ⋄ ( pro j ) . Hence an y lax transformation α : [1] ♯ ⊗ d 𝔻 ⋄ → ℚ ⋄ o ver 𝔹 ⋄ from α 0 to α 1 giv es a natural equiv alence b etw een L ϵ 𝔹 ⋄ ( α 0 ) and L ϵ 𝔹 ⋄ ( α 1 ) . W e can therefore use [1] ♯ ⊗ d ( – ) to deﬁne “homotopies” and “homotopy equiv alences” that are in particular alwa ys ϵ -equiv alences. The following sp ecial case of this will b e useful later: 56 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Lemma 3.6.11. Supp ose F : ℂ ⋄ → 𝔻 ⋄ is a de c or ate d functor over 𝔹 ⋄ such that ther e exists G : 𝔻 ⋄ → ℂ ⋄ over 𝔹 ⋄ such that GF ≃ id ℂ ⋄ as wel l as a c ommutative triangle [1] ♯ ⊗ d 𝔻 ⋄ 𝔻 ⋄ 𝔹 ⋄ ρ wher e ρ is a lax natur al tr ansformation b etwe en id 𝔻 ⋄ and F G . Then for any de c- or ate d functor f : 𝔸 ⋄ → 𝔹 ⋄ , the pul lb ack f ∗ F : f ∗ ℂ ⋄ → f ∗ 𝔻 ⋄ is an ϵ -e quivalenc e over 𝔸 ⋄ . Pr o of. Applying Observ ation 3.6.10 to ρ , we see that L ϵ 𝔹 ⋄ ( G ) b ecomes an inv erse of L ϵ 𝔹 ⋄ ( F ) , so that this is indeed an ϵ -equiv alence. Pulling back ρ along f , w e moreo ver get an analogous natural transformation for f ∗ F as the comp osite [1] ♯ ⊗ d f ∗ 𝔻 ⋄ → f ∗ ([1] ♯ ⊗ d 𝔻 ⋄ ) f ∗ ρ − − → f ∗ 𝔻 ⋄ , so the same argument shows that f ∗ F is also a ϵ -equiv alence. □ As a consequence of Corollary 3.5.5 we also get the following class of ϵ -coﬁbrations: Corollary 3.6.12. Supp ose f : 𝔸 ⋄ → 𝔹 ⋄ is an ϵ -c oﬁbr ation and 𝕂 ⋄ is a de c or ate d ( ∞ , 2) -c ate gory e quipp e d with a marking E such that al l de c or ate d 1-morphisms ar e marke d. Then f ⊗ d , ϵ ♭,E 𝕂 ⋄ is again an ϵ -c oﬁbr ation. In p articular, f × 𝕂 ⋄ is an ϵ -c oﬁbr ation for any 𝕂 ⋄ . □ T o apply certain results prov ed in the setting of scaled simplicial sets in our con text, it will b e useful to ﬁnd an alternative set of generating ϵ -coﬁbrations in terms of orien tals (see Deﬁnition 2.1.2 ), which are easier to work with in that model. Deﬁnition 3.6.13. Let 𝕆 n denote the 2-truncated n -dimensional oriental . W e deﬁne: ▶ A decorated ( ∞ , 2) -category 𝕆 n, ⋄ b y decorating the 1-morphism ( n − 1) → n and every 2-morphism determined b y a subset inclusion of the form { i, n } ⊂ { i, n − 1 , n } . ▶ A decorated ( ∞ , 2) -category Λ n i 𝕆 ⋄ for 0 ≤ i ≤ n whose underlying ( ∞ , 2) - category is given by the colimit colim I ⊂ [ n ] 𝕆 I where I ranges o v er the collection of non-empty subsets with the property that [ n ] \ { i } ⊂ I and where a k - morphism is decorated if and only if its image under the canonical map Λ n i 𝕆 → 𝕆 n is. ▶ A decorated ( ∞ , 2) -category 𝕆 n, † + , whose underlying ( ∞ , 2) -category 𝕆 n + is obtained by inv erting every 2-morphism in 𝕆 n of the form { 0 , j } → { 0 , 1 , j } , and where we decorate the 1-morphism 0 → 1 . ▶ A decorated 2-category Λ n 0 𝕆 † + whose underlying 2-category is obtained from Λ n 0 𝕆 b y collapsing the same 2-morphisms as ab ov e and where the decorations are induced by the functor Λ n 0 𝕆 + → 𝕆 n + . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 57 Deﬁnition 3.6.14. Let [1] ♭, ⊗ k denote the k -fold decorated Gray tensor pro duct of [1] ♭ . W e deﬁne decorated 2-categories ⊏ ⋄ k for k = 1 , 2 , 3 as follows: ▶ F or k = 1 we denote ⊏ ⋄ 1 = [0] . ▶ F or k = 2 w e set ⊏ ⋄ 2 = [1] ♭ ⨿ ∂ [1] ([1] ♯ ⨿ [1] ♯ ) where the map ∂ [1] → [1] ♯ ⨿ [1] ♯ is giv en b y the disjoin t union of the maps selecting the initial vertex. ▶ F or k = 3 , we denote ∂ ([1] ♭, ⊗ 2 ) = [2] ♭ ⨿ 0 < 2 [2] ♭ and deﬁne ⊏ ⋄ 3 = [1] ♭, ⊗ 2 ⨿ ∂ ([1] ♭, ⊗ 2 ) [1] ♯ ⊗ ∂ ([1] ♭, ⊗ 2 ) where the attac hing map ∂ ([1] ♭, ⊗ 2 ) → [1] ♯ ⊗ d , ϵ ∂ ([1] ♭, ⊗ 2 ) is induced b y the map { 0 } → [1] ♯ . W e note that w e hav e natural decorated functors ⊏ ⋄ k → [1] ♯ ⊗ d , ϵ [1] ♭, ⊗ k − 1 for k = 2 , 3 . Remark 3.6.15. In several of the arguments below we will need to construct maps b et w een orien tals and iterated Gra y pro ducts of [1] . F or this purp ose, it is useful to recall that such morphisms may b e describ ed, in the mo del of scaled simplicial sets, as maps of p osets compatible with the relev ant decorations, or scalings. Concretely , the n th orien tal is mo deled b y (the nerv e of ) the poset [ n ] equipp ed with the minimal scaling, while [1] ♭, ⊗ k is given by the k -fold cartesian pro duct of [1] , endo w ed with its Gray tensor pro duct scaling (cf. [ GHL21 ]). Lemma 3.6.16. The maps ⊏ ⋄ k → [1] ♯ ⊗ [1] ♭, ⊗ k − 1 ar e (0 , 1) -c oﬁbr ations for k = 1 , 2 , 3 . Mor e over, their satur ation is the class of (0 , 1) -c oﬁbr ations. Pr o of. Let us observe that the saturated class of the morphisms { 0 } × [1] ♭, ⊗ k − 1 − → [1] ♯ ⊗ d , ϵ [1] ♭, ⊗ k − 1 , k = 1 , 2 , 3 is precisely given by the (0 , 1) -coﬁbrations. Moreov er, for each k we hav e a factor- ization { 0 } × [1] ♭, ⊗ k − 1 → ⊏ ⋄ k → [1] ♯ ⊗ d , ϵ [1] ♭, ⊗ k − 1 . Here it is easy to chec k that the ﬁrst map is in the saturated class of the morphisms ⊏ ⋄ ℓ → [1] ♯ ⊗ d , ϵ [1] ♭, ⊗ ℓ − 1 for  < k , so we see that this is a (0 , 1) -coﬁbration b y working inductiv ely on k . It then follows from cancellation that the second map m ust also b e a (0 , 1) -coﬁbration. This argument also shows that the maps { 0 } × [1] ♭, ⊗ k − 1 − → [1] ♯ ⊗ d , ϵ [1] ♭, ⊗ k − 1 lie in the saturated class generated by these maps, whic h prov es the conv erse. □ Lemma 3.6.17. L et 0 < i < n , then the maps Λ n i 𝕆 ⋄ → 𝕆 n, ⋄ ar e (0 , 1) -c oﬁbr ations. Pr o of. Let [1] × n denote the n -fold cartesian pro duct. There exists a functor r ′ n : [1] × n → [ n ] giv en b y v = { v i } n i =1 7→ n − ( α v − 1) where α v = min { i | v i = 1 } where w e mak e the conv en tion that α v = 0 if v i = 0 for i = 1 , 2 , . . . , n . This map admits a section i ′ n : [ n ] → [1] × n whic h sends j to to the elemen t { v i } i =1 with v i = 0 for i ≤ n − j and v i = 1 otherwise. These maps of partially ordered sets induce func- tors (see Remark 3.6.15 ) of decorated 2-categories i n : 𝕆 n, ⋄ → [1] ♯ ⊗ d , ϵ [1] ⊗ d , ϵ n − 1 ,♭ , r n : [1] ♯ ⊗ d , ϵ [1] ⊗ n − 1 ,♭ → 𝕆 n, ⋄ suc h that r n ◦ i n = id . 58 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI W e conclude that we ha ve a retract diagram 𝕆 n − 1 ,♭ { 0 } ⊗ d , ϵ [1] ⊗ n − 1 ,♭ 𝕆 n − 1 ,♭ 𝕆 n, ⋄ [1] ♯ ⊗ d , ϵ [1] ⊗ n − 1 ,♭ 𝕆 n, ⋄ . By the previous discussion w e know that the maps 𝕆 n − 1 ,♭ → 𝕆 n, ⋄ are (0 , 1) - coﬁbrations. W e conside r the factorization 𝕆 n − 1 ,♭ → Λ n i 𝕆 ⋄ → 𝕆 n, ⋄ and observe that the ﬁrst map is a (0 , 1) -coﬁbration by an easy inductiv e argument on n . The result follows by cancellation. □ Lemma 3.6.18. The morphisms Λ n 0 𝕆 † → 𝕆 n, † + for n = 1 , 2 , 3 ar e (0 , 1) -c oﬁbr ations. Pr o of. It suﬃces to show that there exists a retract diagram, Λ n 0 𝕆 † + ⊏ ⋄ k Λ n 0 𝕆 † + 𝕆 n, † + [1] ♯ ⊗ [1] ♭, ⊗ n − 1 𝕆 n, † + ι n r k for n = 1 , 2 , 3 . W e only deal with the case n = 3 as the remaining cases are similar and easier. The map ι 3 is induced by the functor of p osets i ′ 3 : [3] → [1] × [1] × [1] (see Remark 3.6.15 ) given by (0 , 0 , 0) → (1 , 0 , 0) → (1 , 0 , 1) → (1 , 1 , 1) . W e consider a map T : [1] × [1] × [1] → [1] × [1] × [1] depicted graphically as, (1 , 0 , 0) (1 , 1 , 1) (0 , 0 , 0) (1 , 0 , 0) (1 , 1 , 1) (1 , 1 , 1) (1 , 0 , 1) (1 , 0 , 1) and deﬁne r ′ 3 as its factorization through [3] . The map r ′ 3 induces the desired functor r 3 in our statement. □ Prop osition 3.6.19. The fol lowing c ol le ction of maps gener ates the (0 , 1) -c oﬁbr ations as a satur ate d class: (i) Λ n i 𝕆 ⋄ → 𝕆 n, ⋄ for n = 2 , 3 and 0 < i < n . (ii) Λ n 0 𝕆 † → 𝕆 n, † + for n = 1 , 2 , 3 . Pr o of. By Lemma 3.6.16 it will b e enough to show that the maps in the statemen t are (0 , 1) -coﬁbrations and that the morphisms (3.4) ⊏ ⋄ k → [1] ♯ ⊗ d , ϵ [1] ⊗ k − 1 , k = 1 , 2 , 3 lie in their saturated hull. The ﬁrst claim comes from Lemma 3.6.17 and Lemma 3.6.18 . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 59 No w we address the second claim. The case k = 1 is trivial. F or the case k = 2 , w e can pro duce a ﬁltration which we diagrammatically depict b elow, • •  → • • • •  → • • • •  → • • • • where the ﬁrst step is obtained by taking a pushout along a morphism of type (ii), the second along a morphism of type (i) and the ﬁnal step is obtained by taking a pushout along a morphism of type (ii). (Note the use of barred arrows to denote decorated 1-morphisms.) F or the case k = 3 , we consider the same ﬁltration as ab ov e but after taking the decorated Gray tensor pro duct with [1] ♭ on the right. This reduces our claim to sho wing that the maps Λ 2 1 𝕆 ⋄ ⊗ d , ϵ [1] ♭ → 𝕆 2 , ⋄ ⊗ d , ϵ [1] ♭ , Λ 2 0 𝕆 † + ⊗ d , ϵ [1] ♭ → 𝕆 2 , † + ⊗ d , ϵ [1] ♭ . can b e expressed as an iterated pushouts of morphisms of type (i) and (ii). The corresp onding ﬁltrations can be directly adapted from [ GHL21 , Prop osition 2.16] after recalling that in the mo del structure of scaled simplicial sets the minimally scaled n -simplex corresp onds to the n -th oriental. □ 4. Free fibra tions and pushfor w ards f or ( ∞ , 2) -ca tegories When setting up the theory of ∞ -categories, it turns out to be surprisingly useful to know that the forgetful functor from (co)cartesian ﬁbrations on a ﬁxed base B to ∞ -categories ov er B has a left adjoin t, the fr e e (co)cartesian ﬁbration functor, which has a concrete description as a pullback of the arrow ∞ -category of B . Our ﬁrst main goal in this section is to prov e the ( ∞ , 2) -categorical version of this result, in the general context of partial ﬁbrations. W e start b y showing that the ev aluation functors from 𝔸 r (op)lax ( 𝔹 ) to 𝔹 are ﬁbrations in § 4.1 , and then use these to construct free partial ﬁbrations in § 4.2 . In § 4.3 we extend this result sligh tly to describ e free de c or ate d ﬁbrations, whic h we use in § 4.4 to describ e the left adjoin t to the forgetful functor from ﬁbrations ov er 𝔹 to decorated ( ∞ , 2) -categories ov er 𝔹 ♯♯ . W e then turn to the second main goal of this section, which is to de scribe right adjoin ts to pullbac k. In § 4.5 w e show that pullbac k along a decorated ﬁbration 𝔸 ⋄ → 𝔻 ♯♯ on slices of decorated ( ∞ , 2) -categories has a righ t adjoin t, and this preserv es ﬁbrations of the opp osite v ariance. W e then use this in § 4.6 to identify the righ t adjoin t to pullbac k on partial ﬁbrations along an arbitrary such functor; this includes the case of c ofr e e ﬁbrations. Finally , in § 4.7 w e apply these results to iden tify the ﬁbration for the lo calization of a functor to decorated ( ∞ , 2) -categories. 4.1. Fibrations from (op)lax arrows. If C is an ∞ -category, then ev i : Ar ( C ) → C is a cocartesian ﬁbration for i = 1 and a cartesian ﬁbration for i = 0 , with the (co)cartesian morphisms precisely those that go to equiv alences un der the opp osite ev aluation. Our goal in this section is to generalize this statemen t to decorated ( ∞ , 2) -categories. 60 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Notation 4.1.1. F or ϵ = ( i, j ) and ℂ ⋄ a decorated ( ∞ , 2) -category, let 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ♭ [ ϵ ] denote the decoration of 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) where ▶ an (op)lax square is a decorated 1-morphism if and only if it commutes, its image under ev i is an equiv alence, and its image under ev 1 − i is decorated; ▶ a 2-morphism is decorated if and only if its image under ev i is an equiv alence and its image under ev 1 − i is decorated. Theorem 4.1.2. F or any de c or ate d ( ∞ , 2) -c ate gory ℂ ⋄ and ϵ = ( i, j ) , the functor ev 1 − i : 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ♭ [ ϵ ] → ℂ ⋄ is a p artial ϵ -ﬁbr ation. Remark 4.1.3. The fully decorated case of this theorem has already b een prov ed b y Gagna, Harpaz, and Lanari as [ GHL24 , Theorem 3.0.7]. Remark 4.1.4. Less compactly , this means that ▶ ev 1 : 𝔻𝔸 r oplax ( ℂ ⋄ ) → ℂ is a partial (0 , 1) -ﬁbration, ▶ ev 0 : 𝔻𝔸 r oplax ( ℂ ⋄ ) → ℂ is a partial (1 , 0) -ﬁbration, ▶ ev 1 : 𝔻𝔸 r lax ( ℂ ⋄ ) → ℂ is a partial (0 , 0) -ﬁbration, ▶ ev 0 : 𝔻𝔸 r lax ( ℂ ⋄ ) → ℂ is a partial (1 , 1) -ﬁbration, all with resp ect to the given decoration ℂ ⋄ on ℂ . W e also hav e that ▶ an (op)lax square is an i -cartesian morphism if and only if it commutes and its image under ev i is an equiv alence, ▶ a 2-morphism is j -cartesian if and only if its image under ev i is an equiv alence. W e will prov e this using the criterion of Theorem 3.4.1 , for which we will need the following result on lo calizations of ( ∞ , 2) -categories: Prop osition 4.1.5. Supp ose F : ℂ → ℂ ′ is a lo c alization at c ertain 1- and 2- morphisms. Then F ∗ : 𝔽 un ( ℂ ′ , 𝔻 ) (op)lax → 𝔽 un ( ℂ , 𝔻 ) (op)lax is a lo c al ly ful l inclu- sion for any ( ∞ , 2) -c ate gory 𝔻 . F or this, we in turn need some preliminary results on conserv ativ e functors: Observ ation 4.1.6. Giv en a functor of ( ∞ , 2) -categories F : 𝔸 → 𝔹 , consider the comm utative square 𝔸 ≤ i 𝔸 ≤ j 𝔹 ≤ i 𝔹 ≤ j F ≤ i F ≤ j for 0 ≤ i < j ≤ 2 (with ( – ) ≤ 2 = id ). W e hav e: (i) F is conserv ative on 2-morphisms if and only if the square is a pullback for i = 1 , j = 2 . (ii) F is conserv ative on 1-morphisms if and only if the square is a pullback for i = 0 , j = 1 . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 61 (iii) F is conserv ative on b oth 1- and 2-morphisms if and only if the square is a pullbac k for i = 0 , 1 and j = 2 . Prop osition 4.1.7. (i) The c ommutative squar e ∂ ([1] ⊗ [1]) ⊗ ([1] ⊗ [1]) [1] ⊗ 4 ∂ ([1] ⊗ [1]) ⊗ [1] ([1] ⊗ [1]) ⊗ [1] is a pushout. (ii) F or any ( ∞ , 2) -c ate gory ℂ , the functor 𝔽 un ([1] ⊗ [1] , ℂ ) (op)lax → 𝔽 un ( ∂ ([1] ⊗ [1]) , ℂ ) (op)lax is c onservative on 2-morphisms. (iii) Supp ose F : 𝔸 → 𝔹 is an essential ly surje ctive functor of ( ∞ , 2) -c ate gories. Then 𝔽 un ( 𝔹 , ℂ ) (op)lax → 𝔽 un ( 𝔸 , ℂ ) (op)lax is c onservative on 2-morphisms for al l ℂ . Pr o of. A functor is conserv ativ e on 2-morphisms if and only if it is right orthogonal to C 2 → [1] . This is equiv alen t to b eing right orthogonal to [1] ⊗ [1] → [1] × [1] , as each is a cobase change of the other. F rom [ AGH25 , 2.4.4] w e know that 𝔸 r oplax ( ℂ ) → ℂ × ℂ is conserv ativ e on 2-morphisms for an y ℂ , which is then equiv- alen t to the commutativ e square (4.1) [1] ⊗ [1] × ∂ [1] [1] ⊗ 3 [1] × [1] × ∂ [1] ([1] × [1]) ⊗ [1] b eing a pushout. T o prov e that the square in (i) is a pushout, w e ﬁrst consider ([1] ⊗ [1]) ⨿ 4 [1] ⊗ 2 ⊗ ∂ ([1] ⊗ 2) [1] ⊗ 4 ([1] × [1]) ⨿ 4 [1] × 2 ⊗ ∂ ([1] ⊗ 2 ) [1] × 2 ⊗ [1] ⊗ 2 , where the left square is obtained by gluing 4 copies of ( 4.1 ) and so is a pushout. T o see that the right-hand square is a pushout it therefore suﬃces to show that the outer comp osite square is a pushout. Next, we factor this square instead as ([1] ⊗ [1]) ⨿ 4 [1] ⊗ 3 × ∂ [1] [1] ⊗ 4 ([1] × [1]) ⨿ 4 [1] × 2 ⊗ [1] × ∂ [1] [1] × 2 ⊗ [1] ⊗ 2 . Here the left square is a copro duct of tw o copies of ( 4.1 ) and the right square is ( 4.1 ) tensored with [1] , so that b oth are pushouts. The orthogonality condition in (ii) is now immediate from the square in (i) b e- ing a pushout. T o prov e (iii), ﬁrst recall (e.g. from [ LMGR + 24 , Theorem 5.3.7]) 62 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI that essentially surjectiv e and fully faithful functors form a factorization system on Cat ( ∞ , 2) , and that a functor is fully faithful if and only if it is right orthogonal to ∂ C i → C i for i = 1 , 2 by [ AGH25 , Prop osition 2.5.10]. Hence the essentially surjectiv e functors are the smallest saturated class containing these tw o maps. On the other hand, the class of maps for whic h the desired conclusion holds is clearly saturated, so it suﬃces to pro v e it in these t wo cases. The map ∂ C 2 → C 2 generates the same saturated class as ∂ [1] ⊗ 2 → [1] ⊗ 2 b y [ AGH25 , 2.5.13], so this case follows from part (ii), while the case i = 1 is part of [ AGH25 , 2.4.4]. □ Pr o of of Pr op osition 4.1.5 . W e prov e the lax case; the oplax case follo ws similarly , or b y using equiv alences of the form 𝔽 un ( 𝔸 , 𝔹 ) oplax ≃ 𝔽 un ( 𝔸 op , 𝔹 op ) lax , op . By deﬁnition the functor F is obtained by inv erting certain 1- and 2-morphisms in ℂ . Since the class of functors for whic h the desired conclusion is true is clearly closed under cobase change, it suﬃces to show that it holds for the functors  i : ℂ → τ ≤ i ℂ for i = 0 , 1 given by lo calizing ℂ to an ( ∞ , i ) -category . The condition that  ∗ i : 𝔽 un ( τ ≤ i ℂ , 𝔻 ) lax → 𝔽 un ( ℂ , 𝔻 ) lax is right orthogonal to some functor ϕ : 𝔸 → 𝔹 is equiv alen t to the commutativ e square Map( ℂ , 𝔽 un ( 𝔹 , 𝔻 ) oplax , ≤ i ) Map( ℂ , 𝔽 un ( 𝔹 , 𝔻 ) oplax ) Map( ℂ , 𝔽 un ( 𝔸 , 𝔻 ) oplax , ≤ i ) Map( ℂ , 𝔽 un ( 𝔸 , 𝔻 ) oplax ) b eing a pullback, which is true for all ℂ if and only if 𝔽 un ( 𝔹 , 𝔻 ) oplax , ≤ i 𝔽 un ( 𝔹 , 𝔻 ) oplax 𝔽 un ( 𝔸 , 𝔻 ) oplax , ≤ i 𝔽 un ( 𝔸 , 𝔻 ) oplax is a pullback of ( ∞ , 2) -categories. By Observ ation 4.1.6 , this holds for b oth i = 0 , 1 if and only if ϕ ∗ : 𝔽 un ( 𝔹 , 𝔻 ) oplax → 𝔽 un ( 𝔸 , 𝔻 ) oplax is conserv ativ e on 1- and 2- morphisms. By [ AGH25 , 2.5.8], a functor is a lo cally full inclu sion if and only if it is right orthogonal to ∂ [1] → [0] and ∂ C 2 → C 2 . Applying the preceding discussion to these maps, we see that for the statemen t we wan t it is enough to prov e that the functors 𝔻 → 𝔻 × 2 , 𝔽 un ( C 2 , 𝔻 ) oplax → 𝔽 un ( ∂ C 2 , 𝔻 ) oplax are conserv ativ e on 1- and 2-morphisms. F or the ﬁrst map this holds since the inclusions 𝔻 ≤ i → 𝔻 are monomorphisms, so that we hav e pullbacks 𝔻 ≤ i 𝔻 𝔻 ≤ i, × 2 𝔻 × 2 . F or the second we saw in Observ ation 3.3.3 that it is conserv ative on 1-morphisms, and we just prov ed in Prop osition 4.1.7 that it is so on 2-morphisms. □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 63 Observ ation 4.1.8. The proof of Proposition 4.1.5 sho ws that for an y ( ∞ , 2) - category ℂ , the functor F ∗ : 𝔽 un ( 𝔹 , ℂ ) (op)lax → 𝔽 un ( 𝔸 , ℂ ) (op)lax is conserv ativ e on 1- and 2-morphisms pro vided F is in the saturated class generated b y ∂ C 2 → C 2 and ∂ [1] → [0] . By [ LMGR + 24 , Theorem 5.3.7], the saturated class generated by the ﬁrst map consists of those functors F that are surjectiv e on ob jects with the further prop ert y that 𝔸 ( a, a ′ ) → 𝔹 ( F a, F a ′ ) is surjective on ob jects for all a, a ′ ∈ 𝔸 , so all such maps hav e this conserv ativit y prop erty . It w ould b e interesting to identify also the larger class obtained by adding our second generator, i.e. the class of maps that are left orthogonal to lo cally full inclusions rather than all locally fully faithful functors. Prop osition 4.1.9. The c ommutative squar e 𝔻𝔸 r ϵ -lax ( 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ♭ [ ϵ ] ) 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ℂ ev 1 ev 0 is a pul lb ack of ( ∞ , 2) -c ate gories, giving an e quivalenc e 𝔻𝔸 r ϵ -lax ( 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ♭ [ ϵ ] ) ≃ 𝔻𝔽 un ([2] ♯ , ℂ ⋄ ) ϵ -lax of ( ∞ , 2) -c ate gories. Pr o of. W e prov e the case ϵ = (0 , 1) . Since 𝔻𝔽 un ( – , ℂ ⋄ ) oplax preserv es limits, we can iden tify the pullback in the square as 𝔻𝔽 un ([2] ♯ , ℂ ⋄ ) oplax . On the other hand, from Observ ation 2.3.9 we know that 𝔻𝔸 r oplax ( 𝔻𝔸 r oplax ( ℂ ⋄ ) ♭ [(0 , 1)] ) is a lo cally full sub category of 𝔸 r oplax ( 𝔸 r oplax ( ℂ )) ≃ 𝔽 un ([1] ⊗ [1] , ℂ ) oplax , whose ▶ ob jects are oplax squares that commute and whose top edge is inv ertible, which w e can identify with functors [2] → ℂ , ▶ morphisms can similarly b e identiﬁed with functors [2] ⊗ [1] → ℂ . It therefore suﬃces to show that the functor p ∗ : 𝔽 un ([2] , ℂ ) oplax → 𝔽 un ([1] ⊗ [1] , ℂ ) oplax , given by comp osition with the functor p : [1] ⊗ [1] → [2] that inv erts the 2-morphism and the top edge, is a lo cally full subcategory inclusion, whic h follo ws from Prop osition 4.1.5 . □ Pr o of of The or em 4.1.2 . Com bine Theorem 3.4.1 with Prop osition 4.1.9 . □ W e also hav e the following decorated v ariant of the theorem: Corollary 4.1.10. F or any de c or ate d ( ∞ , 2) -c ate gory ℂ ⋄ and ϵ = ( i, j ) , the functor ev 1 − i : 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ⋄ -ﬁb → ℂ ⋄ is a de c or ate d p artial ϵ -ﬁbr ation. Pr o of. Unpac king the deﬁnitions, we see that 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ⋄ -ﬁb (2) ≃ 𝔻𝔸 r ϵ -lax ( ℂ ⋄ (2) ) , 64 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI where we think of ℂ ⋄ (2) as a decorated ( ∞ , 2) -category with the decorations inherited from ℂ ⋄ , and 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ⋄ -ﬁb , ≤ 1 (1) ≃ Ar ( ℂ ⋄ , ≤ 1 (1) ) . Th us the restricted functor ev 1 − i : 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ⋄ -ﬁb (2) → ℂ ⋄ (2) is also a partial ϵ -ﬁbration b y Theorem 4.1.2 , with its i -cartesian morphisms and j -cartesian 2-morphisms agreeing with those in 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) , while ev 1 − i : 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) ⋄ -ﬁb , ≤ 1 (1) → ℂ ⋄ , ≤ 1 (1) is an i -ﬁbration of ∞ -categories whose i -cartesian morphisms agree with those in 𝔻𝔸 r ϵ -lax ( ℂ ⋄ ) . □ 4.2. F ree partial ﬁbrations. F or a functor of ∞ -categories f : C → B , w e can iden tify the free co cartesian ﬁbration on f as the functor C × B Ar ( B ) → B given b y ev aluation at 1 , where the pullback is formed using f and ev aluation at 0 . This w as ﬁrst prov ed in [ GHN17 ], with impro ved versions of the proof later giv en in [ AMGR17 , Sha21 ]. In this section w e will prov e the analogue of this statement for ( ∞ , 2) -categories, and more generally describ e free p artial ﬁbrations as follows: Theorem 4.2.1. L et 𝔹 ⋄ b e a de c or ate d ( ∞ , 2) -c ate gory. The for getful functor U ϵ 𝔹 ⋄ : PFib ϵ / 𝔹 ⋄ → Cat ( ∞ , 2) / 𝔹 has a left adjoint 𝔽 ree ϵ 𝔹 ⋄ , given for p : 𝔼 → 𝔹 by 𝔽 ree ϵ 𝔹 ⋄ ( p ) := 𝔼 ♭♭ × 𝔹 ♭♭ 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] → 𝔹 ⋄ , wher e the pul lb ack is via ev i and the map to 𝔹 ⋄ is given by ev 1 − i . Remark 4.2.2. The case of the theorem where 𝔹 is fully decorated and ϵ = (1 , 0) has previously b een prov ed by Ab ellán and Stern as [ AS23b , Theorem 3.17]. Construction 4.2.3. More precisely , we deﬁne the functor 𝔽 ree ϵ 𝔹 ⋄ as the comp osite Cat ( ∞ , 2) / 𝔹 ≃ PFib ϵ / 𝔹 ♭♭ ev ∗ i − − → PFib ϵ / 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] ev 1 − i, ! − − − − → PFib ϵ / 𝔹 ⋄ where w e hav e used that partial ϵ -ﬁbrations are closed under base change along an y functor, such as ev i , and under comp osition with a partial ϵ -ﬁbration, such as ev 1 − i (Theorem 4.1.2 ). The unit of the adjunction is easy to deﬁne: Construction 4.2.4. The degeneracy [1] → [0] induces a commutativ e triangle 𝔹 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) 𝔹 s ∗ 0 ev 1 − i of ( ∞ , 2) -categories. By pulling this bac k we obtain a natural transformation η ϵ 𝔹 ⋄ : id → U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 65 That is, for p : 𝔼 → 𝔹 , the map η ϵ ( p ) is the commutativ e triangle 𝔼 𝔼 × 𝔹 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) 𝔹 η ϵ 𝔹 ⋄ ( p ) p with the horizontal map given by comp osing with s 0 in the second factor. When p is a partial ϵ -ﬁbration o ver 𝔹 ⋄ , we can use Theorem 3.4.1 to give an alternativ e iden tiﬁcation of 𝔽 ree ϵ 𝔹 ⋄ ( p ) ; this allows us to deﬁne the counit of the adjunction and prov e one of the triangle identities: Prop osition 4.2.5. Supp ose p : 𝔼 → 𝔹 is a p artial ϵ -ﬁbr ation over 𝔹 ⋄ . (i) The or em 3.4.1 extends to a natur al e quivalenc e 𝔽 ree ϵ 𝔹 ⋄ ( p ) ≃ 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) ♭ [ ϵ ] p ◦ ev 1 − i − − − − − → 𝔹 ⋄ of de c or ate d ( ∞ , 2) -c ate gories. (ii) Under this e quivalenc e the map η ϵ 𝔹 ⋄ : p → U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) c orr esp onds to the de gener acy map s ∗ 0 : 𝔼 → 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) . (iii) ev 1 − i gives a natur al morphism of de c or ate d ( ∞ , 2) -c ate gories 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) ♭ [ ϵ ] → 𝔼 ♮ over 𝔹 ⋄ , and so a natur al tr ansformation ε ϵ 𝔹 ⋄ : 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ → id . (iv) The c omp osite 𝔼 η ϵ 𝔹 ⋄ − − → 𝔽 ree ϵ 𝔹 ⋄ ( p ) ε ϵ 𝔹 ⋄ − − → 𝔼 is the identity, i.e. the c omp osite natur al tr ansformation U ϵ 𝔹 ⋄ η ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ − − − − → U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ ε ϵ 𝔹 ⋄ − − − − → U ϵ 𝔹 ⋄ is the identity of U ϵ 𝔹 ⋄ . Pr o of. W e prov e the case ϵ = (0 , 1) to simplify the notation. Then Prop osition 3.5.4 giv es an equiv alence of decorated ( ∞ , 2) -categories 𝔻𝔸 r oplax ( 𝔼 ♮ ) ⋄ -ﬁb ∼ − → 𝔻𝔸 r oplax ( 𝔹 ⋄ ) ⋄ -ﬁb × 𝔹 ⋄ 𝔼 ♮ . Pulling this back along 𝔼 ♭♭ → 𝔼 ♮ then gives the desired equiv alence 𝔻𝔸 r oplax ( 𝔼 ♮ ) ♭ [(0 , 1)] ∼ − → 𝔽 ree (0 , 1) 𝔹 ⋄ ( p ) in (i). 66 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI P art (ii) is immediate from the commutativ e diagram 𝔼 𝔻𝔸 r oplax ( 𝔼 ♮ ) 𝔼 𝔹 𝔻𝔸 r oplax ( 𝔹 ⋄ ) 𝔹 , s ∗ 0 p = 𝔻𝔸 r oplax ( p ) ev 0 p s ∗ 0 = ev 0 where the right-hand square is a pullback. F or (iii), w e observe that for a decorated 1-morphism in 𝔻𝔸 r oplax ( 𝔼 ♮ ) ♭ [(0 , 1)] , which is of the form • • • • , the cancellation prop erty of co cartesian morphisms implies that the b ottom horizon- tal morphism must also b e co cartesian, so that ev 1 preserv es decorated 1-morphisms. The cancellation prop ert y of cartesian 2-morphisms similarly implies that ev 1 also preserv es decorated 2-morphisms, which completes the pro of. P art (iv) now follows from the fact that under the identication of (ii), the com- p osite 𝔼 → 𝔽 ree (0 , 1) 𝔹 ⋄ ( p ) → 𝔼 is identiﬁed with 𝔼 s ∗ 0 − → 𝔻𝔸 r oplax ( 𝔼 ♮ ) ev 1 − − → 𝔼 , whic h is manifestly the identit y . □ It remains to chec k the other triangle identit y , for which we need to identify the comp osite 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ more explicitly: Observ ation 4.2.6. Suppose ϵ = ( i, j ) and set i ′ =    0 , i = 0 , 2 , i = 1 . W e deﬁne 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) ϵ -lax ,♭ [ ϵ ] to b e equipp ed with the decorations where ▶ a 1-morphism is decorated if it is a strong natural transformation such that its images under ev i ′ and ev 1 are inv ertible and its image under ev 2 − i ′ is decorated, ▶ a 2-morphism is decorated if its images under ev i ′ and ev 1 are inv ertible and its image under ev 2 − i ′ is decorated. F or p : 𝔼 → 𝔹 , iterating the deﬁnition of 𝔽 ree ϵ 𝔹 ⋄ ( – ) as a pullbac k then gives an equiv alence 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) ≃ ( 𝔼 × 𝔹 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ )) ♭♭ × 𝔹 ♭♭ 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] ≃ 𝔼 ♭♭ × 𝔹 ♭♭ 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) ϵ -lax ,♭ [ ϵ ] , (4.2) where the pullback is taken via ev i ′ and the map to 𝔹 is given by ev 2 − i ′ . On the other hand, since 𝔽 ree ϵ 𝔹 ⋄ ( p ) is an ϵ -ﬁbration, Prop osition 4.2.5 implies that we ha v e an equiv alence 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) ≃ 𝔻𝔸 r ϵ -lax ( 𝔽 ree ϵ 𝔹 ⋄ ( p )) ♭ [ ϵ ] . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 67 Unpac king the deﬁnition of 𝔽 ree ϵ 𝔹 ⋄ ( p ) , we see that the right-hand side is equiv alent to the pullback 𝔻𝔸 r ϵ -lax ( 𝔼 ♭♭ ) ♭ [ ϵ ] × 𝔻𝔸 r ϵ -lax ( 𝔹 ♭♭ ) ♭ [ ϵ ] 𝔻𝔸 r ϵ -lax ( 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] ) ♭ [ ϵ ] , whic h w e can iden tify with 𝔼 ♭♭ × 𝔹 ♭♭ 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) ϵ -lax ,♭ [ ϵ ] using Prop osition 4.1.9 . W e claim that the resulting equiv alence 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) ≃ 𝔻𝔸 r ϵ -lax ( 𝔽 ree ϵ 𝔹 ⋄ ( p )) ♭ [ ϵ ] ≃ 𝔼 ♭♭ × 𝔹 ♭♭ 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) ϵ -lax ,♭ [ ϵ ] , (4.3) is the same as ( 4.2 ). Indeed, this second equiv alence arises from the commutativ e diagram 𝔻𝔸 r ϵ -lax ( 𝔽 ree ϵ 𝔹 ⋄ ( p )) ♭ [ ϵ ] 𝔻𝔸 r ϵ -lax ( 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] ) ♭ [ ϵ ] 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] 𝔽 ree ϵ 𝔹 ⋄ ( p ) ♭♭ 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭♭ 𝔹 ♭♭ 𝔼 ♭♭ 𝔹 ♭♭ , where the outer square in the top row is a pullback b y Prop osition 4.2.5 , the top right square by Prop osition 4.1.9 , and the b ottom square by the deﬁnition of 𝔽 ree ϵ 𝔹 ⋄ ( p ) . Our equiv alence ( 4.3 ) then arises from the composite square in the left column b eing a pullback. But this comp osite square can also b e factored as 𝔻𝔸 r ϵ -lax ( 𝔽 ree ϵ 𝔹 ⋄ ( p )) ♭ [ ϵ ] 𝔻𝔸 r ϵ -lax ( 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] ) ♭ [ ϵ ] 𝔻𝔸 r ϵ -lax ( 𝔼 ♭♭ ) ♭ [ ϵ ] 𝔻𝔸 r ϵ -lax ( 𝔹 ♭♭ ) ♭ [ ϵ ] 𝔼 ♭♭ 𝔹 ♭♭ , ∼ ∼ where the low er vertical maps are b oth equiv alences, and this is the square that giv es the ﬁrst equiv alence ( 4.2 ). Prop osition 4.2.7. Under the e quivalenc e ( 4.2 ), we have: (i) the map ε ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ : 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) → 𝔽 ree ϵ 𝔹 ⋄ ( p ) c orr esp onds to the pul lb ack of the c omp osition functor d ∗ 1 : 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) ϵ -lax ,♭ [ ϵ ] → 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ♭ [ ϵ ] , (ii) and the map 𝔽 ree ϵ 𝔹 ⋄ η ϵ 𝔹 ⋄ : 𝔽 ree ϵ 𝔹 ⋄ ( p ) → 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) c orr esp onds to the pul lb ack of the de gener acy functor s ∗ i : 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) → 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) ϵ -lax . 68 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI (iii) The c omp osite 𝔽 ree ϵ 𝔹 ⋄ ( p ) 𝔽 ree ϵ 𝔹 ⋄ η ϵ 𝔹 ⋄ − − − − − − → 𝔽 ree ϵ 𝔹 ⋄ U ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ ( p ) ε ϵ 𝔹 ⋄ 𝔽 ree ϵ 𝔹 ⋄ − − − − − − → 𝔽 ree ϵ 𝔹 ⋄ ( p ) is the identity. Pr o of. Assume ϵ = (0 , 1) . Unpac king the equiv alence ( 4.3 ) we see that ev 1 : 𝔻𝔸 r ϵ -lax ( 𝔽 ree ϵ 𝔹 ⋄ ( p )) ♭ [ ϵ ] → 𝔽 ree ϵ 𝔹 ⋄ ( p ) is given by the identit y on 𝔼 ♭♭ and 𝔹 ♭♭ and by d 1 on 𝔻𝔽 un ([2] ♯ , 𝔹 ⋄ ) , as required for (i). On the other hand, unpacking ( 4.2 ) we get (ii). Putting these together, (iii) is then immediate. □ Pr o of of The or em 4.2.1 . It follo ws from Prop osition 4.2.7 and Prop osition 4.2.5 that the triangle identities hold, so that the natural transformations η ϵ 𝔹 ⋄ and ε ϵ 𝔹 ⋄ are the unit and counit of an adjunction 𝔽 ree ϵ 𝔹 ⋄ ⊣ U ϵ 𝔹 ⋄ . □ Observ ation 4.2.8. The functor U ϵ 𝔹 ⋄ : PFib ϵ / 𝔹 ⋄ → Cat ( ∞ , 2) / 𝔹 ﬁts in a commutativ e triangle Cat ∞ PFib ϵ / 𝔹 ⋄ Cat ( ∞ , 2) / 𝔹 . ( – ) ♭♭ × 𝔹 ⋄ ( – ) × 𝔹 U ϵ 𝔹 ⋄ Moreo ver, the canonical map 𝔽 ree ϵ 𝔹 ⋄ ( 𝕂 × p ) → 𝕂 ♭♭ × 𝔽 ree ϵ 𝔹 ⋄ ( p ) is an equiv alence for an y ( ∞ , 2) -category 𝕂 . It therefore follows from Corollary 2.1.6 that the adjunction 𝔽 ree ϵ 𝔹 ⋄ ⊣ U ϵ 𝔹 ⋄ upgrades to an adjunction of ( ∞ , 2) -categories. In fact, replacing Cat ∞ b y Cat ( ∞ , 2) in the diagram ab ov e, this same argument shows that we can even get an adjunction of ( ∞ , 3) -categories — in particular, for an y functor p : ℂ → 𝔹 and any partial ϵ -ﬁbration q : 𝔼 ⋄ → 𝔹 ⋄ , the adjunction induces equiv alences of ( ∞ , 2) -categories 𝔽 un / 𝔹 ( ℂ , 𝔼 ) ≃ 𝔻𝔽 un / 𝔹 ⋄ ( 𝔽 ree ϵ 𝔹 ⋄ ( p ) , 𝔼 ⋄ ) . As a ﬁrst application of our description of free ﬁbrations, we obtain the follo wing c haracterization of representable (1 , 0) -ﬁbrations: Prop osition 4.2.9. L et p : 𝔼 → 𝔹 b e a 1-ﬁbr e d (1 , 0) -ﬁbr ation. Then the fol lowing ar e e quivalent for an obje ct e ∈ 𝔼 over b = p ( e ) in 𝔹 : (1) Ther e exists an e quivalenc e 𝔹 → b 𝔼 𝔹 ∼ p that takes id b to e . (2) The morphism { e } ♭♭ → 𝔼 ♮ is a (1 , 0) -e quivalenc e over 𝔹 ♯♯ . (3) The morphism { e } ♭♭ → 𝔼 ♮ is a (1 , 0) -e quivalenc e over 𝔼 ♯♯ . (4) F or every obje ct x ∈ 𝔼 ther e exists a c artesian morphism x → e , and every c artesian morphism x → e is an initial obje ct of 𝔼 ( x, e ) . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 69 If these c onditions hold, we say that e exhibits p as a r epr esentable (0 , 1) -ﬁbr ation, which is r epr esente d by b ∈ 𝔹 . Pr o of. By Theorem 4.2.1 , the commutativ e triangle ∗ 𝔼 𝔹 e b p extends uniquely to a morphism of (1 , 0) -ﬁbrations out of the free ﬁbration on ∗ b − → 𝔹 , i.e. to 𝔹 → b 𝔼 𝔹 . F p If the ﬁrst condition holds, the equiv alence in question must therefore b e this functor F , while F is an equiv alence precisely if { e } ♭♭ → 𝔼 ♮ is a (1 , 0) -equiv alence ov er 𝔹 ♯♯ , since 𝔹 → b → 𝔹 is the free (1 , 0) -ﬁbration on { e } . This shows the ﬁrst t w o conditions are equiv alent, while the second and third are equiv alent since Observ ation 3.6.7 implies ﬁrstly that (2) is equiv alent to ha ving a (1 , 0) -equiv alence o ver 𝔼 ♮ , and secondly that these are detected ov er 𝔼 ♯♯ since the ﬁbration factors through this. T o obtain the ﬁnal condition, note that the functor F takes an ob ject f : b ′ → b to the source f ∗ e of the cartesian morphism ov er f with target e , so that F is essen tially surjective if and only if ev ery ob ject of 𝔼 is the source of a cartesian morphism to e . If this holds, F is an equiv alence if and only if it is fully faithful, whic h means that for f : b ′ → b and g : b ′′ → b in 𝔹 the horizontal morphism in the comm utative triangle (where b oth down ward maps are left ﬁbrations) 𝔹 → b ( f , g ) 𝔼 ( f ∗ e, g ∗ e ) 𝔹 ( b ′ , b ′′ ) is an equiv alence. W e ﬁrst consider the case where g = id b . Then we can use Prop osition 3.3.1 to iden tify 𝔹 → b ( f , id b ) as 𝔹 ( b ′ , b ) f / , so we get an equiv alence if and only if the initial ob ject f is mapp ed to an initial ob ject in 𝔼 ( f ∗ e, e ) ; here f is sent to the cartesian morphism ¯ f : f ∗ e → e , so this precisely corresp onds to the condition that ¯ f is an initial ob ject. It remains to see that this gives an equiv alence also for general targets g , for which we observe that g gives a cartesian morphism g → id b , and comp osition with this and its image in 𝔼 gives a commutativ e cub e 𝔹 → b ( f , g ) 𝔼 ( f ∗ e, g ∗ e ) 𝔹 → b ( f , id b ) 𝔼 ( f ∗ e, e ) 𝔹 ( b ′ , b ′′ ) 𝔹 ( b ′ , b ′′ ) 𝔹 ( b ′ , b ) 𝔹 ( b ′ , b ) ∼ = = 70 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Here the b ottom, left and right faces are pullbacks, hence so is the top face, and so the top horizontal map is indeed an equiv alence, as required. □ Observ ation 4.2.10. Under straigh tening, if F : 𝔹 → ℂ at ∞ is the functor cor- resp onding to the 1-ﬁbred (1 , 0) -ﬁbration p : 𝔼 → 𝔹 , the conditions of Proposi- tion 4.2.9 for e ∈ 𝔼 ov er b ∈ 𝔹 are equiv alent to the existence of a natural equiv a- lence 𝔹 ( – , b ) ≃ F under which id b corresp onds to e ∈ F ( b ) ≃ 𝔼 b . Remark 4.2.11. Condition (2) in Prop osition 4.2.9 is the characterization of rep- resen tability studied in [ GHL25 , §4.3], while condition (3) sa ys precisely that the functor of marked ( ∞ , 2) -categories { e } ♭ → ( 𝔼 , C ) is (1 , 0) -c oﬁnal in the sense of § 5.5 , by Prop osition 5.5.4 . 4.3. F ree decorated partial ﬁbrations. In this subsection we consider a v ariant of Theorem 4.2.1 , which will describe free de c or ate d partial ﬁbrations. W e also sho w that the unit maps of the resulting adjunction hav e the prop erty that any pullbac ks thereof are ϵ -equiv alences, which will b e a crucial input to our discussion of cofree ﬁbrations b elow. Prop osition 4.3.1. L et 𝔹 ⋄ b e a de c or ate d ( ∞ , 2) -c ate gory. The for getful functor U ϵ d , 𝔹 ⋄ : 𝔻ℙ𝔽 ib ϵ / 𝔹 ⋄ → 𝔻ℂ at ( ∞ , 2) / 𝔹 ⋄ has a left adjoint 𝔻𝔽 ree ϵ 𝔹 ⋄ , given for p : ℂ ⋄ → 𝔹 ⋄ by 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) := ℂ ⋄ × 𝔹 ⋄ 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb → 𝔹 ⋄ , wher e the pul lb ack is via ev i and the map to 𝔹 ⋄ is given by ev 1 − i . Observ ation 4.3.2. With this notation, we can in terpret Proposition 3.5.4 as giving a natural equiv alence 𝔻𝔽 ree ϵ 𝔹 ⋄ U ϵ d , 𝔹 ⋄ ( p ) ≃ 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) ⋄ -ﬁb for any decorated partial ϵ -ﬁbration p : 𝔼 ⋄ → 𝔹 ⋄ . Pr o of of Pr op osition 4.3.1 . W e’ll show that we get an adjunction on underlying ∞ - categories; this upgrades to ( ∞ , 2) -categories as in Observ ation 4.2.8 . F or p : 𝔼 ⋄ → 𝔹 ⋄ , we can regard 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) as the pullback (4.4) 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb 𝔼 ⋄ × 𝔹 ⋄ 𝔹 ⋄ × 𝔹 ⋄ (ev i , ev 1 − i ) p × id of decorated partial ϵ -ﬁbrations ov er 𝔹 ⋄ (using Corollary 4.1.10 ). Th us 𝔻𝔽 ree ϵ 𝔹 ⋄ do es indeed deﬁne a functor DCat ( ∞ , 2) / 𝔹 ⋄ → DPFib ϵ / 𝔹 ⋄ . T o see that this gives a left adjoint, it suﬃces to chec k that the unit and counit from the previous section also give natural transformations η : id → U ϵ d , 𝔹 ⋄ 𝔻𝔽 ree ϵ 𝔹 ⋄ and  : 𝔻𝔽 ree ϵ 𝔹 ⋄ U ϵ d , 𝔹 ⋄ → id , whic h amounts to c hecking they are compatible with FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 71 the decorations w e now consider; the triangle iden tities will then automatically hold. F or the unit, this follows from observing that the degeneracy giv es a decorated functor 𝔹 ⋄ → 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb . On the other hand, Prop osition 3.5.4 implies that for p : 𝔼 ⋄ → 𝔹 ⋄ a decorated partial ϵ -ﬁbration, the counit at p can b e identiﬁed with the decorated functor ev 1 − i : 𝔻𝔸 r ϵ -lax ( 𝔼 ♮ ) ⋄ -ﬁb → 𝔼 ⋄ , whic h is therefore a morphism of decorated partial ϵ -ﬁbrations; this completes the pro of. □ Observ ation 4.3.3. W e can view the unit map of the adjunction of Prop osi- tion 4.3.1 at p : ℂ ⋄ → 𝔹 ⋄ as a map ℂ ⋄ 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) 𝔹 ⋄ . η p in DCat ( ∞ , 2) / 𝔹 ⋄ . Supp ose q : 𝔼 → 𝔹 is a partial ϵ -ﬁbration with resp ect to 𝔹 ⋄ ; then q : 𝔼 ♮ → 𝔹 ⋄ is a decorated partial ﬁbration, and moreov er any decorated functor 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) → q o v er 𝔹 ⋄ is a morphism of decorated partial ϵ -ﬁbrations (i.e. it automatically preserves (co)cartesian 1 - and 2 -morphisms), so that comp osition with η induces an equiv alence Map / 𝔹 ⋄ ( 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) , 𝔼 ♮ ) ∼ − → Map / 𝔹 ⋄ ( ℂ ⋄ , 𝔼 ♮ ) . In other words, η is an ϵ -equiv alence ov er 𝔹 ⋄ with these decorations. With a bit more work, we can ﬁnd a “homotopy inv erse” of η that lets us upgrade this observ ation to the following statement: Prop osition 4.3.4. L et p : 𝔼 ⋄ → 𝔹 ⋄ b e a de c or ate d p artial ϵ -ﬁbr ation, and c onsider the unit map of the adjunction of Pr op osition 4.3.1 at p as a map 𝔼 ⋄ 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) 𝔹 ⋄ . η p F or any de c or ate d functor f : 𝔸 ⋄ → 𝔹 ⋄ , the pul lb ack f ∗ η : f ∗ 𝔼 ⋄ → f ∗ 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) is an ϵ -e quivalenc e over 𝔸 ⋄ . F or the pro of we make use of a canonical (op)lax transformation asso ciated to the free decorated ﬁbration, which we sp ell out in the case ϵ = (0 , 1) for simplicity: Construction 4.3.5. Consider the decorated functor π (0 , 1) : [1] ♯ ⊗ [1] ♭ → [1] ♯ × [1] ♭ → [1] ♯ , whic h tak es (0 , 0) to 0 and the remaining ob jects to 1 . This induces for any decorated ( ∞ , 2) -category ℂ ⋄ a functor of ( ∞ , 2) -categories π ∗ (0 , 1) : 𝔻𝔸 r oplax ( ℂ ⋄ ) → 𝔻𝔽 un ([1] ♯ ⊗ [1] ♭ , ℂ ⋄ ) oplax ≃ 𝔸 r oplax ( 𝔻𝔸 r oplax ( ℂ ⋄ )) , 72 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI whic h is adjoint to a lax transformation [1] ⊗ 𝔻𝔸 r oplax ( ℂ ⋄ ) → 𝔻𝔸 r oplax ( ℂ ⋄ ); un winding the deﬁnition we see that this transformation go es from the identit y to s ∗ 0 ev 1 . Lemma 4.3.6. Supp ose p : 𝔼 ⋄ → 𝔹 ⋄ is a de c or ate d p artial (0 , 1) -ﬁbr ation. Then the c onstruction ab ove gives a de c or ate d functor [1] ♯ ⊗ d 𝔻𝔸 r oplax ( 𝔼 ♮ ) ⋄ -ﬁb 𝔻𝔸 r oplax ( 𝔼 ♮ ) ⋄ -ﬁb 𝔼 ⋄ ev 1 ◦ pro j ev 1 and so a lax natur al tr ansformation [1] ♯ ⊗ d U ϵ d , 𝔹 ⋄ 𝔻𝔽 ree ϵ 𝔹 ⋄ U ϵ d , 𝔹 ⋄ ( p ) → U ϵ d , 𝔹 ⋄ 𝔻𝔽 ree ϵ 𝔹 ⋄ U ϵ d , 𝔹 ⋄ ( p ) fr om id to ( η ϵ 𝔹 ⋄ U ϵ d , 𝔹 ⋄ ) ◦ ( U ϵ d , 𝔹 ⋄ ε ϵ 𝔹 ⋄ ) via Pr op osition 3.5.4 . Pr o of. It is clear that the source and target functors id and s ∗ 0 ev 1 : 𝔻𝔸 r oplax ( 𝔼 ♮ ) ⋄ -ﬁb → 𝔻𝔸 r oplax ( 𝔼 ♮ ) ⋄ -ﬁb are b oth decorated. W e therefore need to chec k that ▶ for ev ery ob ject f : x → y of 𝔻𝔸 r oplax ( 𝔼 ♮ ) (where f is a co cartesian morphism), the induced map f → s ∗ 0 ev 1 ( f ) = id y is decorated, ▶ for every morphism α : f → g in 𝔻𝔸 r oplax ( 𝔼 ♮ ) , the 2-morphism in the asso ciated naturalit y square is decorated. Unpac king the deﬁnition, we can iden tify the morphism in the ﬁrst p oin t as the square x y y y . f f id id This comm utes, and b oth f and id y are decorated in 𝔼 ⋄ , so this is indeed decorated. In the second p oint, the naturality square unpacks to the cub e x x ′ y y ′ y y ′ y y ′ f g = = = = where we hav e not shown that the back and top faces contain the 2-morphism α ; the other four faces comm ute. In particular, b oth the top and b ottom faces con tain a decorated 2-morphism in 𝔼 ⋄ , so as an oplax square in 𝔻𝔸 r oplax ( 𝔼 ♮ ) this contains a decorated 2-morphism, as required. □ Pr o of of Pr op osition 4.3.4 . W e apply Lemma 3.6.11 using the counit of the ad- junction and the lax transformation from Lemma 4.3.6 (and its v ariants for other v ariances). □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 73 Remark 4.3.7. Supp ose p : E → B is a co cartesian ﬁbration of ∞ -categories. Then our deﬁnitions ab ov e give functors η : E → E × B Ar ( B ) and  : E × B Ar ( B ) → E such that η ≃ id E . In fact, this is part of the data of an adjunction  ⊣ η , which leads to the characterization of co cartesian ﬁbrations ov er B as the functors p such that η has a left adjoint ov er B (see [ R V22 , Theorem 5.2.8]). W e can think of Construction 4.3.5 as extending the deﬁnition of the unit of this adjunction to the ( ∞ , 2) -categorical con text. Note, how ever, that as it is only an (op)lax transformation, it do es not actually exhibit an adjunction of ( ∞ , 2) -categories. It w ould b e interesting to kno w if the characterization of ﬁbrations via adjunctions could b e extended to ( ∞ , 2) - categories via some notion of “lax adjunctions”, but we will not pursue this here. 4.4. F ree ﬁbrations on decorated ( ∞ , 2) -categories. In this subsection w e will use Prop osition 4.3.1 together with the straightening equiv alence for decorated ﬁbrations from Theorem 3.2.8 to get a description of the left adjoint of the forgetful functor from ϵ -ﬁbrations to decorated ( ∞ , 2) -categories ov er the base. W e then sp ecialize this to obtain free 1-ﬁbred ﬁbrations on marke d ( ∞ , 2) -categories, which will pla y an imp ortan t role in our study of colimits and Kan extensions b elo w. Note that w e will later also b e able to describ e the corresp onding ﬁbrations as c ertain lo calizations of free decorated ﬁbration in § 4.7 . W e start by iden tifying the straightening of free decorated ϵ -ﬁbrations, which requires some notation: Notation 4.4.1. F or b ∈ 𝔹 , we introduce the follo wing notation for the decorated (op)lax slices induced from 𝔻𝔸 r ϵ -lax ( 𝔹 ⋄ ) ⋄ -ﬁb : ▶ 𝔹 ⋄ b → is { b } × 𝔹 ⋄ 𝔻𝔸 r oplax ( 𝔹 ⋄ ) ⋄ -ﬁb via ev 0 ; this is 𝔻𝔽 ree (0 , 1) 𝔹 ⋄ ( { b } ) . ▶ 𝔹 ⋄ → b is { b } × 𝔹 ⋄ 𝔻𝔸 r oplax ( 𝔹 ⋄ ) ⋄ -ﬁb via ev 1 ; this is 𝔻𝔽 ree (1 , 0) 𝔹 ⋄ ( { b } ) . ▶ 𝔹 ⋄ b → is { b } × 𝔹 ⋄ 𝔻𝔸 r lax ( 𝔹 ⋄ ) ⋄ -ﬁb via ev 0 ; this is 𝔻𝔽 ree (0 , 0) 𝔹 ⋄ ( { b } ) . ▶ 𝔹 ⋄ → b is { b } × 𝔹 ⋄ 𝔻𝔸 r lax ( 𝔹 ⋄ ) ⋄ -ﬁb via ev 1 ; this is 𝔻𝔽 ree (1 , 1) 𝔹 ⋄ ( { b } ) . Giv en a functor p : ℂ ⋄ → 𝔹 ⋄ and b ∈ 𝔹 we then denote by ℂ ⋄ b → the pullbac k ℂ ⋄ × 𝔹 ⋄ 𝔹 ⋄ b → using the functor to 𝔹 ⋄ induced by ev 1 , and similarly in the other v ariances; w e can then iden tify the ﬁbre at b of the free decorated ﬁbration 𝔻𝔽 ree ϵ 𝔹 ⋄ ( p ) as ℂ ⋄ × 𝔹 ⋄ 𝔻𝔽 ree ϵ 𝔹 ⋄ ( { b } ) , i.e. as ▶ 𝔻𝔽 ree (1 , 0) ( p ) b ≃ ℂ ⋄ b → , ▶ 𝔻𝔽 ree (0 , 1) ( p ) b ≃ ℂ ⋄ → b , ▶ 𝔻𝔽 ree (1 , 1) ( p ) b ≃ ℂ ⋄ b → , ▶ 𝔻𝔽 ree (0 , 0) ( p ) b ≃ ℂ ⋄ → b . Prop osition 4.4.2. Under the str aightening e quivalenc e F un ( 𝔹 ϵ -op , 𝔻ℂ at ( ∞ , 2) ) ≃ DFib ϵ / 𝔹 ♯♯ , 74 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI the de c or ate d fr e e ﬁbr ation 𝔻𝔽 ree ϵ 𝔹 ♯♯ ( p ) of p : ℂ ⋄ → 𝔹 ♯♯ c orr esp onds to the functor b 7→ ℂ ⋄ × 𝔹 ♯♯ 𝔻𝔽 ree ϵ 𝔹 ♯♯ ( { b } ) ≃                ℂ ⋄ b → , ϵ = (1 , 0) , ℂ ⋄ → b , ϵ = (0 , 1) , ℂ ⋄ b → , ϵ = (1 , 1) , ℂ ⋄ → b , ϵ = (0 , 0) , wher e the functoriality in b c omes fr om pul ling b ack the str aightening of ev 1 − i : 𝔻𝔸 r ϵ -lax ( 𝔹 ♯♯ ) → 𝔹 ♯♯ . Pr o of. This follows from the naturality of straightening and the pullback square of decorated ﬁbrations ( 4.4 ). □ Corollary 4.4.3. The left adjoint to the for getful functor Fib ϵ / 𝔹 → DCat ( ∞ , 2) / 𝔹 ♯♯ takes ℂ ⋄ → 𝔹 ♯♯ to the unstr aightening of the functor 𝔹 ϵ -op → ℂ at ( ∞ , 2) given by b 7→ τ d ( ℂ ⋄ × 𝔹 ♯♯ 𝔻𝔽 ree ϵ 𝔹 ♯♯ ( { b } )) ≃                τ d ℂ ⋄ b → , ϵ = (1 , 0) , τ d ℂ ⋄ → b , ϵ = (0 , 1) , τ d ℂ ⋄ b → , ϵ = (1 , 1) , τ d ℂ ⋄ → b , ϵ = (0 , 0) . Pr o of. The forgetful functor factors as Fib ϵ / 𝔹 ( – ) ♮ − − → DFib ϵ / 𝔹 → DCat ( ∞ , 2) / 𝔹 ♯♯ , so its left adjoint factors as 𝔻𝔽 ree ϵ 𝔹 ♯♯ follo wed by the left adjoint of ( – ) ♮ . Corol- lary 3.2.13 shows that ( – ) ♮ corresp onds under straigh tening to comp osition with ( – ) ♭♭ , so its left adjoint corresp onds to comp osition with τ d . □ No w w e sp ecialize our results to describ e free 1-ﬁbr e d ϵ -ﬁbrations: Deﬁnition 4.4.4. Let 𝕄𝔽 ib ϵ / 𝔹 denote the full sub- ( ∞ , 2) -category of 𝔻𝔽 ib ϵ / 𝔹 spanned b y the decorated ϵ -ﬁbrations whose source is in the image of ( – ) ♯ : 𝕄ℂ at ( ∞ , 2) → 𝔻ℂ at ( ∞ , 2) ; we refer to these as marke d ϵ -ﬁbr ations over 𝔹 Observ ation 4.4.5. F or ϵ = ( i, j ) , let 𝟙𝔽 ib ϵ / 𝔹 denote the full sub- ( ∞ , 2) -category of 𝔽 ib ϵ / 𝔹 on the 1-ﬁbr e d ϵ -ﬁbrations, i.e. those whose ﬁbres are ∞ -categories, or equiv alently those where al l 2-morphisms are j -cartesian. W e can then regard 𝟙𝔽 ib ϵ / 𝔹 as a full sub category of 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ , where 𝔼 → 𝔹 corresp onds to 𝔼 ♮ → 𝔹 ♯ with 𝔼 ♮ mark ed by the i -cartesian 1-morphisms (since for 1-ﬁbred ﬁbrations all 2- morphisms are j -cartesian and so they are alwa ys preserved). In fact, this describes 𝟙𝔽 ib ϵ / 𝔹 as precisely the in tersection of 𝔽 ib ϵ / 𝔹 and 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ in 𝔻ℂ at ( ∞ , 2) / 𝔹 ♯♯ (where we embed the latter via ( – ) ♯ ). FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 75 Notation 4.4.6. Let ( 𝔼 , S ) b e a marked ( ∞ , 2) -category. F or a functor p : 𝔼 → 𝔹 , w e let F ϵ 𝔹 ( 𝔼 , S ) : 𝔹 ϵ -op → ℂ at ∞ b e the functor obtained b y ﬁrst unstraightening the free decorated ϵ -ﬁbration 𝔻𝔽 ree ϵ 𝔹 ♯♯ ( p ) , where we view p as a map ( 𝔼 , S ) ♯ → 𝔹 ♯♯ , and then comp osing with τ d . Th us F ϵ 𝔹 ( 𝔼 , S ) is given by b 7→ τ d (( 𝔼 , S ) ♯ × 𝔹 ♯♯ 𝔻𝔽 ree ϵ 𝔹 ♯♯ ( { b } )) ≃                τ d 𝔼 ⋄ b → , ϵ = (1 , 0) , τ d 𝔼 ⋄ → b , ϵ = (0 , 1) , τ d 𝔼 ⋄ b → , ϵ = (1 , 1) , τ d 𝔼 ⋄ → b , ϵ = (0 , 0) , where w e can identify 𝔼 ⋄ b → as ( 𝔼 b → , S b ) ♯ with S b consisting of maps whose pro jection to 𝔼 lies in S , and similarly in the other cases. In particular, these decorated ( ∞ , 2) - categories ha v e al l 2-morphisms decorated, so after lo calizing them this functor indeed takes v alues in ℂ at ∞ . In particular, giv en a marking ( 𝔹 , E ) of 𝔹 , we hav e the functor F ϵ 𝔹 ( 𝔹 , E ) : 𝔹 ϵ -op → ℂ at ∞ , giv en b y b 7→                τ d 𝔹 ⋄ b → , ϵ = (1 , 0) , τ d 𝔹 ⋄ → b , ϵ = (0 , 1) , τ d 𝔹 ⋄ b → , ϵ = (1 , 1) , τ d 𝔹 ⋄ → b , ϵ = (0 , 0) , where 𝔹 ⋄ b → is decorated b y the 1-morphisms that lie o ver E and all 2-morphisms, and similarly in the other 3 cases. Corollary 4.4.7. The ful ly faithful inclusion 𝟙𝔽 ib ϵ / 𝔹  → 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ has a left adjoint, which sends ( ℂ , S ) → 𝔹 ♯ to the unstr aightening of the functor F ϵ 𝔹 ( ℂ , S ) : 𝔹 ϵ -op → ℂ at ∞ . In p articular, for F : 𝔹 ϵ -op → ℂ at ∞ with c orr esp onding ϵ -ﬁbr ation 𝔼 → 𝔹 , we have a natur al e quivalenc e of ∞ -c ate gories Nat 𝔹 ϵ -op , ℂ at ∞ ( F ϵ 𝔹 ( ℂ , S ) , F ) ≃ 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ (( ℂ , S ) , 𝔼 ♮ ) . Pr o of. W e need to show that the adjunction from Corollary 4.4.3 restricts to these full sub categories; the only thing to chec k is that the left adjoint takes a mark ed ( ∞ , 2) -category to a 1-ﬁbred ﬁbration, whic h is clear since 𝔼 ⋄ × 𝔹 ♯♯ 𝔻𝔽 ree ϵ 𝔹 ♯♯ ( { b } ) then has al l 2-morphisms decorated, and so applying τ d ( – ) pro duces an ∞ -category. □ 76 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI 4.5. Smo othness for decorated ( ∞ , 2) -categories. Supp ose p : E → B is a co- cartesian ﬁbration of ∞ -categories. Then p is in particular exp onen tiable, meaning that the pullbac k functor p ∗ : Cat ∞ / B → Cat ∞ / E has a righ t adjoint p ∗ . Moreo v er, this right adjoint has the prop erty that if q : D → E is a cartesian ﬁbration, then so is p ∗ ( q ) o ver B . Our goal in this section is to generalize these results, as well as the closely related notion of smo oth functors from [ Lur09a , §4.1.2], to the setting of decorated ( ∞ , 2) -categories. Deﬁnition 4.5.1. A decorated functor p : 𝔸 ⋄ → 𝔹 ⋄ is said to b e exp onentiable if the pullback functor p ∗ dec : 𝔻ℂ at ( ∞ , 2) / 𝔹 ⋄ → 𝔻ℂ at ( ∞ , 2) / 𝔸 ⋄ admits a right adjoint p dec , ∗ whic h w e call the de c or ate d pushforwar d functor . Prop osition 4.5.2. Supp ose p : 𝔸 ⋄ → 𝔻 ♯♯ is a de c or ate d ϵ -ﬁbr ation. Then p is exp onentiable. F or the pro of we use the following observ ation: Lemma 4.5.3. L et F : I → ( DCat ( ∞ , 2) ) / 𝔻 ♯♯ b e a functor wher e I is an ∞ -c ate gory, and supp ose that the c olimit of F is pr eserve d by the functors u d , ( – ) ≤ (1) and ( – ) (2) . Then given a de c or ate d ϵ -ﬁbr ation p : 𝔸 ⋄ → 𝔻 ♯♯ , the c anonic al map colim I ( p ∗ dec ◦ F ) → p ∗ dec (colim I F ) is an e quivalenc e. Pr o of. Let F ≤ 1 (1) − → F ← − F (2) b e the cospan asso ciated to F (here we are using the description of DCat ( ∞ , 2) giv en in ( 2.1 )) and consider the following commutativ e diagram, where the front part computes p ∗ dec (colim I F ) by our assumption on this colimit: colim I F ≤ 1 (1) colim I F colim I F (2) p ≤ 1 , ∗ (1) (colim I F (1) ) ≤ 1 p ∗ (colim I F ) p ∗ (2) (colim I F (2) ) 𝔻 ≤ 1 𝔻 𝔻 𝔸 ≤ 1 (1) 𝔸 𝔸 (2) p ≤ 1 (1) p p (2) Since p is a decorated ϵ -ﬁbration, the functors p ≤ 1 (1) , p and p (2) are all ϵ -ﬁbrations, and so pullback along them preserve colimits b y Theorem 2.4.10 . The front of the diagram will therefore describ e colim I p ∗ F after lo calizing it to DCat ( ∞ , 2) / 𝔸 ⋄ . Ho wev er, in this case no lo calization is required, so this completes the pro of. □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 77 Lemma 4.5.4. L et p : 𝔸 ⋄ → 𝔻 ♯♯ b e a de c or ate d ϵ -ﬁbr ation. Then for every de c o- r ate d functor q : 𝕏 ⋄ → 𝔸 ⋄ the pr eshe af DCat ( ∞ , 2) / 𝔸 ⋄ ( p ∗ dec ( – ) , 𝕏 ⋄ ) : DCat ( ∞ , 2) / 𝔻 ♯♯ → S , is r epr esentable. Pr o of. W e consider the p oint wise monomorphism of spaces (cf. Proposition 2.6.6 ) Φ : DCat ( ∞ , 2) / 𝔸 ⋄ ( p ∗ dec ( – ) , 𝕏 ⋄ ) → Cat ( ∞ , 2) / 𝔸 ( p ∗ ( – ) , 𝕏 ) ≃ Cat ( ∞ , 2) / 𝔻 ( – , p ∗ 𝕏 ) , where the ﬁnal isomorphism follo ws from the fact that the underlying functor of p is an ordinary ϵ -ﬁbration together with Theorem 2.4.10 . Let p dec , ∗ 𝕏 b e the sub- ( ∞ , 2) -category of p ∗ 𝕏 whose ob jects, morphisms and 2-morphisms are given by decorated functors 𝕋 ♭♭ × 𝔻 ⋄ 𝔸 ⋄ → 𝕏 ⋄ o ver 𝔸 ⋄ with 𝕋 ∈ G = { [0] , [1] , C 2 } , resp ectively . T o see that this deﬁnition do es in fact give a sub- ( ∞ , 2) -category, w e need to verify that these classes of i -morphisms are stable under comp osition in p ∗ 𝕏 . This can b e v eriﬁed by chec king that p ∗ dec preserv es certain colimits of elements in G . It is immediate to verify that those colimits satisfy the h yp othesis of Lemma 4.5.3 . This shows p dec , ∗ 𝕏 is a sub- ( ∞ , 2) -category of p ∗ 𝕏 . No w w e deﬁne the decorations on p dec , ∗ 𝕏 : W e deﬁne p dec , ∗ 𝕏 ⋄ ( i ) for i = 1 , 2 to b e sub- ( ∞ , 2) -category of p ∗ 𝕏 whose ob jects, morphisms and 2-morphisms are giv en b y decorated functors ov er 𝔸 ⋄ of the form 𝕋 ⋄ × 𝔻 ⋄ 𝔸 ⋄ → 𝕏 ⋄ with 𝕋 ⋄ ∈    [0] , [1] ♯ , C ♯♭ 2 , i = 1 , [0] , [1] ♭ , C ♭♯ 2 , i = 2 , resp ectiv ely . The same argumen t as b efore shows that the deﬁnition ab o v e yields sub- ( ∞ , 2) -categories as desired. Moreov er, we obtain canonical functors p ∗ 𝕏 ⋄ (1) − → p ∗ 𝕏 ← − p ∗ 𝕏 ⋄ (2) W e observe that since [0] ♭♭ = [0] ♯♭ = [0] ♭♯ b oth functors ab o v e induce equiv alences on underlying spaces. W e further observe that [1] ♭♭ = [1] ♭♯ and so the right-most morphism is an equiv alence on underlying ∞ -categories. Finally , we note that C ♯♭ 2 ≃ C ♭♭ 2 ⨿ ∂ C ♭♭ 2 ∂ C ♯♭ 2 and that this colimit again satisﬁes the hypothesis of Lemma 4.5.3 , whic h implies that the left-most map of the span ab o ve is lo cally full. T o ﬁnish the pro of, we note that we hav e a p oin t wise monomorphism of spaces DCat ( ∞ , 2) / 𝔻 ♯♯ ( − , p dec , ∗ 𝕏 )  → Cat ( ∞ , 2) / 𝔻 ( − , p ∗ 𝕏 ) whose image agrees with that of Φ . W e conclude that p dec , ∗ 𝕏 is the desired representing ob ject. □ Pr o of of Pr op osition 4.5.2 . By Lemma 4.5.4 , the functor p ∗ dec admits a right adjoint at the lev el of underlying ∞ -categories. As p ∗ dec is Cat ∞ -linear, we can promote p ∗ dec to an ( ∞ , 2) -categorical left adjoint using Corollary 2.1.6 as in the pro of of Theorem 2.4.10 . □ 78 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Observ ation 4.5.5. Supp ose p : 𝔸 ⋄ → 𝔹 ⋄ is exp onentiable. Then for q : 𝔼 ⋄ → 𝔸 ⋄ , the ﬁbre of p dec , ∗ ( q ) at b ∈ 𝔹 is 𝔻𝔽 un / 𝔸 ⋄ ( 𝔸 ⋄ b , 𝔼 ⋄ ) . Indeed, for a decorated ( ∞ , 2) - category 𝕂 ⋄ , we hav e a natural equiv alence Map( 𝕂 ⋄ , ( p dec , ∗ ( q )) b ) ≃ Map / 𝔹 ⋄ ( 𝕂 ⋄ × { b } , p dec , ∗ ( q )) ≃ Map / 𝔸 ⋄ ( 𝕂 ⋄ × 𝔸 ⋄ b , 𝔼 ⋄ ) ≃ Map( 𝕂 ⋄ , 𝔻𝔽 un / 𝔸 ⋄ ( 𝔸 ⋄ b , 𝔼 ⋄ )) . More generally , given a pullback square ℂ ⋄ 𝔸 ⋄ 𝔻 ⋄ 𝔹 ⋄ α q p β where p and q are exp onentiable, we hav e a Beck–Chev alley equiv alence β ∗ p ∗ ≃ q ∗ α ∗ , since the corresp onding mate transformation of left adjoints is ob viously an equiv- alence. Deﬁnition 4.5.6. A decorated functor p : 𝔸 ⋄ → 𝔻 ⋄ is said to b e ϵ - smo oth if p is exp onen tiable, and given a commutativ e diagram 𝕊 ′⋄ 𝕋 ′⋄ 𝔸 ⋄ 𝕊 ⋄ 𝕋 ⋄ 𝔹 ⋄ φ ′ φ where b oth squares are pullbacks and ϕ is an ϵ -coﬁbration (cf. Deﬁnition 3.6.2 ), then ϕ ′ is again an ϵ -coﬁbration. Lemma 4.5.7. Supp ose p : 𝔸 ⋄ → 𝔹 ⋄ is ϵ -smo oth. Then (i) The pul lb ack functor p ∗ dec : 𝔻ℂ at ( ∞ , 2) / 𝔹 ⋄ → 𝔻ℂ at ( ∞ , 2) / 𝔸 ⋄ pr eserves ϵ -e quivalenc es. (ii) The pushforwar d functor p dec , ∗ : 𝔻ℂ at ( ∞ , 2) / 𝔸 ⋄ → 𝔻ℂ at ( ∞ , 2) / 𝔹 ⋄ pr eserves p artial ϵ -ﬁbr ations. (iii) The adjunction p ∗ dec ⊣ p dec , ∗ r estricts on ful l sub c ate gories to an adjunction p ∗ dec : ℙ𝔽 ib ϵ / 𝔹 ⋄ ⇄ ℙ𝔽 ib / 𝔸 ⋄ : p dec , ∗ . Pr o of. The ﬁrst part is immediate from the deﬁnition of smo othness, and implies the second by adjunction. It follo ws that b oth of the functors p ∗ dec and p dec , ∗ preserv e partial ϵ -ﬁbrations, so the adjunction restricts as claimed in the ﬁnal part. □ Theorem 4.5.8. L et p : 𝔸 ⋄ → 𝔹 ♯♯ b e a de c or ate d ϵ -ﬁbr ation. Then p is ϵ -smo oth. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 79 Remark 4.5.9. The cases of Prop osition 4.5.2 and Theorem 4.5.8 where p is an ϵ - ﬁbration (without additional decorations) can b e extracted from Theorem 3.2.3.15 and Prop osition 3.2.5.9 of [ Lou24 ], resp ectively . Loubaton also allows the marking on the target to b e non-maximal, so his result also cov ers some cases we do not consider. F or the pro of we need the following technical input: Prop osition 4.5.10. Supp ose that we ar e given pul lb ack diagr ams of de c or ate d ( ∞ , 2) -c ate gories 𝔸 ⋄ 𝔸 ♦ 𝕆 n, ⋄ ( 𝕆 n ) ♯♯ , p p 𝔹 ⋄ 𝔹 ♦ 𝕆 n, † + ( 𝕆 n + ) ♯♯ , q q wher e p and q ar e de c or ate d (1 , 0) -ﬁbr ations and we use the notation of Deﬁni- tion 3.6.13 . Then we have pushout squar es in 𝔻ℂ at ( ∞ , 2) 𝔸 ⋄ n × 𝕆 [0 ,n − 1] ,♭♭ 𝔸 ⋄ n × 𝕆 n, ⋄ 𝔸 ⋄ × 𝕆 n, ⋄ 𝕆 [0 ,n − 1] ,♭♭ 𝔸 ⋄ , 𝔹 ⋄ n × 𝕆 [0 ,n − 1] , † + 𝔹 ⋄ n × 𝕆 n, † 𝔹 ⋄ × 𝕆 n, † + 𝕆 [0 ,n − 1] , † 𝔹 ⋄ , wher e 𝔸 ⋄ n and 𝔹 ⋄ n denote the c orr esp onding ﬁbr es over the obje ct n ∈ 𝕆 n . Pr o of. Let f ⋄ : P ⋄ n → 𝔸 ⋄ and g ⋄ : Q ⋄ n → 𝔹 ⋄ denote corresp onding functors out of the pushout. Inv oking [ AS23a , Corollary 3.73, Remark 3.74] w e see that the functors f ⋄ and g ⋄ are equiv alences on underlying ( ∞ , 2) -categories. Therefore our question reduces to verifying that the decorations on both sides agree. First, we observe that P ⋄ n and Q ⋄ n con tain all of the decorations of 𝔸 ⋄ and 𝔹 ⋄ that live in the ﬁbre ov er a p oin t i ∈ 𝕆 n . W e pro ceed case by case. ▶ Let u : x → y b e an edge in 𝔸 ⋄ (1) living ov er n − 1 → n . Then w e can express u = β ◦ α where β is a decorated morphism in the ﬁbre ov er n − 1 and α is a decorated edge that factors through P ⋄ n . W e conclude that u also factors through P ⋄ n . The analogous claim for 𝔹 ⋄ is immediate. ▶ Let u, v : x → y b e morphisms in 𝔸 ⋄ that live ov er the morphisms { i, n } and { i, n − 1 , n } , resp ectively , and consider θ : u → v in 𝔸 ⋄ (2) . W e consider the map ϕ : { y } × 𝕆 2 → 𝔸 n × 𝕆 2 → 𝔸 and denote b y ϕ (0) = ˆ x and by ˆ θ the corresp onding map C 2 → 𝔸 . Then it follo ws that there exists a map ι : x → ˆ x such that the whiskering ι ◦ ˆ θ equals θ . Since 𝔸 ⋄ 2 → 𝕆 n, ⋄ (2) is a ﬁbration, we can choose ˆ θ to factor through 𝔸 ⋄ (2) and so the result holds. □ Pr o of of The or em 4.5.8 . W e let  = (0 , 1) without loss of generality . The pro of no w follo ws as a combination of Prop osition 4.5.2 , Prop osition 3.6.19 and Prop osi- tion 4.5.10 . □ The following corollary now follows directly from Theorem 4.5.8 and Lemma 4.5.7 . 80 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Corollary 4.5.11. L et p : 𝔸 ⋄ → 𝔹 ♯♯ b e a de c or ate d ϵ -ﬁbr ation. Then the adjunction p ∗ dec ⊣ p dec , ∗ on slic es of 𝔻ℂ at ( ∞ , 2) r estricts on ful l sub c ate gories to an adjunction p ∗ dec : 𝔽 ib ϵ / 𝔹 → ℙ𝔽 ib ϵ / 𝔸 ⋄ : p dec , ∗ . □ Observ ation 4.5.12. Since the left adjoint is Cat ( ∞ , 2) -linear, the adjunction of ( ∞ , 2) -categories ab ov e can ev en b e upgraded to an adjunction of ( ∞ , 3) -categories using a v arian t of Corollary 2.1.6 . As a useful sp ecial case, we ha ve: Corollary 4.5.13. L et q : ( 𝔸 , I ) → 𝔹 ♯ b e a functor of marke d ( ∞ , 2) -c ate gories such that q ♯ is a de c or ate d ϵ -ﬁbr ation. Then the adjunction q ∗ dec ⊣ q dec , ∗ on slic es of 𝔻ℂ at ( ∞ , 2) r estricts on ful l sub c ate gories to an adjunction q ∗ dec : 𝔽 ib ϵ / 𝔹 → 𝔽 ib ϵ / ( 𝔸 ,I ) : q dec , ∗ . Pr o of. Since 𝔽 ib ϵ / ( 𝔸 ,I ) is a full sub category of ℙ𝔽 ib ϵ / ( 𝔸 ,I ) ♯ and the pullbac k functor q ∗ dec tak es v alues in this full sub category , this is immediate from Corollary 4.5.11 applied to q ♯ . □ 4.6. Pushforw ard of partial ﬁbrations. Our goal in this section is to generalize Corollary 4.5.11 and see that pullback of ﬁbrations along an arbitrary decorated functor f : ℂ ⋄ → 𝔻 ♯♯ has a right adjoint. As sp ecial cases this will pro duce righ t Kan extensions for functors to ℂ at ∞ , which will b e the starting p oin t for our discussion of Kan extensions b elow in § 5.6 , as w ell as c ofr e e ﬁbrations. Deﬁnition 4.6.1. Let f : ℂ ⋄ → 𝔻 ♯♯ b e a decorated functor. W e deﬁne a functor R f : ℙ𝔽 ib ϵ / ℂ ⋄ → ℙ𝔽 ib ϵ / 𝔻 ♯♯ = 𝔽 ib ϵ / 𝔻 as the comp osite ℙ𝔽 ib ϵ / ℂ ⋄ π ∗ − → ℙ𝔽 ib ϵ / ℚ ⋄ φ ∗ − − → ℙ𝔽 ib ϵ / 𝔻 ♯♯ where ℚ ⋄ := ℂ ⋄ × 𝔻 ♯♯ 𝔻𝔸 r ϵ -lax ( 𝔻 ♯♯ ) ⋄ -ﬁb , the ﬁrst functor is given by taking pullbacks along the pro jection π : ℚ ⋄ → ℂ ⋄ (see § 4.3 ) and the second functor is the right adjoint (see Corollary 4.5.11 ) to the pullbac k functor along ϕ = 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( f ) : ℚ ⋄ → 𝔻 ♯♯ , whic h is an ϵ -ﬁbration. Observ ation 4.6.2. By deﬁnition, R f is the restriction to full subcategories of partial ﬁbrations of the functor 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ π ∗ − → 𝔻ℂ at ( ∞ , 2) / ℚ ⋄ φ ∗ − − → 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ . This has a left adjoin t π ! ϕ ∗ giv en b y pullback along ϕ follow ed b y composition with π . This functor takes q : 𝔸 ⋄ → 𝔻 ♯♯ to the pro jection ℂ ⋄ × 𝔻 ♯♯ 𝔻𝔸 r ϵ -lax ( 𝔻 ♯♯ ) ⋄ -ﬁb × 𝔻 ♯♯ 𝔸 ⋄ → ℂ ⋄ , FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 81 whic h is the pullback to ℂ ⋄ of the functor 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( q ) : 𝔻𝔸 r ϵ -lax ( 𝔻 ♯♯ ) ⋄ -ﬁb × 𝔻 ♯♯ 𝔸 ⋄ → 𝔻 ♯♯ , so that π ! ϕ ∗ ≃ f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯ . Observ ation 4.6.3. Consider a partial ϵ -ﬁbration q : 𝔼 ♮ → ℂ ⋄ . By Observ a- tion 4.5.5 , in the situation ab ov e we can identify the ﬁbres of R f ( q ) as R f ( q ) d ≃ 𝔻𝔽 un / ℚ ⋄ ( ℚ ⋄ d , π ∗ 𝔼 ♮ ) ≃ 𝔻𝔽 un / ℂ ⋄ ( ℚ ⋄ d , 𝔼 ♮ ) . Using the notation from 4.4.1 , we get R f ( q ) d ≃                𝔻𝔽 un / ℂ ⋄ ( ℂ ⋄ d → , 𝔼 ♮ ) , ϵ = (0 , 1) , 𝔻𝔽 un / ℂ ⋄ ( ℂ ⋄ → d , 𝔼 ♮ ) , ϵ = (1 , 0) , 𝔻𝔽 un / ℂ ⋄ ( ℂ ⋄ d → , 𝔼 ♮ ) , ϵ = (0 , 0) , 𝔻𝔽 un / ℂ ⋄ ( ℂ ⋄ → d , 𝔼 ♮ ) , ϵ = (1 , 1) , . Theorem 4.6.4. L et f : ℂ ⋄ → 𝔻 ♯♯ b e a de c or ate d functor. Then ther e exists an adjunction of ( ∞ , 2) -c ate gories f ∗ : 𝔽 ib ϵ / 𝔻 − → ← − ℙ𝔽 ib ϵ / ℂ ⋄ : R f wher e f ∗ denotes the pul lb ack functor. Pr o of. Let q : 𝔸 ⋄ → 𝔻 ♯♯ b e an ϵ -ﬁbration. Then for every partial ϵ -ﬁbration p : 𝕏 ⋄ → ℂ ⋄ , we hav e from Observ ation 4.6.2 a natural equiv alence 𝔽 ib ϵ / 𝔻 ( q , R f ( p )) ≃ 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯  ( q ) , p ) . Moreo ver, by pulling back the unit map in Prop osition 4.3.1 , we obtain a morphism χ f : f ∗ q → f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( q ) , whic h is an ϵ -equiv alence by Prop osition 4.3.4 . In other w ords, restriction along χ f yields an equiv alence 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( q ) , p ) ≃ − → 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( f ∗ q , p ) ≃ ℙ𝔽 ib ϵ / ℂ ⋄ ( f ∗ q , p ) . The result now follows. □ W e note tw o imp ortan t sp ecial cases: F or any functor of ( ∞ , 2) -categories f : 𝔸 → 𝔹 , we get: Corollary 4.6.5. The pul lb ack functor f ∗ : 𝔽 ib ϵ / 𝔹 → 𝔽 ib ϵ / 𝔸 has a right adjoint R f . □ T ogether with our explicit description of this right adjoint, this will b e the start- ing p oint for our discussion of Kan extensions for ( ∞ , 2) -categories below in § 5.6 . The following v ariant will similarly lead to a notion of p artial ly lax Kan extensions: 82 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Corollary 4.6.6. Supp ose f : ( ℂ , I ) → 𝔻 ♯ is a functor of marke d ( ∞ , 2) -c ate gories. Then the pul lb ack functor f ∗ : 𝔽 ib ϵ / 𝔻 → 𝔽 ib ϵ / ( ℂ ,I ) has a right adjoint R f . Pr o of. W e can identify 𝔽 ib ϵ / ( ℂ ,I ) as the full sub- ( ∞ , 2) -category of ℙ𝔽 ib ϵ / ( ℂ ,I ) ♯ spanned b y the ϵ -ﬁbrations. Since the pullback functor from 𝔽 ib ϵ / 𝔻 tak es v alues in this full sub category , the right adjoint from Theorem 4.6.4 restricts to a righ t adjoint of f ∗ , as required. □ On the other hand, taking the identit y as a functor 𝔹 ♭♭ → 𝔹 ♯♯ w e get the existence of cofree ﬁbrations: Corollary 4.6.7. The for getful functor 𝔽 ib ϵ / 𝔹 → ℂ at ( ∞ , 2) / 𝔹 has a right adjoint. □ More generally , for any decorated ( ∞ , 2) -category 𝔹 ⋄ w e get a right adjoint to the forgetful functor 𝔽 ib ϵ / 𝔹 → ℙ𝔽 ib ϵ / 𝔹 ⋄ . F or later use, it will b e useful to also identify the unit and counit of the adjunction of Theorem 4.6.4 a bit more explicitly: Observ ation 4.6.8. Let f : ℂ ⋄ → 𝔻 ♯♯ b e a decorated functor and let q : 𝕏 ⋄ → 𝔻 ♯♯ b e an ϵ -ﬁbration. Then the map f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( q ) → f ∗ q , obtained b y pulling bac k the counit of the adjunction in Prop osition 4.3.1 , corresp onds via the adjunction of Observ ation 4.6.2 to a map  : q → R f ( f ∗ q ) . Note that η is sent under the equiv alence 𝔽 ib ϵ / 𝔻 ( q , R f ( f ∗ q )) ≃ − → ℙ𝔽 ib ϵ / ℂ ⋄ ( f ∗ q , f ∗ q ) to the iden tit y on f ∗ q , which identiﬁes  as the unit of the adjunction in Theo- rem 4.6.4 . Given p ∈ 𝔽 ib ϵ / 𝔻 , we can apply the previous discussion, together with the fact that forgetful functor in Prop osition 4.3.1 is fully faithful, to identify the comp osite 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ ( p, q ) η ∗ − → 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ ( p, R f ( f ∗ q )) ≃ − → 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( f ∗ p, f ∗ q ) , with the pullback functor along f . Moreov er, we can now obtain a comm utativ e diagram (4.5) 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ ( p, q ) 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ ( p, R f ( f ∗ q )) 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ ( f ! f ∗ p, q ) 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( f ∗ p, f ∗ q ) . η ∗ ξ ∗ ≃ ≃ where ξ ∗ is the unit of f ! ⊣ f ∗ . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 83 Observ ation 4.6.9. Let f , q b e as b efore, and let p ∈ 𝔻ℂ at ( ∞ , 2) / 𝔻 ♯♯ . W e consider the commutativ e diagram (4.6) 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( p, f ∗ R f ( q )) 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( p, q ) 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( f ! p ) , q ) 𝔻ℂ at ( ∞ , 2) / ℂ ⋄ ( p, q ) ≃ ≃ ψ ∗ where the b ottom is given by precomp osition along the morphism ψ : p − → f ∗ f ! p − → f ∗ 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( f ! p ) where the ﬁrst maps is the unit of f ! ⊣ f ∗ and the second is the pullback of the unit of the adjunction in Prop osition 4.3.1 . Note that this diagram is natural in p , so by the Y oneda lemma w e obtain a map  : f ∗ R f ( q ) → q which we claim can b e iden tiﬁed with the counit of the adjunction in Theorem 4.6.4 . In order to do so, we claim that the comp osite 𝔽 ib ϵ / 𝔻 ( p, R f ( q )) − → ℙ𝔽 ib ϵ / ℂ ⋄ ( f ∗ p, f ∗ R f ( q )) ϵ ∗ − → ℙ𝔽 ib ϵ / ℂ ⋄ ( f ∗ p, q ) can b e identiﬁed with the natural isomorphism in the pro of of Theorem 4.6.4 . The result follows from the commutativit y of the diagram f ∗ p f ∗ f ! f ∗ p f ∗ 𝔻𝔽 ree ϵ / ℂ ⋄ ( f ! f ∗ p ) f ∗ p f ∗ 𝔻𝔽 ree ϵ / ℂ ⋄ ( p ) ≃ after some minor unrav eling of the deﬁnitions. 4.7. Lo calization of ﬁbrations. Giv en a functor of ∞ -categories F : C → MCat ∞ , Hinic h show ed in [ Hin16 , §2] that we can identify the co cartesian ﬁbration for the functor τ m ( F ) (where we inv ert the marked morphisms) as a lo calization of the ﬁbration for the underlying functor u m ( F ) : C → Cat ∞ ; an alternativ e pro of of this v ery useful comparison has also b een given by Nikolaus and Scholze [ NS18 , Prop o- sition A.14]. Our goal in this section is to pro ve the following ( ∞ , 2) -categorical extension of this result; in fact, w e will see that our results on cofree ﬁbrations allo w us to prov e this by a m uc h easier argumen t than the existing ones in the ∞ -categorical case. Theorem 4.7.1. Supp ose 𝔼 ⋄ → 𝔹 ♯♯ is a de c or ate d (0 , 1) -ﬁbr ation, c orr esp onding to a functor F : 𝔹 → 𝔻ℂ at ( ∞ , 2) . If 𝔼 ⋄ | 𝔹 ♭♭ denotes the pul lb ack to 𝔹 ♭♭ (i.e. we ke ep only the de c or ations in 𝔼 ⋄ that map to e quivalenc es), then the lo c alization τ d ( 𝔼 ⋄ | 𝔹 ♭♭ ) → 𝔹 is a (0 , 1) -ﬁbr ation, and c orr esp onds to the functor τ d ( F ) : 𝔹 → ℂ at ( ∞ , 2) . W e’ll derive this as a sp ecial case of the following more general observ ation: Prop osition 4.7.2. Supp ose p : 𝔼 ⋄ → 𝔹 ♯♯ is a de c or ate d (0 , 1) -ﬁbr ation, c orr e- sp onding to a functor F : 𝔹 → 𝔻ℂ at ( ∞ , 2) , and let 𝔼 ′ → 𝔹 b e the (0 , 1) -ﬁbr ation for τ d ( F ) . Then for any de c or ate d functor f : ℂ ⋄ → 𝔹 ♯♯ , the pul lb ack of the c anonic al map η : 𝔼 ⋄ → 𝔼 ′ ♮ induc es an e quivalenc e DF un / ℂ ⋄ ( f ∗ 𝔼 ′ ♮ , ℚ ⋄ ) ∼ − → DFun / ℂ ⋄ ( f ∗ 𝔼 ⋄ , ℚ ⋄ ) 84 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI for any p artial (0 , 1) -ﬁbr ation ℚ ⋄ → ℂ ⋄ . In other wor ds, f ∗ η exhibits f ∗ 𝔼 ′ ♮ as the lo c alization L (0 , 1) ℂ ⋄ ( f ∗ 𝔼 ⋄ ) to a p artial (0 , 1) -ﬁbr ation. Pr o of. Using the adjunction f ∗ : 𝔽 ib (0 , 1) / 𝔹 ♯♯ ⇄ ℙ𝔽 ib (0 , 1) / ℂ ⋄ : R f from Theorem 4.6.4 and (decorated) straigh tening, we get natural equiv alences of ∞ -categories DF un / ℂ ⋄ ( f ∗ 𝔼 ′ ♮ , ℚ ⋄ ) ≃ DF un / 𝔹 ♯♯ ( 𝔼 ′ ♮ , R f ( ℚ ⋄ ) ♮ ) ≃ Nat 𝔹 , ℂ at ( ∞ , 2) ( τ d ( F ) , Q ) ≃ Nat 𝔹 , 𝔻ℂ at ( ∞ , 2) ( F , Q ♭♭ ) ≃ 𝔻𝔽 ib (0 , 1) / 𝔹 ♯♯ ( 𝔼 ⋄ , R f ( ℚ ⋄ ) ♮ ) , where Q denotes the straightening of R f ( ℚ ⋄ ) . Now we observe that for ℙ → 𝔹 a (0 , 1) -ﬁbration, morphisms of decorated ﬁbrations 𝔼 ⋄ → ℙ ♮ are just morphisms of decorated ( ∞ , 2) -categories ov er 𝔹 ♯♯ (as an y decorated functor to ℙ ♮ in particular preserv es cocartesian morphisms and cartesian 2-morphisms). W e can therefore use Observ ation 4.6.2 to get an equiv alence 𝔻𝔽 ib (0 , 1) / 𝔹 ♯♯ ( 𝔼 ⋄ , R f ( ℚ ⋄ ) ♮ ) ≃ DF un / 𝔹 ♯♯ ( 𝔼 ⋄ , R f ( ℚ ⋄ ) ♮ ) ≃ DF un / ℂ ⋄ ( f ∗ 𝔻𝔽 ree (0 , 1) 𝔹 ♯♯ ( p ) , ℚ ⋄ ) . No w w e can use that the unit map p → 𝔻𝔽 ree (0 , 1) 𝔹 ♯♯ ( p ) pulls bac k to a (0 , 1) - equiv alence along any map by Prop osition 4.3.4 ; since ℚ ⋄ is a partial (0 , 1) -ﬁbration this gives an equiv alence DF un / ℂ ⋄ ( f ∗ 𝔻𝔽 ree (0 , 1) 𝔹 ♯♯ ( p ) , ℚ ⋄ ) ≃ DF un / ℂ ⋄ ( f ∗ 𝔼 ⋄ , ℚ ⋄ ) , as required. □ Pr o of of The or em 4.7.1 . Let 𝔼 ′ → 𝔹 b e the (0 , 1) -ﬁbration for τ d ( F ) ; we will prov e that it has the universal prop erty of the lo calization τ d ( 𝔼 ⋄ | 𝔹 ♭♭ ) : Applying Prop osi- tion 4.7.2 to the iden tity functor 𝔹 ♭♭ → 𝔹 ♯♯ , and noting that a functor to 𝔹 ♭♭ is a partial (0 , 1) -ﬁbration if and only if it is of the form ℂ ♭♭ → 𝔹 ♭♭ , w e get a natural equiv alence DF un / 𝔹 ♭♭ ( 𝔼 ⋄ | 𝔹 ♭♭ , ℂ ♭♭ ) ≃ DF un / 𝔹 ♭♭ ( 𝔼 ′ ♭♭ , ℂ ♭♭ ) , since 𝔼 ′ ♮ | 𝔹 ♭♭ is decorated by the co cartesian morphisms and cartesian 2-morphisms that lie ov er equiv alences, i.e. by the equiv alences. F or an ( ∞ , 2) -category 𝔻 we therefore hav e natural equv alences F un ( 𝔼 ′ , 𝔻 ) ≃ F un / 𝔹 ( 𝔼 ′ , 𝔻 × 𝔹 ) ≃ DF un / 𝔹 ♭♭ ( 𝔼 ′ ♭♭ , ( 𝔻 × 𝔹 ) ♭♭ ) ≃ DF un / 𝔹 ♭♭ ( 𝔼 ⋄ | 𝔹 ♭♭ , ( 𝔻 × 𝔹 ) ♭♭ ) ≃ DF un ( 𝔼 ⋄ | 𝔹 ♭♭ , 𝔻 ♭♭ ) ≃ F un ( τ d ( 𝔼 ⋄ | 𝔹 ♭♭ ) , 𝔻 ) , It follows that we hav e an equiv alence τ d ( 𝔼 ⋄ | 𝔹 ♭♭ ) ≃ 𝔼 ′ , as required. □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 85 Corollary 4.7.3. (1) The left adjoint to the inclusion 𝔽 ib (0 , 1) / 𝔹  → 𝔻𝔽 ib (0 , 1) / 𝔹 ♯♯ sends 𝔼 ⋄ → 𝔹 ♯♯ to τ d ( 𝔼 ⋄ | 𝔹 ♭♭ ) → 𝔹 . (2) The left adjoint L (0 , 1) 𝔹 ♯♯ to the inclusion 𝔽 ib (0 , 1) / 𝔹  → 𝔻ℂ at ( ∞ , 2) / 𝔹 ♯♯ sends ℂ ⋄ → 𝔹 ♯♯ to τ d ( 𝔻𝔽 ree (0 , 1) 𝔹 ♯♯ ( ℂ ⋄ ) | 𝔹 ♭♭ ) . (3) The left adjoint to the inclusion 𝟙𝔽 ib (0 , 1) / 𝔹  → 𝕄𝔽 ib (0 , 1) / 𝔹 ♯ sends ( 𝔼 , S ) → 𝔹 ♯ to τ d (( 𝔼 , S ) ♯ | 𝔹 ♭♭ ) → 𝔹 , wher e we lo c alize the morphisms fr om S and al l 2-morphisms in e ach ﬁbr e. (4) The left adjoint L (0 , 1);1 𝔹 ♯ to the inclusion 𝟙𝔽 ib (0 , 1) / 𝔹  → 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ sends ( ℂ , S ) → 𝔹 ♯ to τ d ( 𝔻𝔽 ree (0 , 1) 𝔹 ♯♯ (( ℂ , S ) ♯ | 𝔹 ♭♭ )) . Pr o of. The ﬁrst statement follo ws from Corollary 3.2.13 together with Theorem 4.7.1 , and in (2) the righ t adjoin t factors through 𝔻𝔽 ib (0 , 1) / 𝔹 ♯♯ , so that the left adjoin t factors as the decorated free ﬁbration from Prop osition 4.3.1 follow ed by the left adjoint from (1). The last statemen ts follow from restricting the ﬁrst t w o to the mark ed case. □ 5. (Co)limits and Kan extensions in ( ∞ , 2) -ca tegories In this section w e will apply our results on ﬁbrations from the previous section to dev elop the theory of (co)limits and Kan extensions for ( ∞ , 2) -categories. In § 5.1 w e introduce one natural notion of (co)limits in ( ∞ , 2) -categories, the (partially) lax and oplax (co)limits, and compare some deﬁnitions of these. W e then introduce another notion of (co)limits, the weighte d (co)limits, in § 5.2 ; w e prov e that these t yp es of (co)limits are equiv alen t, in the sense that any weigh ted (co)limit can b e reexpressed as a partially lax or oplax (co)limit, and vice v ersa. In § 5.3 w e then lo ok at such (co)limits in ℂ at ( ∞ , 2) and ℂ at ∞ , and sho w that these hav e simple descriptions in terms of ﬁbrations. Next w e collect some useful results on the existence and preserv ation of (co)limits in § 5.4 . In § 5.5 we will then see that the formalism of decorated ( ∞ , 2) -categories and our w ork on free ﬁbrations make it easy to understand coﬁnal functors of marked ( ∞ , 2) -categories. After this w e turn to Kan extensions of ( ∞ , 2) -categories in § 5.6 , and then set up a Bousﬁeld–Kan form ula for weigh ted colimits in § 5.7 . Finally , w e giv e a ﬁbrational pro of of a functorial version of the Y oneda lemma and use this to prov e that presheav es of ∞ -categories gives the free co completion of an ( ∞ , 2) -category in § 5.8 , and then apply this to compare some characterizations of presentable ( ∞ , 2) -categories in § 5.9 . 86 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI 5.1. Lax and oplax (co)limits. Partially (op)lax (co)limits were ﬁrst introduced for strict 2-categories by Descotte, Dubuc, and Szyld in [ DDS18 ] under the name σ -(c o)limits , and ha v e previously b een studied in the ∞ -categorical context by Berman [ Ber24 ] for diagrams in ℂ at ∞ indexed by an ∞ -category, and in greater generalit y by the ﬁrst author in [ AG22 ] and by Gagna–Harpaz–Lanari in [ GHL25 ]. In this section we will ﬁrst deﬁne partially (op)lax colimits in ( ∞ , 2) -categories b y a universal prop ert y in terms of partially (op)lax transformations, and derive sev eral alternative characterizations; in particular, we will see that our deﬁnition is equiv alent to that of [ GHL25 ]. Notation 5.1.1. W e denote by ( − ) : ℂ → 𝔽 un ( 𝕀 , ℂ ) the diagonal functor, i.e. the one that transp oses to the pro jection 𝕀 × ℂ → ℂ on to the second factor. W e call this functor the c onstant diagr am functor. Deﬁnition 5.1.2. Let ( 𝕀 , E ) b e a marked ( ∞ , 2) -category and F : 𝕀 → ℂ a functor of ( ∞ , 2) -categories. W e say that an ob ject of ℂ is: ▶ the E -(op)lax limit of F , denoted b y lim E - (op)lax 𝕀 F , if there exists an equiv a- lence of functors ℂ ( c, lim E - (op)lax 𝕀 F ) ≃ Nat E - (op)lax 𝕀 , ℂ ( c, F ) that is natural in c ; ▶ the E -(op)lax c olimit of F , denoted b y colim E - (op)lax 𝕀 F , if there exists an equiv- alence of functors ℂ (colim E - (op)lax 𝕀 F , c ) ≃ Nat E - (op)lax 𝕀 , ℂ ( F , c ) that is natural in c . In the minimally marked case ( 𝕀 , E ) = 𝕀 ♭ w e refer to E -(op)lax (co)limits just as (op)lax (c o)limits , denoted lim (op)lax 𝕀 F and colim (op)lax 𝕀 F . On the other hand, in the maximally marked case ( 𝕀 , E ) = 𝕀 ♯ w e refer to them as str ong 3 (co)limits, denoted lim 𝕀 F and colim 𝕀 F ; here there is no distinction b etw een lax and oplax, as b oth univ ersal prop erties are equiv alent to (co)represen ting the (co)preshea ves Nat 𝕀 , ℂ ( c, F ) and Nat 𝕀 , ℂ ( F , c ) . Lemma 5.1.3. F or a functor F : 𝕀 → ℂ and an obje ct x ∈ ℂ , the fol lowing ar e e quivalent: ▶ x is the E -lax c olimit of F in ℂ . ▶ x is the E -oplax limit of F op in ℂ op . ▶ x is the E -oplax c olimit of F co in ℂ co . ▶ x is the E -lax limit of F coop in ℂ coop . Pr o of. This follows from the deﬁnition and the equiv alences among lax and oplax functor ( ∞ , 2) -categories that arise from the order-reversing symmetry of the Gray tensor pro duct under b oth ( – ) op and ( – ) co (see [ AGH25 , Observ ation 2.2.11]). □ 3 W e will see later in Example 5.2.5 that these precisely corresp ond to c onic al colimits in the sense of enriched category theory . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 87 Observ ation 5.1.4. Suppose that ℂ admits all partially (op)lax (co)limits indexed b y a marked ( ∞ , 2) -category ( 𝕀 , E ) . Then it follows that taking E -(op)lax (co)limits giv e adjoin ts to the constant diagram functor ( − ) : ℂ → 𝔽 un ( 𝕀 , ℂ ) E - (op)lax : ( − ) : ℂ − → ← − 𝔽 un ( 𝕀 , ℂ ) E - (op)lax : lim E - (op)lax 𝕀 , colim E - (op)lax 𝕀 : 𝔽 un ( 𝕀 , ℂ ) E - (op)lax − → ← − ℂ : ( − ) . Con versely , the existence of such adjoints implies that ℂ has all partially (op)lax (co)limits indexed by ( 𝕀 , E ) . T o compare our deﬁnition of (op)lax (co)limits to that giv en in [ GHL25 ], w e ﬁrst need to derive a description of the ∞ -categories Nat E -(op)lax 𝔸 , 𝔹 ( F , G ) , for which we use the following notation: Deﬁnition 5.1.5. Let p : 𝔸 → ℂ , q : 𝔹 → ℂ b e ( i, j ) -ﬁbrations and let ( ℂ , E ) b e a mark ed ( ∞ , 2) -category. F or i = 0 we write Fun E -co c / ℂ ( 𝔸 , 𝔹 ) for the full sub category of F un / ℂ ( 𝔸 , 𝔹 ) spanned by functors that preserve the cocartesian 1-morphisms that lie o v er E as well as all j -cartesian 2-morphisms; for i = 1 we deﬁne Fun E -cart / ℂ ( 𝔸 , 𝔹 ) similarly . Prop osition 5.1.6. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory. F or functors F, G : 𝔸 → 𝔹 , we have a natur al e quivalenc e Nat E - lax 𝔸 , 𝔹 ( F , G ) ≃ F un ( E ) / 𝔸 × 𝔸 ( 𝔸 , ( F , G ) ∗ 𝔸 r oplax ( 𝔹 )) , wher e the right-hand side denotes the ∞ -c ate gory of se ctions that send the e dges in E to c ommutative squar es in ( F , G ) ∗ 𝔸 r oplax ( 𝔹 ) . Dual ly, we have an e quivalenc e in the E -oplax c ase Nat E - oplax 𝔸 , 𝔹 ( F , G ) op ≃ F un ( E ) / 𝔸 × 𝔸 ( 𝔸 , ( F , G ) ∗ 𝔸 r lax ( 𝔹 )) . In p articular, for an obje ct b ∈ 𝔹 we have: Nat E - lax 𝔸 , 𝔹 ( b, G ) ≃ F un E -co c / 𝔸 ( 𝔸 , G ∗ 𝔹 b → ) , Nat E - oplax 𝔸 , 𝔹 ( b, G ) op ≃ F un E -co c / 𝔸 ( 𝔸 , G ∗ 𝔹 b → ) , Nat E - lax 𝔸 , 𝔹 ( F , b ) ≃ F un E -cart / 𝔸 ( 𝔸 , F ∗ 𝔹 → b ) , Nat E - oplax 𝔸 , 𝔹 ( F , b ) op ≃ F un E -cart / 𝔸 ( 𝔸 , F ∗ 𝔹 → b ) . Pr o of. W e will ﬁrst deal with the E -lax case in full detail and then explain how to adapt the argument to derive the results in the E -oplax case. Note that w e hav e a natural equiv alence 𝔸 r oplax ( 𝔽 un ( 𝔸 , 𝔹 ) lax ) ≃ 𝔽 un ( 𝔸 , 𝔸 r oplax ( 𝔹 )) lax from [ AGH25 , Lemma 2.2.9]. Identifying Nat lax 𝔸 , 𝔹 ( F , G ) with the ﬁbre o ver F , G on the left, we get Nat lax 𝔸 , 𝔹 ( F , G ) ≃ 𝔽 un / 𝔸 × 𝔸 ( 𝔸 , ( F , G ) ∗ 𝔸 r oplax ( 𝔹 )) lax . W e claim that the right-hand side is in fact the ∞ -category Fun / 𝔸 × 𝔸 ( 𝔸 , ( F , G ) ∗ 𝔸 r oplax ( 𝔹 )) . T o see this it will suﬃce to show that for any commutativ e diagram [1] ⊗ 𝔸 ( F , G ) ∗ 𝔸 r oplax ( 𝔹 ) 𝔸 × 𝔸 , f p 𝔸 88 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI the map f factors through the cartesian pro duct [1] × 𝔸 , where p 𝔸 is giv en by the pro jection to 𝔸 and follow ed by the diagonal map. This follows from the fact that p 𝔸 factors through [1] × 𝔸 together with [ AGH25 , Prop osition 2.4.3]. T o deal with the E -lax case, we observ e that Nat E - lax 𝔸 , 𝔹 ( F , G ) is a full sub category of Nat lax 𝔸 , 𝔹 ( F , G ) . Chasing through the equiv alences ab o ve one sees that an E -lax natural transformation is identiﬁed with a section that sends the edges in E to a comm utative square in ( F , G ) ∗ 𝔸 r oplax ( 𝔹 ) . In the oplax case w e use the equiv alence 𝔸 r lax ( 𝕏 ) ≃ ( 𝔸 r oplax ( 𝕏 op )) op to identify the ﬁbre at ( x, y ) ∈ 𝕏 × 𝕏 as 𝕏 op ( y , x ) op ≃ 𝕏 ( x, y ) op . W e can therefore identify Nat oplax 𝔸 , 𝔹 ( F , G ) op as a ﬁbre in 𝔸 r lax ( 𝔽 un ( 𝔸 , 𝔹 ) oplax ) ≃ 𝔽 un ( 𝔸 , 𝔸 r lax ( 𝔹 )) oplax , and pro ceed as in the lax case. □ Corollary 5.1.7. F or F : 𝔸 → 𝔹 and b ∈ 𝔹 , we have natur al e quivalenc es Nat E - lax 𝔸 , 𝔹 ( b, F ) ≃ Nat E - lax 𝔸 , ℂ at ∞ ( ∗ , 𝔹 ( b, F ( – ))) , (5.1) Nat E - lax 𝔸 , 𝔹 ( F , b ) ≃ Nat E - oplax 𝔸 op , ℂ at ∞ ( ∗ , 𝔹 ( F ( – ) , b )) , (5.2) Nat E - oplax 𝔸 , 𝔹 ( b, F ) ≃ Nat E - oplax 𝔸 , ℂ at ∞ ( ∗ , 𝔹 ( b, F ( – ))) , (5.3) Nat E - oplax 𝔸 , 𝔹 ( F , b ) ≃ Nat E - lax 𝔸 op , ℂ at ∞ ( ∗ , 𝔹 ( F ( – ) , b )) . (5.4) Pr o of. F rom Prop osition 5.1.6 we hav e Nat E - lax 𝔸 , 𝔹 ( b, F ) ≃ Fun E -co c / 𝔸 ( 𝔸 , F ∗ 𝔹 oplax b → ) , where the right-hand side is a mapping ∞ -category in 𝔽 ib (0 , 1) / ( 𝔸 ,E ) . Under straighten- ing, the iden tity of 𝔸 corresp onds to the constan t functor ∗ and 𝔹 b → to 𝔹 ( b, – ) b y [ Lur09b , Prop osition 4.1.8]; the pullback along F therefore corresp onds to 𝔹 ( b, F ( – )) , as required. A totally analogous argument shows that equiv alence ( 5.2 ) also holds. W e now lo ok at Nat E - oplax 𝔸 , 𝔹 ( b, F ) op ≃ F un E -co c / 𝔸 ( 𝔸 , F ∗ 𝔹 b → ) and observe that by taking opp osite ∞ -categories we can pro duce an equiv alence F un E -co c / 𝔸 ( 𝔸 , F ∗ 𝔹 b → ) ≃ F un E -cart / 𝔸 op ( 𝔸 op , ( F ∗ 𝔹 b → ) op ) op . Finally w e see that ( F ∗ 𝔹 b → ) op is the (1 , 0) -ﬁbration that classiﬁes the functor 𝔹 ( b, F ( – )) , so from Theorem 2.4.12 we get F un E -cart / 𝔸 op ( 𝔸 op , ( F ∗ 𝔹 b → ) op ) op ≃ Nat E - oplax 𝔸 , ℂ at ∞ ( ∗ , 𝔹 ( b, F ( − ))) op , and the case ( 5.3 ) follows. The remaining veriﬁcation for ( 5.4 ) is prov ed by a dual argumen t. □ Corollary 5.1.8. F or F : 𝔸 → 𝔹 , the E -(op)lax (c o)limits of F ar e uniquely char- acterize d by the fol lowing (c o)r epr esentability pr op erties in terms of p artial ly (op)lax limits in ℂ at ∞ : ▶ 𝔹 ( b, lim E - lax 𝔸 F ) ≃ lim E - lax 𝔸 𝔹 ( b, F ) , ▶ 𝔹 (colim E - lax 𝔸 F , b ) ≃ lim E - oplax 𝔸 op 𝔹 ( F , b ) , ▶ 𝔹 ( b, lim E - oplax 𝔸 F ) ≃ lim E - oplax 𝔸 𝔹 ( b, F ) , FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 89 ▶ 𝔹 (colim E - oplax 𝔸 F , b ) ≃ lim E - lax 𝔸 op 𝔹 ( F , b ) . Pr o of. W e prov e the ﬁrst statement; the rest follow similarly . Here we hav e 𝔹 ( b, lim E - lax 𝔸 F ) ≃ Nat E - lax 𝔸 , 𝔹 ( b, F ) ≃ Nat E - lax 𝔸 , ℂ at ∞ ( ∗ , 𝔹 ( b, F )) ≃ F un ( ∗ , lim E - lax 𝔸 𝔹 ( b, F )) ≃ lim E - lax 𝔸 𝔹 ( b, F ) , where the ﬁrst and third equiv alences follo w from the deﬁnition of E -lax limits, the second from Corollary 5.1.7 , and the last is obvious. □ Com bining this with [ GHL25 , Corollary 5.1.7], we get the following comparison with the deﬁnitions of Gagna–Harpaz–Lanari: Corollary 5.1.9. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory. The E -(op)lax (c o)limits of a functor F : 𝔸 → 𝔹 as deﬁne d ab ove agr e e with the inner and outer (c o)limits of F deﬁne d in [ GHL25 ] as fol lows: 4 ▶ the E -lax limit of F is the inner limit of F , ▶ the E -oplax limit of F is the outer limit of F , ▶ the E -lax c olimit of F is the outer c olimit of F , ▶ the E -oplax c olimit of F is the inner c olimit of F . □ Remark 5.1.10. By Observ ation 4.2.10 , exhibiting an ob ject c ∈ ℂ as the E - oplax colimit of F : 𝕀 → ℂ amounts to exhibiting the (0 , 1) -ﬁbration ℂ E - oplax F → → ℂ as representable by an ob ject ov er c as in Prop osition 4.2.9 . By Lemma 2.5.6 w e can think of suc h an ob ject as a functor 𝕁 ▷ E -oplax → ℂ that satisﬁes the conditions of Prop osition 4.2.9 . Such partially lax cones and co cones can b e iden tiﬁed with the inner and outer (co)limit cones used in [ GHL25 , §6]. 5.2. W eigh ted (co)limits in ( ∞ , 2) -categories. In V -enric hed ( ∞ -)category theory the general notion of (co)limits is that of V - weighte d (co)limits. In this sec- tion we ﬁrst recall the deﬁnition of Cat ∞ -w eighted (co)limits in ( ∞ , 2) -categories and giv e an easy pro of of their description as partially (op)lax (co)limits from [ GHL25 ] using our new description of the latter. W e then sho w that an y partially (op)lax (co)limit can also b e expressed as an explicit weigh ted (co)limit. This al- lo ws us to conclude that the theories of partially lax, partially oplax, and weigh ted (co)limits are equiv alen t, so that they all give rise to the same notion of (co)complete ( ∞ , 2) -categories and (co)contin uous functors among these; w e thus provide a mo del- indep enden t treatment of the main results in [ GHL25 , §6.2]. Deﬁnition 5.2.1. F or F : 𝔸 → 𝔹 and W : 𝔸 → ℂ at ∞ , the W -weighte d limit of F , if it exists, is the ob ject lim W 𝔸 F of 𝔹 characterized by the universal prop erty 𝔹 ( b, lim W 𝔸 F ) ≃ Nat 𝔸 , ℂ at ∞ ( W , 𝔹 ( b, F )) . 4 Here the discrepancy in the pairing b etw een lax/oplax and inner/outer for limits and colimits is due to the switch from lax to oplax in ( 5.2 ), which does not o ccur in ( 5.1 ). 90 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Dually , for F : 𝔸 → 𝔹 and W : 𝔸 op → ℂ at ∞ , the W -weighte d c olimit of F , if it exists, is the ob ject colim W 𝔸 F characterized by the univ ersal prop ert y 𝔹 (colim W 𝔸 F , b ) ≃ Nat 𝔸 op , ℂ at ∞ ( W , 𝔹 ( F , b )) . Observ ation 5.2.2. F or a functor F : 𝔸 → ℂ at ∞ , we hav e F un ( C , lim W 𝔸 F ) ≃ Nat 𝔸 , ℂ at ∞ ( W , Fun ( C , F )) ≃ Nat 𝔸 , ℂ at ∞ ( W × C , F ) ≃ F un ( C , Nat 𝔸 , ℂ at ∞ ( W , F )) . By the Y oneda Lemma we can thus describ e weigh ted limits in ℂ at ∞ b y a natural equiv alence lim W 𝔸 F ≃ Nat 𝔸 , ℂ at ∞ ( W , F ) . W e can therefore also characterize the weigh ted (co)limits of a functor F : 𝔸 → 𝔹 b y natural equiv alences 𝔹 ( b, lim W 𝔸 F ) ≃ lim W 𝔸 𝔹 ( b, F ) for W : 𝔸 → ℂ at ∞ , and 𝔹 (colim W 𝔹 F , b ) ≃ lim W 𝔸 op 𝔹 ( F , b ) for W : 𝔸 op → ℂ at ∞ . Example 5.2.3. Giv en an ∞ -category A and an ob ject c in an ( ∞ , 2) -category ℂ , w e can consider the colimit of [0] c − → ℂ weigh ted by [0] A − → ℂ at ∞ . W e denote this by A ⊠ c := colim A [0] c ; its universal prop erty is that there is a natural equiv alence ℂ ( A ⊠ c, c ′ ) ≃ F un ( A , ℂ ( c, c ′ )) , so this is precisely the tensoring of c by A . Similarly , the weigh ted limit c A := lim A [0] c is the c otensoring of c by A , satisfying ℂ ( c ′ , c A ) ≃ F un ( A , ℂ ( c, c ′ )) . Observ ation 5.2.4. Supp ose X is an ∞ -group oid. Then for W : X → ℂ at ∞ and F : X → ℂ , we hav e colim W X F ≃ colim x ∈ X W ( x ) ⊠ F ( x ) , since by deﬁnition we hav e ℂ (colim W X F , c ) ≃ Nat X, ℂ at ∞ ( W , ℂ ( F , c )) ≃ lim x ∈ X F un ( W ( x ) , ℂ ( F ( x ) , c )) ≃ lim x ∈ X ℂ ( W ( x ) ⊠ F ( x ) , c ) ≃ ℂ (colim x ∈ X W ( x ) ⊠ F ( x ) , c ) , where the second equiv alence holds since 𝔽 un ( X , ℂ ) ≃ lim X ℂ . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 91 Example 5.2.5. Giv en a functor F : 𝕁 → ℂ we can alwa ys consider the weigh t ∗ : 𝕁 → ℂ at ∞ that is constant at the terminal ob ject. In this case the universal prop ert y of the c onic al limit lim ∗ 𝕁 F is the same as that of the strong limit lim 𝕁 F : there are natural equiv alences ℂ ( c, lim ∗ 𝕁 F ) ≃ Nat 𝕁 , ℂ at ∞ ( ∗ , ℂ ( c, F )) ≃ Nat 𝕁 , ℂ ( c, F ) ≃ ℂ ( c, lim 𝕁 F ) via Corollary 5.1.7 . In the case where F : J → ℂ is a functor from an ∞ -category, then we also hav e ℂ ( c, lim ∗ J F ) ≃ ≃ Nat J , ℂ ( c, F ) ≃ ≃ Nat J , ℂ ≤ 1 ( c, F ≤ 1 ) since 𝔽 un ( J , ℂ ) ≤ 1 ≃ F un ( J , ℂ ≤ 1 ) . Thus in this case the conical limit lim ∗ J F is also the ordinary limit of F viewed as a diagram in the underlying ∞ -category ℂ ≤ 1 . In particular, if we kno w that ℂ has a certain conical/strong limit indexed o ver an ∞ -category, to identify it we only hav e consider the limit in ℂ ≤ 1 . W e now prov e that we can express weigh ted limits as b oth lax and oplax limits; this was previously shown as [ GHL25 , Prop osition 6.2.3]. Prop osition 5.2.6. F or W : 𝔸 → ℂ at ∞ and F : 𝔸 → 𝔹 , let p : 𝕎 → 𝔸 b e the (0 , 1) -ﬁbr ation for W and q : 𝕎 ′ → 𝔸 op b e the (1 , 0) -ﬁbr ation. If C is the c ol le ction of p -c o c artesian morphisms in 𝕎 and C ′ that of q -c artesian morphisms in 𝕎 ′ , then we have e quivalenc es lim W 𝔸 F ≃ lim C -lax 𝕎 F ◦ p ≃ lim C ′ -oplax 𝕎 ′ op F ◦ q op . Pr o of. W e ﬁrst consider the lax case. Let b ∈ 𝔹 and observ e that we hav e natural equiv alences 𝔹 ( b, lim W 𝔸 F ) ≃ Nat 𝔸 , ℂ at ∞ ( W , 𝔹 ( b, F ( − ))) ≃ F un coc / 𝔸 ( 𝕎 , F ∗ 𝔹 b → ) ≃ F un C -co c / 𝕎 ( 𝕎 , ( F ◦ p ) ∗ 𝔹 b → ) , where the second equiv alence is given by straightening and the third equiv alence is simply given by base change. W e further see F un C -co c / 𝕎 ( 𝕎 , ( F ◦ p ) ∗ 𝔹 b → ) ≃ Nat C -lax 𝕎 , ℂ at ∞ ( ∗ , 𝔹 ( b, F ◦ p ( − ))) ≃ Nat C -lax 𝕎 , 𝔹 ( b, F ◦ p ) , where the ﬁrst equiv alence is given by Theorem 2.4.12 and the second uses Corol- lary 5.1.7 . Therefore, b oth univ ersal properties coincide, as desired. The oplax case is prov ed similarly , using straightening for (1 , 0) -ﬁbrations instead. □ Dualizing, we similarly hav e: Corollary 5.2.7. F or W : 𝔸 op → ℂ at ∞ and F : 𝔸 → 𝔹 , let p : 𝕎 → 𝔸 b e the (1 , 0) -ﬁbr ation for W and q : 𝕎 ′ → 𝔸 op b e the (0 , 1) -ﬁbr ation. If C is the c ol le ction of p -c artesian morphisms in 𝕎 and C ′ that of q -c o c artesian morphisms in 𝕎 ′ , we have colim W 𝔸 F ≃ colim C -lax 𝕎 F ◦ p ≃ colim C ′ -oplax 𝕎 ′ op F ◦ q op . 92 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI W e now wan t to express partially (op)lax (co)limits as w eighted (co)limits: Prop osition 5.2.8. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider functors F : 𝔸 → 𝔹 and D : 𝔹 → ℂ . Then we have e quivalenc es lim E - lax 𝔸 D F ≃ lim F (0 , 1) 𝔹 ( 𝔸 ,E ) 𝔹 D , colim E - lax 𝔸 D F ≃ colim F (1 , 0) 𝔹 ( 𝔸 ,E ) 𝔹 D , lim E - oplax 𝔸 D F ≃ lim F (0 , 0) 𝔹 ( 𝔸 ,E ) op 𝔹 D , colim E - oplax 𝔸 D F ≃ colim F (1 , 1) 𝔹 ( 𝔸 ,E ) op 𝔹 F , whenever either side exists in ℂ , wher e the weights F ϵ 𝔹 ( 𝔸 , E ) ar e as deﬁne d in Notation 4.4.6 . Pr o of. F or the lax limit, we hav e equiv alences Nat E - lax 𝔸 , ℂ ( c, D F ) ≃ Nat E - lax 𝔸 , ℂ at ∞ ( ∗ , ℂ ( c, D F ( – ))) ≃ 𝟙𝔽 ib (0 , 1) / ( 𝔸 ,E ) ( 𝔸 , F ∗ D ∗ ℂ c → ) . Here the right-hand side is equiv alently the mapping ∞ -category 𝕄ℂ at ( ∞ , 2) / 𝔸 ♯ (( 𝔸 , E ) , F ∗ D ∗ ( ℂ c → ) ♮ ) ≃ 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ (( 𝔸 , E ) , D ∗ ( ℂ c → ) ♮ ) where F ∗ D ∗ ( ℂ c → ) ♮ is the canonical marking b y cocartesian 1-morphisms. By Corol- lary 4.4.7 , this is naturally equiv alent to Nat 𝔹 , ℂ at ∞ ( F (0 , 1) 𝔹 ( 𝔸 , E ) , ℂ ( c, D )) , so that w e ha ve Nat E - lax 𝔸 , ℂ ( c, D F ) ≃ Nat 𝔹 , ℂ at ∞ ( F (0 , 1) 𝔹 ( 𝔸 , E ) , ℂ ( c, D )) . By the Y one da lemma, it follows that an ob ject of ℂ represents the left-hand side, i.e. is an E -lax limit of D F , if and only if it represents the right-hand side, i.e. is a F (0 , 1) 𝔹 ( 𝔸 , E ) -weigh ted limit of F . The other cases are prov ed similarly . □ As a sp ecial case, our identiﬁcation of free ﬁbrations gives the following more explicit version of [ GHL25 , Corollary 6.2.11]: Corollary 5.2.9. L et ( ℂ , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider a functor F : ℂ → 𝔻 . Then we have e quivalenc es lim E - lax ℂ F ≃ lim F (0 , 1) ℂ ( ℂ ,E ) ℂ F , colim E - lax ℂ F ≃ colim F (1 , 0) ℂ ( ℂ ,E ) ℂ F , lim E - oplax ℂ F ≃ lim F (0 , 0) ℂ ( ℂ ,E ) ℂ F , colim E - oplax ℂ F ≃ colim F (1 , 1) ℂ ( ℂ ,E ) ℂ F . whenever either side exists in 𝔻 . □ Com bining Corollary 5.2.9 and Prop osition 5.2.6 , we hav e shown that partially lax and oplax limits can b oth b e expressed as weigh ted limits, and vice versa, so that in a sense all three types of limits are equiv alent. In particular, this implies: Corollary 5.2.10. The fol lowing ar e e quivalent for an ( ∞ , 2) -c ate gory ℂ : (1) ℂ has al l smal l p artial ly lax (c o)limits. (2) ℂ has al l smal l p artial ly oplax (c o)limits. (3) ℂ has al l smal l Cat ∞ -weighte d (c o)limits. □ Deﬁnition 5.2.11. W e sa y that ℂ is (c o)c omplete if an y of the conditions in Corollary 5.2.10 hold. Giv en a functor f : ℂ → 𝔻 of ( ∞ , 2) -categories, we will sa y that f is (c o)c ontinuous if it preserves all small partially (op)lax (co)limits (and therefore all small Cat ∞ -w eighted (co)limits). FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 93 W e note the following standard decomp osition results for weigh ted (co)limits: Prop osition 5.2.12. Supp ose we have W : ℂ → ℂ at ∞ and F : ℂ → 𝔻 . If we c an write ℂ ≃ colim j ∈ J ℂ j for some diagr am J → Cat ( ∞ , 2) and write W j , F j for the r estrictions of W and F to ℂ j , then we have lim W ℂ F ≃ lim J op lim W j ℂ j F j , or mor e pr e cisely if the right-hand side exists in 𝔻 then it is the W -weighte d limit of F . Pr o of. Since 𝔽 un ( ℂ , 𝔻 ) ≃ lim j ∈ J op 𝔽 un ( ℂ j , 𝔻 ) , we hav e 𝔻 ( d, lim W ℂ F ) ≃ Nat ℂ , ℂ at ∞ ( W , 𝔻 ( d, F )) ≃ lim j ∈ J op Nat ℂ j , ℂ at ∞ ( W j , 𝔻 ( d, F j )) ≃ lim j ∈ J op 𝔻 ( d, lim W j ℂ j F j ) ≃ 𝔻 ( d, lim j ∈ J op lim W j ℂ j F j ) , as required. □ Prop osition 5.2.13. F or Φ : 𝕁 → ℙ𝕊 h ( ℂ ) , W ∈ ℙ𝕊 h ( 𝕁 ) and F : ℂ → 𝔻 , we have colim colim W 𝕁 Φ ℂ F ≃ colim W 𝕁 (colim Φ( – ) ℂ F ) , if the right-hand side exists. Pr o of. W e hav e 𝔻 (colim W 𝕁 (colim Φ( – ) ℂ F ) , d ) ≃ Nat 𝕁 op , ℂ at ∞ ( W , 𝔻 (colim Φ( – ) ℂ F , d )) ≃ Nat 𝕁 op , ℂ at ∞ ( W , Nat ℂ op , ℂ at ∞ (Φ , 𝔻 ( F , d ))) ≃ Nat ℂ op , ℂ at ∞ (colim W 𝕁 Φ , 𝔻 ( F , d )) ≃ 𝔻 (colim colim W 𝕁 Φ ℂ F , d ) , i.e. the functors corepresented by the tw o ob jects are equiv alent. □ By combining Prop osition 5.2.8 and Prop osition 5.2.13 we obtain the following result on decomp osition of partially (op)lax colimits. Prop osition 5.2.14. L et ( 𝕀 , E ) b e a de c or ate d ( ∞ , 2) -c ate gory and c onsider a di- agr am d : 𝕀 → ℂ . Supp ose further that we ar e given a diagr am τ : J → MCat ( ∞ , 2) indexe d by an ∞ -c ate gory J such that the c olimit of τ in MCat ( ∞ , 2) is ( 𝕀 , E ) , and that ℂ admits str ong c olimits of shap e J . W rite ( 𝕀 j , E j ) = τ ( j ) and let d j b e the r estriction of d along the c anonic al map 𝕀 j → 𝕀 . If the p artial ly (op)lax c olimits of d and e ach of the d j exists in ℂ , then we have an e quivalenc e colim E - (op)lax 𝕀 d ≃ colim j ∈ J colim E j -(op)lax 𝕀 j d j . Pr o of. F rom Prop osition 5.2.8 we hav e a natural equiv alence colim E j -(op)lax 𝕀 j d j ≃ colim F ϵ ( 𝕀 j ,E j ) 𝕀 d for the appropriate v alue of ϵ . W e can therefore rewrite the righ t-hand side as colim colim j ∈ J F ϵ ( 𝕀 j ,E j ) 𝕀 d. 94 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI It remains to observe that we hav e an equiv alence F ϵ ( 𝕀 , E )) ≃ colim j ∈ J F ϵ ( 𝕀 j , E j ); this follows from the free 1-ﬁbred ﬁbration b eing a left adjoint. □ 5.3. (Co)limits of ( ∞ , 2) -categories in terms of ﬁbrations. In this section w e will use straigh tening to provide a ﬁbrational description of partially (op)lax (co)limits in ℂ at ( ∞ , 2) . This extends (and recov ers) the known characterization for diagrams in ℂ at ∞ due to Berman [ Ber24 , Theorem 4.4]. The analogous results for ( ∞ , ∞ ) -categories hav e b een prov ed by Loubaton; see [ Lou24 , Examples 4.2.3.12– 13]. Notation 5.3.1. Let ( 𝔸 , E ) b e a mark ed ( ∞ , 2) -category and p : 𝔼 → 𝔸 be an ϵ -ﬁbration for ϵ = ( i, j ) . W e write 𝔼 ♮ | E := 𝔼 ♮ × 𝔸 ♯♯ ( 𝔸 , E ) ♯ for the decorated ( ∞ , 2) -category where we decorate only those i -cartesian mor- phisms that lie ov er E together with all of the j -cartesian 2-morphisms. Prop osition 5.3.2. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider a func- tor F : 𝔸 → ℂ at ( ∞ , 2) with c orr esp onding ϵ -ﬁbr ation 𝔼 ϵ → 𝔸 ϵ -op . The E -(op)lax c olimits of F c an then b e describ e d as colim E - lax 𝔸 F ≃ τ d ( 𝔼 ♮ (0 ,j ) | E ) , colim E - oplax 𝔸 F ≃ τ d ( 𝔼 ♮ (1 ,j ) | E ) for b oth j = 0 , 1 , wher e the lo c alization τ d ( 𝔼 ♮ ( i,j ) | E ) is obtaine d by inverting the i -c artesian 1-morphisms over E and all of the j -c artesian 2-morphisms in 𝔼 ( i,j ) . Pr o of. W e consider the case ϵ = (0 , 1) without loss of generality . It follows from Theorem 2.4.12 that we hav e a natural equiv alence of functors Nat E - lax 𝔸 , ℂ at ( ∞ , 2)  F , ( − )  ≃ F un E -co c / 𝔸 ( 𝔼 (0 , 1) , ( − ) × 𝔸 ) . Here the right-hand side at 𝔹 is the ∞ -category of functors 𝔼 (0 , 1) → 𝔹 × 𝔸 ov er 𝔸 that preserv e cartesian 2-morphisms and co cartesian morphisms ov er E . In 𝔹 × 𝔸 these are the 2-morphisms and morphisms whose pro jection to 𝔹 are equiv alences, so under the equiv alence F un / 𝔸 ( 𝔼 (0 , 1) , 𝔹 × 𝔸 ) ≃ Fun ( 𝔼 (0 , 1) , 𝔹 ) the full sub category Fun E -co c / 𝔸 ( 𝔼 (0 , 1) , ( − ) × 𝔸 ) corresp onds to F un ( τ d ( 𝔼 ♮ (0 , 1) | E ) , 𝔹 ) . In other words, we hav e a natural equiv alence Nat E - lax 𝔸 , ℂ at ( ∞ , 2)  F , ( − )  ≃ F un ( τ d ( 𝔼 ♮ (0 , 1) | E ) , – ) , whic h precisely identiﬁes τ d ( 𝔼 ♮ (0 , 1) | E ) as the E -lax colimit of F . □ As a sp ecial case where we can omit the localization, we get a description of ﬁbrations ov er ∞ -categories as (op)lax colimits (ﬁrst pro ved in [ GHN17 ] for functors to ℂ at ∞ ): FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 95 Corollary 5.3.3. Supp ose A is an ∞ -c ate gory. F or a functor F : A → ℂ at ( ∞ , 2) with c orr esp onding 0 -ﬁbr ation 𝔼 0 → A and 1 -ﬁbr ation 𝔼 1 → A op , we have 𝔼 0 ≃ colim lax A F , 𝔼 1 ≃ colim oplax A F . Remark 5.3.4. Note that our result do es not giv e a similar description of ﬁbrations o ver ( ∞ , 2) -categories, as the equiv alence of Prop osition 5.3.2 alwa ys requires us to in vert the (co)cartesian 2-morphisms; to a v oid this presumably requires an ( ∞ , 3) - categorical notion of (op)lax colimits. W e also get a description of (op)lax colimits of ∞ -categories: Corollary 5.3.5. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider a functor F : 𝔸 → ℂ at ∞ with c orr esp onding 1-ﬁbr e d ϵ -ﬁbr ation 𝔼 ϵ → 𝔸 ϵ -op . Then the E - (op)lax c olimits of F in ℂ at ∞ c an b e describ e d as colim E - lax 𝔸 F ≃ τ d ( 𝔼 ♮ (0 ,j ) | E ) , colim E - oplax 𝔸 F ≃ τ d ( 𝔼 ♮ (1 ,j ) | E ) for j = 0 , 1 , wher e τ d ( 𝔼 ♮ ( i,j ) | E ) is obtaine d by inverting the i -c artesian 1-morphisms over E to gether with all 2-morphisms (sinc e al l 2-morphisms ar e j -c artesian in a 1-ﬁbr e d ( i, j ) -ﬁbr ation). □ W e now turn to partially (op)lax limits ; to describ e these we in tro duce the follo wing notation: Deﬁnition 5.3.6. Let ( 𝔸 , E ) b e a marked ( ∞ , 2) -category and let 𝔼 , 𝔽 → 𝔸 b e t wo ϵ -ﬁbrations ov er 𝔸 for ϵ = ( i, j ) . F or i = 0 , we denote by 𝔽 un E -co c / 𝔸 ( 𝔼 , 𝔽 ) := 𝔻𝔽 un / ( 𝔸 ,E ) ♯ ( 𝔼 ♮ | E , 𝔽 ♮ | E ) the ( ∞ , 2) -category characterized by the universal prop ert y F un ( 𝕏 , 𝔽 un E -co c / 𝔸 ( 𝔼 , 𝔽 )) ≃ Fun E -co c / 𝔸 ( 𝔼 × 𝕏 , 𝔽 ) , i.e. the full sub- ( ∞ , 2) -category of 𝔽 un / 𝔸 ( 𝔼 , 𝔽 ) spanned by the functors that pre- serv e co cartesian morphisms ov er E and j -cartesian 2-morphisms. (In other words, 𝔽 un E -co c / 𝔸 ( – , – ) denotes the mapping ( ∞ , 2) -category of the natural ( ∞ , 3) -category structure on Fib (0 ,j ) / ( 𝔸 ,E ) .) F or i = 1 , we deﬁne 𝔽 un E -cart / 𝔸 ( 𝔼 , 𝔽 ) similarly . Observ ation 5.3.7. If 𝔽 → 𝔸 is a 1-ﬁbred ϵ -ﬁbration for ϵ = (0 , j ) , then the ( ∞ , 2) - category 𝔽 un E -co c / 𝔸 ( 𝔼 , 𝔽 ) is actually an ∞ -category: any 2-morphism 𝔼 × C 2 → 𝔽 m ust b e given at each x ∈ 𝔼 by a 2-morphism in 𝔽 that maps to an equiv alence in 𝔸 , and is therefore inv ertible since all 2-morphisms in 𝔽 are j -cartesian. Prop osition 5.3.8. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider a functor F : 𝔸 → ℂ at ( ∞ , 2) with c orr esp onding ϵ -ﬁbr ation 𝔼 ϵ → 𝔸 ϵ -op . Then the E -(op)lax limits of F in ℂ at ( ∞ , 2) c an b e describ e d as lim E - lax 𝔸 F ≃ 𝔽 un E -co c / 𝔸 ϵ -op ( 𝔸 ϵ -op , 𝔼 (0 ,j ) ) , lim E - oplax 𝔸 F ≃ 𝔽 un E -cart / 𝔸 ϵ -op ( 𝔸 ϵ -op , 𝔼 (1 ,j ) ) for j = 0 , 1 . 96 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI Pr o of. Without loss of generality let us assume that ϵ = (0 , 1) . W e note that w e ha ve natural equiv alences F un ( – , 𝔽 un E -co c / 𝔸 ( 𝔸 , 𝔼 (0 , 1) )) ≃ F un E -co c / 𝔸 ( 𝔸 × ( – ) , 𝔼 (0 ,j ) ) ≃ Nat E - lax 𝔸 , ℂ at ( ∞ , 2)  ( − ) , F  , where the ﬁrst equiv alence is formal and the second uses Theorem 2.4.12 . Thus the ( ∞ , 2) -category 𝔽 un E -co c / 𝔸 ( 𝔸 , 𝔼 (0 , 1) ) has the universal prop erty of the E -lax limit. □ Ov er an ∞ -category we get a description of (op)lax limits as sections of ﬁbrations (again due to [ GHN17 ] for functors to ℂ at ∞ ): Corollary 5.3.9. L et A b e an ∞ -c ate gory. F or a functor F : A → ℂ at ( ∞ , 2) with c orr esp onding i -ﬁbr ation 𝔼 i → A for i = 0 , 1 , we have lim lax A F ≃ 𝔽 un / A ( A , 𝔼 0 ) , lim oplax A F ≃ 𝔽 un / A op ( A op , 𝔼 1 ) . □ Corollary 5.3.10. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider a functor F : 𝔸 → ℂ at ∞ with c orr esp onding 1-ﬁbr e d ϵ -ﬁbr ation 𝔼 ϵ → 𝔸 ϵ -op . Then the E - (op)lax limits of F in ℂ at ∞ c an b e describ e d as lim E - lax 𝔸 F ≃ Fun E -co c / 𝔸 ϵ -op ( 𝔸 ϵ -op , 𝔼 (0 ,j ) ) , lim E - oplax 𝔸 F ≃ Fun E -cart / 𝔸 ϵ -op ( 𝔸 ϵ -op , 𝔼 (1 ,j ) ) for j = 0 , 1 . □ Remark 5.3.11. The ﬁbrational description of (op)lax limits in ℂ at ∞ from Corol- lary 5.3.10 is taken as a deﬁnition in [ AMGR24 , §B.6], where these are called left- and right-lax limits; it thus follows from Corollary 5.3.10 that these agree with other notions of lax limits in the literature. W e also note the sp ecialization of our results to ordinary (co)limits in ℂ at ( ∞ , 2) indexed ov er ∞ -categories: Corollary 5.3.12. Supp ose we have a functor F : A → ℂ at ( ∞ , 2) , wher e A is an ∞ -c ate gory, with asso ciate d 0 -ﬁbr ation 𝔼 0 → A and 1-ﬁbr ation 𝔼 1 → A op . (i) The c olimit of F c an b e describ e d as colim A F ≃ τ d ( 𝔼 ♮ i ) for i = 0 , 1 , wher e τ d ( 𝔼 ♮ i ) denotes the lo c alization that inverts al l (c o)c artesian 1 -morphisms. (ii) The limit of F c an b e describ e d as lim A F ≃ 𝔽 un coc / A ( A , 𝔼 0 ) ≃ 𝔽 un cart / A op ( A op , 𝔼 1 ) , i.e. as the ( ∞ , 2) -c ate gories of (c o)c artesian se ctions of these ﬁbr ations. □ Using Prop osition 3.2.14 , we can similarly identify partially (op)lax (co)limits in 𝔻ℂ at ( ∞ , 2) : Corollary 5.3.13. L et ( 𝔸 , E ) b e a marke d ( ∞ , 2) -c ate gory and c onsider a functor F : 𝔸 → 𝔻ℂ at ( ∞ , 2) with c orr esp onding de c or ate d ϵ -ﬁbr ation π : 𝔼 ⋄ ϵ → 𝔸 ♯♯, ϵ -op . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 97 (1) The E -(op)lax c olimits of F in 𝔻ℂ at ( ∞ , 2) c an b e describ e d as the pushouts 𝔼 ♮ (0 ,j ) | E τ d ( 𝔼 ♮ (0 ,j ) | E ) ♭♭ 𝔼 ⋄ (0 ,j ) | E colim E - lax 𝔸 F , 𝔼 ♮ (1 ,j ) | E τ d ( 𝔼 ♮ (1 ,j ) | E ) ♭♭ 𝔼 ⋄ (1 ,j ) | E colim E - oplax 𝔸 F , wher e 𝔼 ⋄ ϵ | E denotes the pul lb ack 𝔼 ⋄ × 𝔸 ♯♯, ϵ -op ( 𝔸 ϵ -op , E ) ♯ . In other wor ds, we invert the i -c artesian morphisms over E and al l j -c artesian 2-morphisms in 𝔼 , and e quip this with the de c or ation induc e d by 𝔼 ⋄ ϵ | E . (2) The E -(op)lax limits of F in 𝔻ℂ at ( ∞ , 2) c an b e describ e d as lim E - lax 𝔸 F ≃ 𝔻𝔽 un E -co c / ( 𝔸 ,E ) ♯ (( 𝔸 , E ) ♯ , 𝔼 ⋄ (0 , 1) | E ) ≃ 𝔻𝔽 un E -co c / ( 𝔸 co ,E ) ♯ (( 𝔸 co , E ) ♯ , 𝔼 ⋄ (0 , 0) | E ) lim E - oplax 𝔸 F ≃ 𝔻𝔽 un E -cart / ( 𝔸 op ,E ) ♯ (( 𝔸 op , E ) ♯ , 𝔼 ⋄ (1 , 0) | E ) ≃ 𝔻𝔽 un E -cart / ( 𝔸 coop ,E ) ♯ (( 𝔸 coop , E ) ♯ , 𝔼 ⋄ (1 , 1) | E ) , wher e the right-hand side denotes the de c or at e d ful l sub- ( ∞ , 2) -c ate gory of 𝔻𝔽 un / ( 𝔸 ϵ -op ,E ) ♯ (( 𝔸 ϵ -op , E ) ♯ , 𝔼 ⋄ ϵ | E ) sp anne d by the se ctions that take 1-morphisms in E to i -c artesian morphisms and al l 2-morphisms to j -c artesian 2-morphisms in 𝔼 ϵ . Pr o of. W e prov e the statement for (0 , 1) -ﬁbrations and abbreviate 𝔼 ⋄ := 𝔼 ⋄ (0 , 1) . By Prop osition 3.2.14 , we hav e a natural equiv alence Nat E - lax 𝔸 , 𝔻ℂ at ( ∞ , 2) ( F , ℂ ⋄ ) ≃ DF un E -co c / ( 𝔸 ,E ) ♯ ( 𝔼 ⋄ | E , ℂ ⋄ × ( 𝔸 , E ) ♯ ) . Here the righ t-hand side is the ∞ -category of decorated functors o v er ( 𝔸 , E ) ♯ that also preserve co cartesian morphisms ov er E and cartesian 2-morphisms; w e can iden tify these as the decorated functors 𝔼 ⋄ | E → ℂ ⋄ that tak e the latter to equiv alences, whic h shows that this copresheaf is corepresen ted by the pushout 𝔼 ⋄ | E ⨿ 𝔼 ♮ | E τ d ( 𝔼 ♮ | E ) ♭♭ , as required. T o identify the E -lax limit of F , we similarly consider the natural equiv alence Nat E - lax 𝔸 , 𝔻ℂ at ( ∞ , 2) ( ℂ ⋄ , F ) ≃ DFun E -co c / ( 𝔸 ,E ) ♯ ( ℂ ⋄ × ( 𝔸 , E ) ♯ , 𝔼 ⋄ | E ) . W e can identify this as the ∞ -category of decorated functors from ℂ ⋄ in to 𝔻𝔽 un / ( 𝔸 ,E ) ♯ (( 𝔸 , E ) ♯ , 𝔼 ⋄ | E ) whose v alue at each ob ject takes the morphisms in E to co cartesian morphisms in 𝔼 and all 2-morphisms to cartesian 2-morphisms in 𝔼 , as required. □ Finally , by combining our description of partially (op)lax (co)limits in ℂ at ( ∞ , 2) with the in terpretation of w eigh ted (co)limits as partially (op)lax ones from the previous subsection, w e also get concrete descriptions of w eighted (co)limits in ℂ at ( ∞ , 2) . F or limits, Prop osition 5.2.6 gives: Corollary 5.3.14. F or W : 𝔸 → ℂ at ∞ and F : 𝔸 → ℂ at ( ∞ , 2) , let ▶ 𝕎 → 𝔸 b e the (0 , 1) -ﬁbr ation for W , 98 FERNANDO ABELLÁN, RUNE HAUGSENG, AND LOUIS MAR TINI ▶ 𝕎 ′ → 𝔸 op b e the (1 , 0) -ﬁbr ation for W , ▶ 𝔼 → 𝔸 b e the (0 , 1) -ﬁbr ation for F , ▶ 𝔼 ′ → 𝔸 op b e the (1 , 0) -ﬁbr ation for F . Then we have lim W 𝔸 F ≃ 𝔽 un coc / 𝔸 ( 𝕎 , 𝔼 ) ≃ 𝔽 un cart / 𝔸 op ( 𝕎 ′ , 𝔼 ′ ) . □ Dually , Corollary 5.2.7 gives the following: Corollary 5.3.15. F or W : 𝔸 op → ℂ at ∞ and F : 𝔸 → ℂ at ( ∞ , 2) , let ▶ 𝕎 → 𝔸 b e the (1 , 0) -ﬁbr ation for W , ▶ 𝕎 ′ → 𝔸 op b e the (0 , 1) -ﬁbr ation for W , ▶ 𝔼 → 𝔸 b e the (0 , 1) -ﬁbr ation for F , ▶ 𝔼 ′ → 𝔸 op b e the (1 , 0) -ﬁbr ation for F . Then we have colim W 𝔸 F ≃ τ d ( 𝔼 ♮ × 𝔸 ♯♯ ( 𝕎 , C ) ♯ ) ≃ τ d ( 𝔼 ′ ♮ × 𝔸 op ,♯♯ ( 𝕎 ′ , C ′ ) ♯ ) . wher e C c onsists of the c artesian morphisms in 𝕎 and C ′ of the c o c artesian mor- phisms in 𝕎 ′ . Remark 5.3.16. In the sp ecial case where F and W are diagrams of ordinary categories, this formula for Cat -weigh ted colimits in Cat app ears in [ Lam17 ]. Using Corollary 5.3.13 , we get a similar description of weigh ted colimits in 𝔻ℂ at ( ∞ , 2) : Corollary 5.3.17. F or W : 𝔸 op → ℂ at ∞ and F : 𝔸 → 𝔻ℂ at ( ∞ , 2) , let ▶ 𝕎 → 𝔸 b e the (1 , 0) -ﬁbr ation for W , ▶ 𝕎 ′ → 𝔸 op b e the (0 , 1) -ﬁbr ation for W , ▶ 𝔼 ⋄ → 𝔸 ♯♯ b e the de c or ate d (0 , 1) -ﬁbr ation for F , ▶ 𝔼 ′⋄ → 𝔸 op ,♯♯ b e the de c or ate d (1 , 0) -ﬁbr ation for F . Then the W -weighte d c olimit of F is given by the pushouts 𝔼 ♮ × 𝔸 ♯♯ ( 𝕎 , C ) ♯ τ d ( 𝔼 ♮ × 𝔸 ♯♯ ( 𝕎 , C ) ♯ ) ♭♭ 𝔼 ⋄ × 𝔸 ♯♯ ( 𝕎 , C ) ♯ colim W 𝔸 F , 𝔼 ′ ♮ × 𝔸 op ,♯♯ ( 𝕎 ′ , C ′ ) ♯ τ d ( 𝔼 ′ ♮ × 𝔸 op ,♯♯ ( 𝕎 ′ , C ′ ) ♯ ) ♭♭ 𝔼 ′⋄ × 𝔸 op ,♯♯ ( 𝕎 ′ , C ′ ) ♯ colim W 𝔸 F , wher e C c onsists of the c artesian morphisms in 𝕎 and C ′ of the c o c artesian mor- phisms in 𝕎 ′ . □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 99 5.4. Existence and preserv ation of (co)limits. In this section we collect a few useful results giving criteria for partially (op)lax (co)limits to exist and b e preserved. Prop osition 5.4.1. L et L : ℂ − → ← − 𝔻 : R b e an adjunction of ( ∞ , 2) -c ate gories. Then the left adjoint L pr eserves al l p artial ly (op)lax c olimits that happ en to ex- ists in ℂ . Dual ly, the right adjoint R pr eserves al l p artial ly (op)lax limits that happ en to exist in 𝔻 . Pr o of. W e give the proof for the left adjoin t; the remaining case is formally dual. Observ e that b y a marked v ariant of Prop osition 2.1.12 , for every marked ( ∞ , 2) - category ( 𝕀 , E ) the functor 𝔽 un ( 𝕀 , – ) E - (op)lax preserv es adjunctions. Thus p ostcom- p osition with the adjunction L ⊣ R yields an adjunction L ∗ : 𝔽 un ( 𝕀 , ℂ ) E - (op)lax − → ← − 𝔽 un ( 𝕀 , 𝔻 ) E - (op)lax : R ∗ . Giv en a functor F : 𝕀 → ℂ admitting an E -(op)lax colimit, we can then pro duce the following chain of natural equiv alences Nat E - (op)lax 𝕀 , 𝔻 ( L ∗ F , d ) ≃ Nat E - (op)lax 𝕀 , ℂ ( F , Rd ) ≃ ℂ  colim E - (op)lax ℂ F , Rd  ≃ 𝔻  L  colim E - (op)lax ℂ F  , d  , whic h precisely shows that L (colim E - (op)lax ℂ F ) has the univ ersal property of the colimit colim E - (op)lax ℂ L ∗ F , as required. □ Prop osition 5.4.2. L et ( 𝕀 , E ) b e a marke d ( ∞ , 2) -c ate gory and ℂ an ( ∞ , 2) -c ate gory that admits E -lax (c o)limits over 𝕀 . (i) F or any marke d ( ∞ , 2) -c ate gory ( 𝕊 , T ) , the ( ∞ , 2) -c ate gory 𝔽 un ( 𝕊 , ℂ ) T -oplax admits E -lax (c o)limits for al l functors 𝕀 → 𝔽 un ( 𝕊 , ℂ ) T -oplax that take the morphisms in E to str ong tr ansformations, and these ar e c ompute d p ointwise in ℂ . (ii) These (c o)limits ar e pr eserve d by the functor f ∗ : 𝔽 un ( 𝕊 , ℂ ) T -oplax → 𝔽 un ( 𝕊 ′ , ℂ ) T ′ -oplax for any functor of marke d ( ∞ , 2) -c ate gories f : ( 𝕊 ′ , T ′ ) → ( 𝕊 , T ) . (iii) Dual ly, if ℂ has E -oplax (c o)limits over 𝕀 ,then 𝔽 un ( 𝕊 , ℂ ) T -lax admits p artial ly oplax (c o)limits of such functors, and these ar e again c ompute d p ointwise and pr eserve d by al l r estriction functors. Pr o of. W e will only deal with lax limits, since the cases of oplax limits and (op)lax colimits are formally dual. Our assumptions guarantee the existence of an adjunc- tion L = const : ℂ ⇄ 𝔽 un ( 𝕀 , ℂ ) E - lax : R = lim E - lax 𝕀 . Since 𝔽 un ( 𝕊 , – ) T -oplax preserv es adjunctions by Prop osition 2.1.12 , this induces an adjunction L ∗ : 𝔽 un ( 𝕊 , ℂ ) T -oplax − → ← − 𝔽 un ( 𝕊 , 𝔽 un ( 𝕀 , ℂ ) E - (op)lax ) T -oplax : R ∗ 100 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI where the left adjoint is given b y p ost-comp osition with the constant diagram func- tor and the righ t adjoint is given by p ostcomp osition along the E -(op)lax limit functor. Let 𝔽 un ′ ( 𝕊 , 𝔽 un ( 𝕀 , ℂ ) E - lax ) T -oplax denote the full sub- ( ∞ , 2) -category on functors that take the morphisms in T to strong transformations; the constant di- agram functor L ∗ factors through this, so we can restrict the adjunction to this sub- ( ∞ , 2) -category. Moreov er, it follo ws from Corollary 2.3.6 that we can identify this with the ( ∞ , 2) -category 𝔽 un ′ ( 𝕀 , 𝔽 un ( 𝕊 , ℂ ) T -oplax ) E - lax of functors that take the morphisms in E to strong natural transformations. W e th us hav e an adjunction 𝔽 un ( 𝕊 , ℂ ) T -oplax ⇆ 𝔽 un ′ ( 𝕀 , 𝔽 un ( 𝕊 , ℂ ) T -oplax ) E -lax , where the left adjoint is still the constan t d iagram functor. The righ t adjoint, which b y construction computes E -lax limits ov er 𝕀 ob ject wise in 𝕊 , therefore gives E -lax limits in 𝔽 un ( 𝕊 , ℂ ) T -oplax for all functors in 𝔽 un ′ ( 𝕀 , 𝔽 un ( 𝕊 , ℂ ) T -oplax ) E -lax , which pro ves (i). Now (ii) follows since this construction is clearly natural in ( 𝕊 , T ) . □ W e note the following useful sp ecial case, where the marking is maximal: Corollary 5.4.3. L et ( 𝕀 , E ) b e a marke d ( ∞ , 2) -c ate gory and ℂ an ( ∞ , 2) -c ate gory that admits E -(op)lax (c o)limits over 𝕀 . (i) F or any ( ∞ , 2) -c ate gory 𝕊 , the ( ∞ , 2) -c ate gory 𝔽 un ( 𝕊 , ℂ ) admits E -lax (c o)limits for al l functors 𝕀 → 𝔽 un ( 𝕊 , ℂ ) , and these ar e c ompute d p ointwise in ℂ . (ii) These (c o)limits ar e pr eserve d by the functor f ∗ : 𝔽 un ( 𝕊 , ℂ ) → 𝔽 un ( 𝕊 ′ , ℂ ) for any functor of ( ∞ , 2) -c ate gories f : 𝕊 ′ → 𝕊 . Observ ation 5.4.4. Suppose ℂ is a (co)complete ( ∞ , 2) -category. Then Corol- lary 5.4.3 implies that for an y 𝕊 , the ( ∞ , 2) -category 𝔽 un ( 𝕊 , ℂ ) is also (co)complete, with its partially (op)lax colimits computed p oin twise in ℂ . Moreov er, taking f in Prop osition 5.4.2 (ii) to b e the map ( 𝕊 , T ) → 𝕊 ♯ , we see that the inclusion 𝔽 un ( 𝕊 , ℂ ) → 𝔽 un ( 𝕊 , ℂ ) T -oplax preserv es all partially lax (co)limits, and so is (co)contin uous (though the target do es not necessarily admit al l (co)limits). In fact, we will see later in Corollary 5.6.7 that this functor has a right or a left adjoint when ℂ is complete or co complete, resp ectiv ely . Prop osition 5.4.5. L et ℂ b e a ( ∞ , 2) -c ate gory. Then the Y one da emb e dding h ℂ : ℂ → ℙ𝕊 h ( ℂ ) pr eserves al l p artial ly (op)lax limits that exist in ℂ . Pr o of. Let ( 𝕀 , E ) b e a marked ( ∞ , 2) -category and let D : 𝕀 → ℂ be a diagram. Since h ℂ is fully faithful, it follows that we hav e natural equiv alences Nat ℂ op , ℂ at ∞  h ℂ ( – ) , h ℂ (lim E - (op)lax 𝕀 D )  ≃ ℂ  – , lim E - (op)lax 𝕀 D  ≃ Nat E - (op)lax 𝕀 , ℂ  ( – ) , D  As a consequence of [ AGH25 , Corollary 2.8.13], the functor h ℂ , ∗ : 𝔽 un ( 𝕀 , ℂ ) E - (op)lax → 𝔽 un ( 𝕀 , ℙ𝕊 h ( ℂ )) E - (op)lax FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 101 is also fully faithful, so we obtain a natural equiv alence Nat E - (op)lax 𝕀 , ℂ  ( – ) , D  ≃ Nat E - (op)lax 𝕀 , ℙ𝕊 h ( ℂ ) ( h ℂ , h ℂ D ) . The previous discussion together with the universal prop ert y of the limit implies that we hav e a canonical morphism h ℂ (lim E - (op)lax 𝕀 D ) → lim E - (op)lax 𝕀 h ℂ D . Inv oking again the Y oneda lemma, we see that our map is p oint wise an equiv alence, which concludes the pro of by the conserv ativity noted in Observ ation 4.1.8 . □ Prop osition 5.4.6. L et ℂ b e an ( ∞ , 2) -c ate gory that admits p artial ly (op)lax limits of shap e ( 𝕀 , E ) and supp ose that we ar e given another marke d ( ∞ , 2) -c ate gory ( 𝕊 , T ) and a functor F : 𝕊 → 𝔽 un ( 𝕀 , ℂ ) E - (op)lax such that the T -(op)lax limit of F exists in 𝔽 un ( 𝕀 , ℂ ) E - (op)lax , and such that ℂ admits p artial ly (op)lax limits indexe d by ( 𝕊 , T ) . Then we have an e quivalenc e in ℂ lim E - (op)lax 𝕀 G ≃ lim T -(op)lax 𝕊 H . wher e G := lim T -(op)lax 𝕊 F is the limit of F in 𝔽 un ( 𝕀 , ℂ ) E - (op)lax and H is the c om- p osite functor lim E - (op)lax 𝕀 ◦ F . Remark 5.4.7. It is tempting to write this equiv alence as lim E - (op)lax 𝕀 lim T -(op)lax 𝕊 F ≃ lim T -(op)lax 𝕊 lim E - (op)lax 𝕀 F , but this is p oten tially misleading as we are not claiming that the limit of F in 𝔽 un ( 𝕀 , ℂ ) E - (op)lax is computed ob jectwise by a limit in ℂ . This is the case under certain conditions, how ev er, as we saw ab ov e in Prop osition 5.4.2 . Pr o of of Pr op osition 5.4.6 . F or c ∈ ℂ , we hav e natural equiv alences ℂ ( c, lim E - (op)lax 𝕀 G ) ≃ Nat E - (op)lax 𝕀 , ℂ ( c 𝕀 , G ) ≃ Nat T -(op)lax 𝕊 , 𝔽 un ( 𝕀 , ℂ ) E - (op)lax ( c 𝕀 𝕊 , F ) , where we subscript the constant diagrams with their domains for clarity . Since ℂ admits E -(op)lax limits ov er 𝕀 , we hav e an adjunction L := ( – ) 𝕀 : ℂ ⇄ 𝔽 un ( 𝕀 , ℂ ) E - (op)lax : lim E - (op)lax 𝕀 =: R, and since 𝔽 un ( 𝕊 , – ) T -(op)lax preserv es adjunctions by Proposition 2.1.12 , this in- duces by comp osition an adjunction L ∗ : 𝔽 un ( 𝕊 , ℂ ) T -(op)lax ⇄ 𝔽 un ( 𝕊 , 𝔽 un ( 𝕀 , ℂ ) E - (op)lax ) T -(op)lax : R ∗ The constant diagram c 𝕀 𝕊 is also L ∗ ( c 𝕊 ) , so using this adjunction we get a natural equiv alence Nat T -(op)lax 𝕊 , 𝔽 un ( 𝕀 , ℂ ) E - (op)lax ( c 𝕀 𝕊 , F ) ≃ Nat T -(op)lax 𝕊 , ℂ ( c 𝕊 , R ∗ F ) ≃ ℂ ( c, lim T -(op)lax 𝕊 H ) , as required. □ 102 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI 5.5. Coﬁnal functors. In this section we use our work so far to giv e a treatment of coﬁnality for ( ∞ , 2) -categories, giving mo del-indep endent pro ofs of results that previously app eared in [ AS23b , AG22 , GHL25 ]. Related results on coﬁnality for ( ∞ , ∞ ) -categories also app ear in [ Lou24 , §4.2.3]. W e start by deﬁning coﬁnal maps by an orthogonality prop ert y and considering some consequences of this b efore we c haracterize coﬁnality in terms of the preser- v ation of (co)limits in Prop osition 5.5.8 ; we also derive a more concrete criterion for coﬁnality in Theorem 5.5.13 . Deﬁnition 5.5.1. A functor of mark ed ( ∞ , 2) -categories F : ( 𝕀 , S ) → ( 𝕁 , T ) is called ϵ -c oﬁnal if it is left orthogonal to all 1-ﬁbred ϵ -ﬁbrations 𝔼 ♮ → 𝔹 ♯ . Deﬁnition 5.5.2. W e write L ϵ ;1 𝔹 ♯ for the left adjoint to the inclusion 𝟙𝔽 ib ϵ / 𝔹  → 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ (whic h w e described explicitly in Corollary 4.7.3 ). W e sa y that a morphism of mark ed ( ∞ , 2) -categories F : ( 𝕀 , S ) → ( 𝕁 , T ) ov er 𝔹 ♯ is a marke d ϵ -e quivalenc e ov er 𝔹 ♯ if L ϵ ;1 𝔹 ♯ ( F ) is an equiv alence. Lemma 5.5.3. F or a morphism of marke d ( ∞ , 2) -c ate gories F : ( 𝕀 , S ) → ( 𝕁 , T ) over 𝔹 ♯ , the fol lowing ar e e quivalent: (1) F is a marke d ϵ -e quivalenc e over 𝔹 ♯ . (2) F is an ϵ -e quivalenc e over 𝔹 ♯♯ when viewe d as a functor ( 𝕀 , S ) ♯ → ( 𝕁 , T ) ♯ . (3) F or b ∈ 𝔹 , the induc e d functor of ∞ -c ate gories F ϵ 𝔹 ( 𝕀 , S )( b ) → F ϵ 𝔹 ( 𝕁 , T )( b ) is an e quivalenc e. Pr o of. The ﬁrst t w o conditions are equiv alent since any map from ( 𝕀 , S ) ♯ to a ﬁbra- tion ov er 𝔹 ♯♯ factors through its underlying 1-ﬁbred ﬁbration. The equiv alence of the ﬁrst and third conditions follows from the description of the straightening of L ϵ ;1 𝔹 ♯ ( 𝕀 , S ) from Corollary 4.4.7 . □ Prop osition 5.5.4. The fol lowing ar e e quivalent for F : ( 𝕀 , S ) → ( 𝕁 , T ) : (1) F is ϵ -c oﬁnal. (2) F is a marke d ϵ -e quivalenc e over 𝕁 ♯ . (3) F is an ϵ -e quivalenc e over 𝕁 ♯♯ when viewe d as a functor ( 𝕀 , S ) ♯ → ( 𝕁 , T ) ♯ . (4) L ϵ ;1 𝕁 ♯ ( F ) : L ϵ ;1 𝕁 ♯ ( 𝕀 , S ) → L ϵ ;1 𝕁 ♯ ( 𝕁 , T ) is an e quivalenc e. (5) F or every b ∈ 𝕁 , the functor of ∞ -c ate gories F ϵ 𝔹 ( 𝕀 , S )( b ) → F ϵ 𝔹 ( 𝕁 , T )( b ) is an e quivalenc e. Pr o of. Since any functor ( 𝕁 , S ) → 𝔹 ♯ factors through 𝕁 ♯ , and 1-ﬁbred ϵ -ﬁbrations are closed under pullback, (1) holds if and only if for any 1-ﬁbred ϵ -ﬁbration 𝔼 ♮ → 𝕁 ♯ , FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 103 there is a unique lift in any square ( 𝕀 , S ) 𝔼 ♮ ( 𝕁 , T ) 𝕁 ♯ . F This is equiv alent to F b eing a lo cal equiv alence for 1-ﬁbred ϵ -ﬁbrations when view ed as a map in 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ , so (1) is equiv alent to (2). The equiv alen ce of this with the remaining conditions is then a sp ecial case of Lemma 5.5.3 . □ Observ ation 5.5.5. W e ha v e the follo wing immediate prop erties of ϵ -coﬁnal maps: ▶ ϵ -coﬁnal functors are closed under cobase c hange, comp osition, retracts, and colimits. ▶ If F is ϵ -coﬁnal then GF is ϵ -coﬁnal if and only if G is. ▶ If F : ℂ ⋄ → 𝔻 ⋄ is an ϵ -coﬁbration, then the underlying functor of mark ed ( ∞ , 2) -categories is ϵ -coﬁnal. In particular, by Corollary 3.6.9 all lo calizations of ( ∞ , 2) -categories give ϵ -coﬁnal maps for any ϵ . Observ ation 5.5.6. The following are equiv alent for a functor of mark ed ( ∞ , 2) - categories F : ( 𝕀 , S ) → ( 𝕁 , T ) (cf. Observ ation 2.4.7 ): ▶ F is (1 , 0) -coﬁnal. ▶ F op is (0 , 0) -coﬁnal. ▶ F co is (1 , 1) -coﬁnal. ▶ F coop is (0 , 1) -coﬁnal. W e next characterize our four notions of coﬁnality in terms of partially (op)lax (co)limits: Deﬁnition 5.5.7. W e say that a functor of marked ( ∞ , 2) -categories F : ( 𝕀 , S ) → ( 𝕁 , T ) is (op)lax c olimit-c oﬁnal if for every functor D : 𝕁 → ℂ the T -(op)lax colimit of D exists if and only if the S -(op)lax colimit of D F exists, and the canonical map colim S -(op)lax 𝕀 D F → colim T -(op)lax 𝕁 F is an equiv alence. Dually , w e say that F (op)lax limit-coﬁnal if the corresp onding condition for partially (op)lax limits holds. Prop osition 5.5.8. F or a functor of marke d ( ∞ , 2) -c ate gories F : ( 𝕀 , S ) → ( 𝕁 , T ) we have ▶ F is (1 , 0) -c oﬁnal if and only if it is lax c olimit-c oﬁnal. ▶ F is (0 , 1) -c oﬁnal if and only if it is lax limit-c oﬁnal. ▶ F is (1 , 1) -c oﬁnal if and only if it is oplax c olimit-c oﬁnal. ▶ F is (0 , 0) -c oﬁnal if and only if it is oplax limit-c oﬁnal. Pr o of. W e prov e the ﬁrst case; the other 3 follo w from this by applying Lemma 5.1.3 and Observ ation 5.5.6 . Consider ﬁrst the represen table functor 𝕁 ( j, – ) : 𝕁 → ℂ at ∞ ; here Corollary 5.3.5 implies that the canonical map colim S -lax 𝕀 𝕁 ( j, F ( – )) → colim T -lax 𝕁 𝕁 ( j, – ) 104 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI is equiv alen t to F (1 , 0) 𝕁 ( 𝕀 , S )( j ) → F (1 , 0) 𝕁 ( 𝕁 , T )( j ) , so if F is lax ﬁnal then this is an equiv alence for all j , i.e. F is (1 , 0) -coﬁnal. On the other hand, Prop osition 5.2.8 identiﬁes the comparison map on lax colimits as the map colim F (1 , 0) 𝕁 ( 𝕀 ,S ) 𝕁 D → colim F (1 , 0) 𝕁 ( 𝕁 ,T ) 𝕁 D induced by the transformation of weigh ts F (1 , 0) 𝕁 ( 𝕀 , S ) → F (1 , 0) 𝕁 ( 𝕁 , T ) . Th us the map on lax colimits is an equiv alence when this transformation is a natural equiv alence, i.e. when F is (1 , 0) -coﬁnal. □ Remark 5.5.9. Loubaton prov es a version of Prop osition 5.5.8 for ( ∞ , ∞ ) -categories as [ Lou24 , Theorem 4.2.3.21]. Prop osition 5.5.10. Consider a diagr am of marke d ( ∞ , 2) -c ate gories (5.5) ( ℂ , P ) ( 𝔻 , Q ) ( 𝔻 , R ) ( 𝕀 , S ) ( 𝕁 , T ) 𝕁 ♯ f ′ p ′ p f wher e b oth squar es ar e pul lb acks (so Q c onsists of the morphisms in R that lie over T ). If p is ϵ -smo oth when viewe d as a morphism ( 𝔻 , R ) ♯ → 𝕁 ♯♯ and f is ϵ -c oﬁnal, then f ′ is also ϵ -c oﬁnal. Pr o of. By Prop osition 5.5.4 f is an ϵ -equiv alence ov er 𝕁 ♯♯ when view ed as a dec- orated functor. It therefore follows from Lemma 4.5.7 that f ′ is an ϵ -equiv alence o ver ( 𝔻 , R ) ♯ . It is then also an ϵ -equiv alence o ver 𝔻 ♯♯ b y Observ ation 3.6.7 , and hence ϵ -coﬁnal as required. □ Sp ecializing this using Theorem 4.5.8 , we get: Corollary 5.5.11. Consider a diagr am ( 5.5 ) of marke d ( ∞ , 2) -c ate gories as ab ove, with b oth squar es pu l lb acks. If p is a marke d ϵ -ﬁbr ation and f is ϵ -c oﬁnal, then f ′ is also ϵ -c oﬁnal. □ Our next goal is to give a more concrete characterization of coﬁnality , which is based on the following prop erties of F (1 , 0) 𝕁 ( 𝕁 , T ) : Lemma 5.5.12. F or a marke d ( ∞ , 2) -c ate gory ( 𝕁 , T ) , we have: (i) F or every j ∈ 𝕁 , the ∞ -c ate gory F (1 , 0) 𝕁 ( 𝕁 , T )( j ) has an initial obje ct (ii) The image in F (1 , 0) 𝕁 ( 𝕁 , T )( j ) of any marke d morphism f : j → j ′ in T is initial. (iii) F or every marke d morphism f : j → j ′ in T , the functor F (1 , 0) 𝕁 ( 𝕁 , T )( f ) pr e- serves initial obje cts. Pr o of. By deﬁnition, F (1 , 0) 𝕁 ( 𝕁 , T )( j ) is the ∞ -category obtained from 𝕁 j → b y inv ert- ing the commuting triangles under j given by mark ed morphisms in 𝕁 as well as all 2-morphisms. W e ﬁrst consider the lo calization τ ≤ 1 𝕁 j → that in verts all 2-morphisms; w e claim id j is an initial ob ject here. Indeed, for a morphism f : j → j ′ w e can iden- tify the mapping ∞ -category 𝕁 j → (id j , f ) as 𝕁 ( j, j ′ ) f / using Prop osition 3.3.1 ; this FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 105 has an initial ob ject, and so ( τ ≤ 1 𝕁 j → )(id j , f ) ≃ ∥ 𝕁 ( j, j ′ ) f / ∥ is con tractible, which means that id j is an initial ob ject. No w consider the comp osition { id j } → τ ≤ 1 𝕁 j → (id j , f ) → F (1 , 0) 𝕁 ( 𝕁 , T )( j ); here the ﬁrst functor is limit-coﬁnal since id j is initial, while the second functor is also limit-coﬁnal as it is a lo calization of ∞ -categories. The comp osite is therefore also limit-coﬁnal, which means that id j is also an initial ob ject of F (1 , 0) 𝕁 ( 𝕁 , T )( j ) . This prov es (i). If f : j → j ′ is a marked morphism, the commuting triangle j j j ′ = f f is sent to an equiv alence in F (1 , 0) 𝕁 ( 𝕁 , T )( j ) , and so the image of f is equiv alent to that of id j , and so must also b e initial, giving (ii). Part (iii) follows as the image of id j ′ under F (1 , 0) 𝕁 ( 𝕁 , T )( f ) is the image of f , which is initial. □ Theorem 5.5.13. A functor of marke d ( ∞ , 2) -c ate gories f : ( 𝕀 , E ) → ( 𝕁 , T ) is (1 , 0) -c oﬁnal (i.e. lax c olimit-c oﬁnal) if and only if the fol lowing c onditions hold: (i) F or every j ∈ 𝕁 , ther e exists an initial obje ct in F (1 , 0) 𝕁 ( 𝕀 , E )( j ) . (ii) The initial obje ct of F (1 , 0) 𝕁 ( 𝕀 , E )( j ) is sent by F (1 , 0) 𝕁 ( f ) to an initial obje ct in F (1 , 0) 𝕁 ( 𝕁 , T )( j ) . (iii) Given an obje ct x ∈ 𝕀 , the obje ct of F (1 , 0) 𝕁 ( 𝕀 , E )( f ( x )) that is the image of ( x, id f ( x ) ) is initial. (iv) F or every marke d morphism ϕ : j → j ′ in T , the functor F ϵ 𝕁 ( 𝕀 , E )( ϕ ) : F ϵ 𝕁 ( 𝕀 , E )( j ′ ) → F ϵ 𝕁 ( 𝕀 , E )( j ) pr eserves initial obje cts. Similarly, in the other thr e e c ases we have: ▶ f is (1 , 1) -c oﬁnal (oplax c olimit-c oﬁnal) if and only if the same c onditions hold with (1 , 0) r eplac e d by (1 , 1) . ▶ f is ϵ -c oﬁnal with ϵ = (0 , 1) or (0 , 0) ((op)lax limit-c oﬁnal) if and only if the analo gous c onditions hold with (1 , 0) r eplac e d by ϵ and “initial obje ct” r eplac e d by “terminal obje ct”. Pr o of. By Prop osition 5.5.4 , the functor f is (1 , 0) -coﬁnal if and only if F ϵ 𝕁 ( f ) is a natural equiv alence. It therefore follo ws from Lemma 5.5.12 that the conditions are necessary . W e are left with pro ving that they are also suﬃcient for F ϵ 𝕁 ( f ) to b e an equiv alence. Let q : ℚ → 𝕁 b e the 1-ﬁbred (1 , 0) -ﬁbration corresp onding to F (1 , 0) 𝕁 ( 𝕀 , E ) and ℙ → 𝕁 b e that corresp onding to F (1 , 0) 𝕁 ( 𝕁 , T ) , with Φ : ℚ → ℙ the morphism of 1-ﬁbred (1 , 0) -ﬁbrations corresp onding to F ϵ 𝕁 ( f ) ; we also hav e the canonical maps η 𝕁 : 𝕁 → ℙ and η 𝕀 : 𝕀 → ℚ that exhibit these as free 1-ﬁbred (0 , 1) -ﬁbrations ov er 𝕁 on ( 𝕁 , T ) and ( 𝕀 , E ) , resp ectively . W e consider the full sub- ( ∞ , 2) -category ℚ 0 ⊆ ℚ of ﬁbrewise initial ob jects. Assumption (i) implies that the restriction ℚ 0 → 𝕁 is 106 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI essen tially surjective; it is also fully faithful since for ob jects x, y ∈ ℚ 0 o ver j, j ′ in 𝕁 , the ﬁbre of the co cartesian ﬁbration ℚ ( x, y ) → 𝕁 ( j, j ′ ) at f : j → j ′ is equiv alent to ℚ j ( x, f ∗ j ′ ) , which is contractible since by assumption x is initial in ℚ j . Hence w e hav e an equiv alence ℚ 0 ∼ − → 𝕁 , and its inv erse gives a section s : 𝕁 → ℚ of q that pic ks out the ﬁbrewise initial ob jects. Moreo v er, condition (iv) implies that s takes the marked morphisms in T to cartesian morphisms. W e can therefore extend s o ver the free 1-ﬁbred (1 , 0) -ﬁbration ℙ to a morphism of 1-ﬁbred (1 , 0) -ﬁbrations ¯ s : ℙ → ℚ that straightens to a natural transformation σ : F (1 , 0) 𝕁 ( 𝕁 , T ) → F (1 , 0) 𝕁 ( 𝕀 , E ) . Moreo ver, the comp osite 𝕁 s − → ℚ Φ − → ℙ picks out the ﬁbrewise initial ob jects in ℙ b y condition (ii), so that by uniqueness the comp osite Φ ◦ ¯ s is the identit y , hence so is F ϵ 𝕁 ( f ) ◦ σ . On the other hand, by the freeness of ℚ the comp osite ¯ s ◦ Φ is uniquely determined by the functor 𝕀 → ℚ Φ − → ℙ ¯ s − → ℚ . Here the comp osite of the ﬁrst tw o maps is equiv alent to the comp osite 𝕀 f − → 𝕁 η 𝕁 − → ℙ , so the full comp osite agrees with 𝕀 f − → 𝕁 s − → ℚ . T o identify this with η 𝕀 , consider the commutativ e triangle 𝕀 ℚ 𝕁 . η 𝕀 f q By assumption (iii), η 𝕀 factors through ˜ η 𝕀 : 𝕀 → ℚ 0 , so that we can identify f with the comp osite 𝕀 ˜ η 𝕀 − → ℚ 0 ∼ − → 𝕁 . It follows that s ◦ f is equiv alent to η 𝕀 , and so ¯ s ◦ Φ m ust be the identit y , and hence so is σ ◦ F ϵ 𝕁 ( f ) . W e ha ve thus shown that σ is an in verse of F ϵ 𝕁 ( f ) ; hence the latter is an equiv alence, as we wan ted to prov e. □ As a consequence, w e get the follo wing simpler suﬃcien t (but not necessary) conditions for coﬁnality of a map from the p oin t: Corollary 5.5.14. Supp ose ( 𝕁 , T ) is a marke d ( ∞ , 2) -c ate gory and j is an obje ct of 𝕁 for which the fol lowing c onditions hold: ▶ for every obje ct j ′ of 𝕁 , ther e exists a marke d morphism j ′ → j , ▶ and every marke d morphism j ′ → j is initial in 𝕁 ( j ′ , j ) . Then the functor { j } ♭ → ( 𝕁 , T ) is c oﬁnal. Pr o of. W e apply Theorem 5.5.13 . In this case F (1 , 0) 𝕁 ( { j } ) is the functor 𝕁 ( – , j ) , so the tw o assumptions together giv e condition (i) as w ell as (iii), since id j is alwa ys mark ed, and (iv), since mark ed morphisms are closed under comp osition. F or condition (ii), w e recall from Lemma 5.5.12 that the initial ob ject of F (1 , 0) 𝕁 ( 𝕁 , T )( j ′ ) is the image of id j ′ , and a mark ed morphism ϕ : j ′ → j gives a morphism id j ′ → ϕ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 107 whose image in F (1 , 0) 𝕁 ( 𝕁 , T )( j ′ ) is an equiv alence; thus the initial ob ject ϕ in 𝕁 ( j ′ , j ) is indeed sent to an initial ob ject, as required. □ Remark 5.5.15. W e saw in Prop osition 4.2.9 that if p : 𝔼 → 𝔹 is a 1-ﬁbred (1 , 0) - ﬁbration and e is an ob ject of 𝔼 , then the functor { e } ♭ → ( 𝔼 , C ) , where C denotes the cartesian morphisms, is (1 , 0) -coﬁnal if and only if the conditions of Corol- lary 5.5.14 are satisﬁed (and b oth are equiv alent to e exhibiting p as a representable ﬁbration). Thus in this particular case these a priori stronger conditions are actually necessary for coﬁnality . 5.6. Kan extensions. In this section we apply our results on pushforward of ﬁ- brations from § 4.6 to construct Kan extensions of ( ∞ , 2) -categories. This includes not only ordinary Kan extensions, whic h can also b e constructed using enriched ∞ -category theory [ Hei24 ], but also p artial ly lax Kan extensions, which originally app eared in [ Ab e23 ] in a mo del-categorical approach. 5 W e start b y considering the case of righ t Kan extensions in ℂ at ( ∞ , 2) , which follo ws immediately from our previous results via straightening: Prop osition 5.6.1. L et f : ( ℂ , I ) → 𝔻 ♯ b e a functor of marke d ( ∞ , 2) -c ate gories. Then ther e exists an adjunction of ( ∞ , 2) -c ate gories f ∗ : 𝔽 un ( 𝔻 , ℂ at ( ∞ , 2) ) − → ← − F un ( ℂ , ℂ at ( ∞ , 2) ) I -(op)lax : f I -(op)lax ∗ wher e f ∗ is the obvious r estriction functor. F or F : ℂ → ℂ at ( ∞ , 2) , we c an describ e f I -(op)lax ∗ F by the p ointwise formula f I -lax ∗ F ( d ) ≃ lim I d -lax ℂ d →  ℂ d → → ℂ F − → ℂ at ( ∞ , 2)  f I -oplax ∗ F ( d ) ≃ lim I d -oplax ℂ d →  ℂ d → → ℂ F − → ℂ at ( ∞ , 2)  , wher e I d denotes the morphisms that pr oje ct over ℂ to morphisms in I and to c ommuting triangles in 𝔻 . Pr o of. The existence of the adjunction follows by com bining Corollary 4.6.6 with the straightening equiv alence of Theorem 2.4.12 : in the lax case by straightening the adjunction for f in the case ϵ = (0 , 1) or that for f co in the case ϵ = (0 , 0) , and in the oplax case b y straightening the adjunction for f coop with ϵ = (1 , 1) or f op with ϵ = (1 , 0) . T o identify the v alues of f I -lax ∗ F w e then combine Obs erv ation 4.6.3 with the ﬁbrational description of I -(op)lax limits in ℂ at ( ∞ , 2) from Prop osition 5.3.8 . □ W e call the functor f I -(op)lax ∗ the I -(op)lax right Kan extension functor along f . Our goal is no w to use the Y oneda em bedding to extend these Kan extensions to more general targets than ℂ at ( ∞ , 2) , for which we start with the following observ a- tions: 5 Loubaton also deﬁnes the notion of partially lax Kan extension for ( ∞ , ∞ ) -categories in [ Lou24 , §4.2.4], but do es not seem to prov e they exist except in the non-lax case (his Corollary 4.2.4.9). 108 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI Observ ation 5.6.2. Let f : ( ℂ , I ) → 𝔻 ♯ b e a mark ed functor. Then for every ( ∞ , 2) -category 𝔸 , restriction along f yields a commutativ e diagram 𝔽 un ( 𝔻 , 𝔸 ) 𝔽 un ( ℂ , 𝔸 ) I -(op)lax 𝔽 un ( 𝔸 op × 𝔻 , ℂ at ∞ ) 𝔽 un ( 𝔸 op × ℂ , ℂ at ∞ ) I -(op)lax f ∗ ( f × id) ∗ where the vertical functors are induced by the Y oneda embedding and I contains all pairs of morphisms whose comp onent in ℂ lies in I . F rom [ AGH25 , Corollary 2.8.5] w e see that the vertical functors are fully faithful with essential image giv en b y those functors F : 𝕏 × 𝔸 op → ℂ at ∞ , with 𝕏 ∈ { ℂ , 𝔻 } , suc h that F ( − , x ) is representable for every x ∈ 𝕏 . Observ ation 5.6.3. Let ( 𝔸 , I ) b e a marked ( ∞ , 2) -category and let J denote the marking of ( 𝔸 , I ) × 𝔹 ♯ . Applying Corollary 2.3.6 to ( 𝔸 , I ) , 𝔹 ♯ and ℂ at ♯ ( ∞ , 2) w e get an equiv alence 𝔽 un ( 𝔹 × 𝔸 , ℂ at ( ∞ , 2) ) J -lax ≃ 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ at ( ∞ , 2) )) I -lax , as well as a fully faithful functor 𝔽 un ( 𝔸 , 𝔽 un ( 𝔹 , ℂ at ( ∞ , 2) )) I -lax → 𝔽 un ( 𝔹 , 𝔽 un ( 𝔸 , ℂ at ( ∞ , 2) ) I -lax ) , with essential image given b y those functors that send every morphism in 𝔹 to a strong natural transformation, or equiv alently factor through the cartesian product. Lemma 5.6.4. Given a marke d ( ∞ , 2) -c ate gory ( 𝔸 , I ) and an ( ∞ , 2) -c ate gory 𝔹 , let π 𝔹 : ( 𝔸 , I ) × 𝔹 ♯ → 𝔹 ♯ b e the pr oje ction and write J for the marking of ( 𝔸 , I ) × 𝔹 ♯ . Then the right adjoint π 𝔹 , ∗ : 𝔽 ib ϵ / ( 𝔸 × 𝔹 ,J ) → 𝔽 ib ϵ / 𝔹 to pul lb ack along π 𝔹 fr om Cor ol lary 4.5.13 c an b e identiﬁe d under str aightening with the c omp osite 𝔽 un (( 𝔸 × 𝔹 ) ϵ -op , ℂ at ( ∞ , 2) ) J -(op)lax  → 𝔽 un ( 𝔹 ϵ -op , 𝔽 un ( 𝔸 ϵ -op , ℂ at ( ∞ , 2) ) I -(op)lax ) → 𝔽 un ( 𝔹 ϵ -op , ℂ at ( ∞ , 2) ) wher e the ﬁrst functor is the inclusion fr om Observation 5.6.3 and the se c ond is given by c omp osing with lim I -(op)lax 𝔸 ϵ -op : 𝔽 un ( 𝔸 ϵ -op , ℂ at ( ∞ , 2) ) I -(op)lax → ℂ at ( ∞ , 2) . Pr o of. W e deal with the case ϵ = (0 , 1) . Let p and q b e the (0 , 1) -ﬁbrations for functors F : 𝔹 → ℂ at ( ∞ , 2) and G : 𝔸 × 𝔹 → ℂ at ( ∞ , 2) , resp ectively . W e then hav e a natural identiﬁcation of mapping ∞ -categories 𝔽 ib (0 , 1) / 𝔹 ( p, π 𝔹 , ∗ ( q )) ≃ 𝔽 ib (0 , 1) / ( 𝔸 × 𝔹 ,J ) ( p × 𝔸 , q ) ≃ Nat J -lax 𝔸 × 𝔹 , ℂ at ( ∞ , 2) ( F ◦ π 𝔸 , G ) , using the straightening equiv alence Theorem 2.4.12 , where π 𝔸 denotes the pro jec- tion to 𝔸 . By Observ ation 5.6.3 we hav e a natural identiﬁcation Nat J -lax 𝔸 × 𝔹 , ℂ at ( ∞ , 2) ( F ◦ π 𝔸 , G ) ≃ Nat 𝔹 , 𝔽 un ( 𝔸 , ℂ at ( ∞ , 2) ) I -lax ( F ′ , G ′ ) FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 109 where G ′ ( b )( – ) ≃ G ( – , b ) and similarly for F ′ , so that we can identify F ′ as the functor b 7→ F ( b ) given b y comp osing with the constan t diagram functor o ver 𝔸 . This functor in F then has a right adjoint, giv en b y taking I -lax limits o v er 𝔸 p oin t wise, so that we hav e a natural equiv alence 𝔽 ib (0 , 1) / 𝔹 ( p, π 𝔹 , ∗ ( q )) ≃ Nat 𝔹 , ℂ at ( ∞ , 2) ( F , lim I -lax 𝔸 G ) . Th us the straightening of π 𝔹 , ∗ ( q ) is the functor lim I -lax 𝔸 G , as required. □ Theorem 5.6.5. L et f : ( ℂ , I ) → 𝔻 ♯ b e a marke d functor and let 𝔹 b e an ( ∞ , 2) - c ate gory. Then the fol lowing holds: ▶ If 𝔹 admits I d -lax limits over ℂ d → for al l d , then we have an adjunction f ∗ : 𝔽 un ( 𝔻 , 𝔹 ) − → ← − F un ( ℂ , 𝔹 ) I -lax : f I -lax ∗ wher e f I -lax ∗ F ( d ) ≃ lim I d -lax ℂ d → F . ▶ If 𝔹 admits I d -lax c olimits over ℂ → d for al l d , then we have an adjunction f I -lax ! : F un ( ℂ , 𝔹 ) I -lax − → ← − 𝔽 un ( 𝔻 , 𝔹 ) : f ∗ wher e f I -lax ! F ( d ) ≃ colim I d -lax ℂ → d F . ▶ If 𝔹 admits I d -oplax limits over ℂ d → for al l d , then we have an adjunction f ∗ : 𝔽 un ( 𝔻 , 𝔹 ) − → ← − F un ( ℂ , 𝔹 ) I -oplax : f I -oplax ∗ wher e f I -oplax ∗ F ( d ) ≃ lim I d -oplax ℂ d → F . ▶ If 𝔹 admits I d -oplax c olimits over ℂ → d for al l d , then we have an adjunction f I -oplax ! : F un ( ℂ , 𝔹 ) I -oplax − → ← − 𝔽 un ( 𝔻 , 𝔹 ) : f ∗ wher e f I -oplax ! F ( d ) ≃ colim I d -oplax ℂ → d F . When they exist, we r efer to the functors f I -(op)lax ∗ and f I -(op)lax ! as the I -(op)lax right and left Kan extension functors along f , r esp e ctively. Pr o of. Applying the inv olutions co and op we reduce to verifying the ﬁrst claim. This amounts to showing that the commutativ e square in Observ ation 5.6.2 is hor- izon tally adjointable. Note that the b ottom horizontal morphism admits a right adjoin t by Prop osition 5.6.1 , so we may reduce again to sho wing that this right adjoin t restricts appropriately . In terms of ﬁbrations, w e thus wan t to sho w that the functor R g : 𝔽 ib (0 , 1) / ( 𝔹 op × ℂ ,J ) → 𝔽 ib (0 , 1) / 𝔹 op × 𝔻 , where g : ( 𝔹 op × ℂ , J ) → ( 𝔹 op × 𝔻 ) ♯ is the marked functor ( 𝔹 op ) ♯ × f , has the follo wing prop erty: given a (0 , 1) -ﬁbration p : 𝕏 → 𝔹 op × ℂ suc h that for ev ery c ∈ ℂ , the pullbac k 𝔼 c → 𝔹 op straigh tens to a representable presheaf on 𝔹 , then the ﬁbration R g ( p ) : 𝕐 → 𝔹 op × 𝔻 has the analogous prop erty for every d ∈ 𝔻 . W e th us w an t to identify the pullbac k i ∗ R g ( p ) for i : 𝔹 op × { d } → 𝔹 op × 𝔻 . Recall that by deﬁnition, R g is the comp osite ϕ ∗ π ∗ for functors ( 𝔹 op × ℂ , J ) ♯ π ← − ℚ ⋄ φ − → ( 𝔹 op × 𝔻 ) ♯♯ , 110 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI using the notation from Deﬁnition 4.6.1 . If we consider the pullback ℚ ⋄ d ℚ ⋄ 𝔹 op ,♯♯ × { d } ( 𝔹 op × 𝔻 ) ♯♯ , j φ d φ i then Observ ation 4.5.5 gives a base c hange equiv alence i ∗ ϕ ∗ ≃ ϕ d, ∗ j ∗ , so that w e ha ve i ∗ R g ≃ ϕ d, ∗ r ∗ where r is the comp osite ℚ ⋄ d j − → ℚ ⋄ π − → ( 𝔹 op × ℂ , J ) ♯ . Unpac king the notation, we see that ℚ ⋄ d is ℂ ⋄ d → × 𝔹 op ,♯♯ with r b eing s × id 𝔹 op where s is the canonical functor ℂ ⋄ d → → ( ℂ , I ) ♯ and ϕ d is the pro jection to 𝔹 op . The desired prop ert y therefore follo ws from Lemma 5.6.4 and our assumption on 𝔹 . □ In the maximally marked case, this sp ecializes to giv e ordinary Kan extensions of ( ∞ , 2) -categories: Corollary 5.6.6. F or a functor of ( ∞ , 2) -c ate gories f : ℂ → 𝔻 and an ( ∞ , 2) - c ate gory 𝔹 , we have: ▶ If 𝔹 admits 𝔻 ( d, f ( – )) -weighte d limits over ℂ for al l d , then we have an ad- junction f ∗ : 𝔽 un ( 𝔻 , 𝔹 ) − → ← − F un ( ℂ , 𝔹 ) : f ∗ wher e f ∗ F ( d ) ≃ lim 𝔻 ( d,f ( – )) ℂ F . ▶ If 𝔹 admits 𝔻 ( F ( – ) , d ) -weighte d c olimits over ℂ for al l d , then we have an adjunction f ! : F un ( ℂ , 𝔹 ) − → ← − 𝔽 un ( 𝔻 , 𝔹 ) : f ∗ wher e f ! F ( d ) ≃ colim 𝔻 ( F ( – ) ,d ) ℂ F . Pr o of. W e prov e the ﬁrst case. F rom the maximally marked case of Theorem 5.6.5 , w e see that the righ t adjoint exists and is given either b y taking partially lax limits ov er ℂ d → or by partially oplax limits ov er ℂ d → , in b oth cases marked by the cartesian morphisms ov er ℂ , if these exist. Here ℂ d → → ℂ is the (0 , 1) -ﬁbration for the functor 𝔻 ( d, f ( – )) while ℂ d → → ℂ is the (0 , 0) -ﬁbration for 𝔻 ( d, f ( – )) op , so that ( ℂ d → ) op → ℂ op is the (1 , 0) -ﬁbration for 𝔻 ( d, f ( – )) . It then follows from Prop osition 5.2.6 that both of these compute the 𝔻 ( d, f ( – )) -weigh ted limit of F . □ A t the other extreme, taking f to b e the identit y gives the follo wing sp ecial case: Corollary 5.6.7. L et 𝔹 b e an ( ∞ , 2) -c ate gory and ( ℂ , I ) a marke d ( ∞ , 2) -c ate gory. Then we have: ▶ If 𝔹 admits I c -lax limits over ℂ c → for al l c ∈ ℂ , then we have an adjunction id ∗ : 𝔽 un ( ℂ , 𝔹 ) − → ← − F un ( ℂ , 𝔹 ) I -(op)lax : id I -lax ∗ wher e id I -lax ∗ F ( c ) ≃ lim I d -lax ℂ c → F . FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 111 ▶ If 𝔹 admits I c -lax c olimits over ℂ → c for al l c ∈ ℂ , then we have an adjunction id I -lax ! : F un ( ℂ , 𝔹 ) I -lax − → ← − 𝔽 un ( ℂ , 𝔹 ) : id ∗ wher e id I -lax ! F ( c ) ≃ colim I c -lax ℂ → c F . ▶ If 𝔹 admits I c -oplax limits over ℂ c → for al l c ∈ ℂ , then we have an adjunction id ∗ : 𝔽 un ( ℂ , 𝔹 ) − → ← − F un ( ℂ , 𝔹 ) I -(op)lax : id I -oplax ∗ wher e id I -oplax ∗ F ( c ) ≃ lim I c -oplax ℂ c → F . ▶ If 𝔹 admits I c -oplax c olimits over ℂ → c for al l c ∈ ℂ , then we have an adjunction id I -oplax ! : F un ( ℂ , 𝔹 ) I -oplax − → ← − 𝔽 un ( ℂ , 𝔹 ) : id ∗ wher e id I -oplax ! F ( c ) ≃ colim I c -oplax ℂ → c F . □ Observ ation 5.6.8. Let f : ( ℂ , I ) → 𝔻 ♯ b e a marked functor and assume that 𝔹 has I d -lax limits ov er ℂ d → . W e make the following claims: ▶ One can identify the unit of the adjunction f ∗ ⊣ f I -lax ∗ at an ob ject c with the canonical map on I -lax limits induced via restriction along the mark ed functor ( ℂ d → , I d ) → 𝔻 ♯ d → . ▶ One can identify the counit of the adjunction f ∗ ⊣ f I -lax ∗ at an ob ject c with the canonical map on I -lax limits induced via restriction along the marked functor ( ℂ f ( c ) → , I f ( c ) ) → 𝔻 ♯ f ( c ) → . T o see this, recall that we sho w ed in Theorem 5.6.5 that the commutativ e square in Observ ation 5.6.2 is horizontally adjointable. In particular, this allo ws us to reduce to 𝔹 = ℂ at ∞ . The ﬁrst claim from Observ ation 4.6.8 by setting p = 𝔻𝔽 ree ϵ 𝔻 ♯♯ ( t d ) for t d : [0] → 𝔻 ♯♯ selecting the ob ject d . The second claim follows similarly from Observ ation 4.6.9 , by setting p to b e a decorated functor with source the terminal category . In particular, if f is fully faithful, we ha ve that ℂ f ( c ) → ≃ ℂ c → ; since the inclusion of id c is (0 , 1) -coﬁnal by Prop osition 4.2.9 , this characterization of the adjunction counit allows us to conclude that f I -lax ∗ is a fully faithful functor. Prop osition 5.6.9. L et ( 𝕀 , E ) b e a marke d ( ∞ , 2) -c ate gory and let ℂ b e an ( ∞ , 2) - c ate gory that admits p artial ly (op)lax c olimits of shap e ( 𝕀 , E ) . Then the left Kan extension along ι : 𝕀 ♯♯  → ( 𝕀 ▷ E - (op)lax ) ♯♯ exists, is ful ly faithful and has as essential image the ful l sub- ( ∞ , 2) -c ate gory on E -(op)lax c olimit c ones. Pr o of. Since the inclusion ι is fully faithful b y Prop osition 2.5.8 , the left Kan ex- tension ι ! will b e fully faithful if it exists by Observ ation 5.6.8 . T o show that ι ! do es exist, w e will apply Corollary 5.6.6 and verify that the corresp onding w eighted colimits exist. W e start by considering the pullback diagram 𝕀 ( j ) =  𝕀 ▷ E - (op)lax  → j × 𝕀 ▷ E - (op)lax 𝕀  𝕀 ▷ E - (op)lax  → j 𝕀 𝕀 ▷ E - (op)lax , p ι π j 112 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI and note that 𝕀 ( j ) can b e identiﬁed with 𝕀 → j whenev er j  = ∗ , where ∗ denotes the cone p oin t. In particular this weigh ted colimit exists and is giv en by ev aluating our functor at the ob ject j . If j = ∗ , then Observ ation 2.5.9 tells us that 𝕀 ( ∗ ) → 𝕀 is the (1 , 0) -ﬁbration F (1 , 0) 𝕀 ( 𝕀 , E ) . W e conclude that the corresp onding weigh ted colimit exists by our assumptions after inv oking Prop osition 5.2.8 . T o ﬁnish the pro of, we must show that a functor ˆ F : 𝕀 ▷ E - (op)lax → ℂ deﬁnes an E - (op)lax colimit cone for F ≃ ι ∗ ˆ F if and only if ˆ F ≃ ι ! F . F rom Observ ation 5.6.8 w e know that the counit transformation can b e iden tiﬁed with the map b etw een colimits colim ♮ - (op)lax 𝕀 ( ∗ ) ˆ F ◦ π ◦ ι → colim ♮ - (op)lax  𝕀 ▷ E - (op)lax  → ∗ ˆ F ◦ π ≃ ˆ F ( ∗ ) where the last equiv alence comes from identifying the colimit ov er the slice with ev aluation at the ob ject corresp onding to the identit y map, exactly as in the case of 𝕀 ( j ) for j  = ∗ . Next, w e observe that the unit of the adjunction in Corollary 4.4.7 giv es a section s : ( 𝕀 , E ) → 𝕀 ( ∗ ) of p . This adjunction immediately implies that s is a marked (1 , 0) -equiv alence o v er 𝕀 ♯ , i.e. (1 , 0) -coﬁnal. Com bining this with the equiv alence ˆ F ◦ j ◦ p ≃ ˆ F ◦ π ◦ ι , we can identify the counit transformation with the map colim E - (op)lax ( 𝕀 ,E ) ˆ F ◦ j − → ˆ F ( ∗ ) . By construction, this map is induced, via the univ ersal prop erty of the E - (op)lax - colimit, by the cone determined b y ˆ F . W e conclude that this map is an equiv alence precisely when ˆ F is a colimit cone. □ 5.7. A Bousﬁeld–Kan formula for weigh ted colimits. Our goal in this section is to use our results on Kan extensions to obtain a Bousﬁeld–Kan type (or “bar construction”) description of weigh ted colimits in ( ∞ , 2) -categories; for a general enric hment this is also done by Heine [ Hei24 , Theorem 3.44] using very diﬀeren t metho ds. The starting point for this is a description of mapping ∞ -categories in functor ( ∞ , 2) -categories, which we deduce from the monadicit y theorem via the follo wing observ ations: Prop osition 5.7.1. Supp ose 𝔻 is a c o c omplete ( ∞ , 2) -c ate gory and f : ℂ ′ → ℂ is an essential ly surje ctive functor of smal l ( ∞ , 2) -c ate gories. Then the adjunction f ! : F un ( ℂ ′ , 𝔻 ) ⇄ F un ( ℂ , 𝔻 ) : f ∗ is monadic, and f ∗ pr eserves al l c olimits. Pr o of. It follows from Lemma 3.3.8 that f ∗ is conserv ativ e. Moreov er, it preserves all colimits by Corollary 5.4.3 . It therefore follows from the monadicity theorem [ Lur17 , Theorem 4.7.3.5] that the adjunction is monadic. □ As a sp ecial case, we hav e: Corollary 5.7.2. Supp ose 𝔻 is a c o c omplete ( ∞ , 2) -c ate gory and ℂ is a smal l ( ∞ , 2) -c ate gory. Then the adjunction i ! : F un ( ℂ ≃ , 𝔻 ) ⇄ F un ( ℂ , 𝔻 ) : i ∗ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 113 is monadic, wher e i is the c or e inclusion ℂ ≃ → ℂ . □ Prop osition 5.7.3. F or functors of ( ∞ , 2) -c ate gories F , G : ℂ → 𝔻 , the ∞ -c ate gory of natur al tr ansformations fr om F to G c an b e expr esse d as a limit of the form Nat ℂ , 𝔻 ( F , G ) ≃ lim [ n ] ∈ ∆ lim ( c 0 ,...,c n ) ∈ ( ℂ ≃ ) × n +1 F un ( ℂ ( c 0 , . . . , c n ) , 𝔻 ( F ( c 0 ) , G ( c n )) wher e ℂ ( c 0 , . . . , c n ) := ℂ ( c 0 , c 1 ) × ℂ ( c 1 , c 2 ) × · · · × ℂ ( c n − 1 , c n ) . Pr o of. By em b edding 𝔻 in (large) presheav es if necessary (whic h gives a fully faith- ful functor on functor ( ∞ , 2) -categories b y [ AGH25 , Corollary 2.7.14]), we ma y without loss of generality assume that ℂ is small and 𝔻 is co complete (with resp ect to some universe). Since the adjunction i ! ⊣ i ∗ for i : ℂ ≃ → ℂ is then monadic by Corollary 5.7.2 , the ob ject F has a simplicial free resolution as a colimit F ≃ colim [ n ] ∈ ∆ op i ! T n i ∗ F in Fun ( ℂ , 𝔻 ) , where T := i ∗ i ! (e.g. by the pro of of [ Lur17 , Lemma 4.7.3.13]). Since 𝔽 un ( ℂ , 𝔻 ) is a co complete ( ∞ , 2) -category by Corollary 5.4.3 , this is also an ( ∞ , 2) - categorical colimit, so that we get an equiv alence Nat ℂ , 𝔻 ( F , G ) ≃ lim [ n ] ∈ ∆ Nat ℂ , 𝔻 ( i ! T n i ∗ F , G ) ≃ lim [ n ] ∈ ∆ Nat ℂ ≃ , 𝔻 ( T n i ∗ F , i ∗ G ) ≃ lim [ n ] ∈ ∆ lim c ∈ ℂ ≃ 𝔻 (( T n i ∗ F )( c ) , G ( c )) , where the ﬁrst equiv alence uses that the left Kan extension i ! is an ( ∞ , 2) -categorical left adjoint and the last equiv alence uses that ℂ ≃ is an ∞ -group oid, so that w e hav e an equiv alence 𝔽 un ( ℂ ≃ , 𝔻 ) ≃ lim ℂ ≃ 𝔻 . No w Observ ation 5.2.4 and the weigh ted colimit formula for left Kan extensions from Corollary 5.6.6 imply that for W : ℂ ≃ → 𝔻 , we hav e ( i ! W )( c ) ≃ colim ℂ ( – ,c ) ℂ ≃ W ≃ colim x ∈ ℂ ≃ W ( x ) ⊠ ℂ ( x, c ) , where ⊠ denotes the tensoring of 𝔻 ov er ∞ -categories, so that ( T n W )( c ) ≃ colim x ∈ ℂ ≃ T n − 1 W ( x ) ⊠ ℂ ( x, c ) ≃ colim ( x 1 ,...,x n ) ∈ ( ℂ ≃ ) × n W ( x 1 ) ⊠ ℂ ( x 1 , . . . , x n , c ) . T aking this colimit out and rewriting the resulting limit, we then get Nat ℂ , 𝔻 ( F , G ) ≃ lim [ n ] ∈ ∆ lim c ∈ ℂ ≃ 𝔻 (colim ( x 1 ,...,x n ) ∈ ( ℂ ≃ ) × n F ( x 1 ) ⊠ ℂ ( x 1 , . . . , x n , c ) , G ( c )) ≃ lim [ n ] ∈ ∆ lim ( x 0 ,...,x n ) ∈ ( ℂ ≃ ) × ( n +1) 𝔻 ( F ( x 0 ) ⊠ ℂ ( x 0 , . . . , x n ) , G ( x n )) ≃ lim [ n ] ∈ ∆ lim ( x 0 ,...,x n ) ∈ ( ℂ ≃ ) × ( n +1) F un ( ℂ ( x 0 , . . . , x n ) , 𝔻 ( F ( x 0 ) , G ( x n ))) , as required. □ Corollary 5.7.4. F or a functor of ( ∞ , 2) -c ate gories F : ℂ → 𝔻 , wher e ℂ is smal l, we have: (i) F or any weight W : ℂ op → ℂ at ∞ , the weighte d c olimit colim W ℂ F exists and is c ompute d as colim [ n ] ∈ ∆ op colim ( c 0 ,...,c n ) ∈ ( ℂ ≃ ) × ( n +1) F ( c 0 ) ⊠ ( ℂ ( c 0 , . . . , c n ) × W ( c n )) 114 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI if this exists in 𝔻 . (ii) F or any weight W : ℂ → ℂ at ∞ , the weighte d limit lim W ℂ F exists and is c om- pute d as lim [ n ] ∈ ∆ lim ( c 0 ,...,c n ) ∈ ( ℂ ≃ ) × ( n +1) F ( c n ) W ( c 0 ) × ℂ ( c 0 ,...,c n ) if this exists in 𝔻 . Pr o of. W e prov e the colimit case; the limit case is prov ed similarly , or follows by applying ( – ) op . F or d ∈ 𝔻 , we hav e 𝔻 (colim W ℂ F , d ) ≃ Nat ℂ op , ℂ at ∞ ( W , 𝔻 ( F , d )) ≃ lim [ n ] ∈ ∆ lim ( c 0 ,...,c n ) ∈ ( ℂ ≃ ) × ( n +1) F un ( W ( c 0 ) × ℂ op ( c 0 , . . . , c n ) , 𝔻 ( F ( c n ) , d )) ≃ lim [ n ] ∈ ∆ lim ( c 0 ,...,c n ) ∈ ( ℂ ≃ ) × ( n +1) 𝔻 ( F ( c 0 ) ⊠ ( ℂ ( c 0 , . . . , c n ) × W ( c n )) , d ) ≃ 𝔻 (colim [ n ] ∈ ∆ op colim ( c 0 ,...,c n ) ∈ ( ℂ ≃ ) × ( n +1) F ( c 0 ) ⊠ ( ℂ ( c 0 , . . . , c n ) × W ( c n ) , d )) , as required, where in the third equiv alence we ha ve reordered the list of ob jects to pass from ℂ op to ℂ . □ Corollary 5.7.5. (i) An ( ∞ , 2) -c ate gory ℂ is c o c omplete if and only if ℂ is tensor e d over Cat ∞ and ℂ ≤ 1 is a c o c omplete ∞ -c ate gory. (ii) An ( ∞ , 2) -c ate gory ℂ is c omplete if and only if ℂ is c otensor e d over Cat ∞ and ℂ ≤ 1 is a c omplete ∞ -c ate gory. □ 5.8. F ree co completion. Our goal in this section is to identify ℙ𝕊 h ( ℂ ) as the free co completion of a small ( ∞ , 2) -category ℂ . This is a sp ecial case of Heine’s very general construction of free cocompletions of enriched ∞ -categories under a class of colimits in [ Hei24 , §3.11]; the analogue for ( ∞ , ∞ ) -categories has also b een prov ed b y Loubaton [ Lou24 , Corollary 4.2.4.8]. F or our pro of we need a functorial version of the Y oneda lemma, which we will derive from our results on ﬁbrations using the follo wing construction: Construction 5.8.1. Given a functor Φ : 𝔸 op × 𝔹 → ℂ at ∞ , w e can view it as a functor Φ ′ : 𝔸 op → 𝔽 un ( 𝔹 , ℂ at ∞ ) ≃ 𝟙𝔽 ib (0 , 1) / 𝔹 . Iden tifying the latter with a full sub- ( ∞ , 2) -category of 𝕄ℂ at ( ∞ , 2) / 𝔹 ♯ , we can straighten this to a morphism of marked (1,0)-ﬁbrations ( 𝔼 , I ) 𝔸 ♯ × 𝔹 ♯ 𝔸 ♯ , ( p,q ) p meaning that p -cartesian 1-morphisms and p -co cartesian 2-morphisms in 𝔼 map to equiv alences in 𝔹 . Giv en a 1-morphism ϕ : x → y in 𝔼 o v er ( f : a → a ′ , g : b → b ′ ) in 𝔸 × 𝔹 , we can factor ϕ as x φ ′ − → f ∗ y ¯ f − → y where ¯ f is p -cartesian o v er f and so lies ov er ( f , id b ′ ) ; then ϕ ′ lies o v er (id a , g ) and so factors as x → g ! x → f ∗ y through a q a -co cartesian morphism in 𝔼 a ; the morphism ϕ lies in the marking I precisely if the resulting morphism g ! x → f ∗ y is an FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 115 equiv alence, i.e. if ϕ factors as a q a -co cartesian morphism follow ed by a p -cartesian morphism. W e refer to ( p, q ) : ( 𝔼 , I ) → 𝔸 ♯ × 𝔹 ♯ as the marke d (1 , 0) -biﬁbr ation asso ciated to Φ . If we instead view Φ as a functor 𝔹 → 𝔽 un ( 𝔸 op , ℂ at ∞ ) ≃ 𝟙𝔽 ib (1 , 0) / 𝔸 and straigh ten this ov er 𝔹 , w e similarly obtain the marke d (0 , 1) -biﬁbr ation 6 ( q ′ , p ′ ) : ( 𝔼 ′ , I ′ ) → 𝔹 ♯ × 𝔸 ♯ for Φ . Prop osition 5.8.2. Given a functor Φ : 𝔸 op × 𝔹 → ℂ at ∞ , let ( p, q ) : ( 𝔼 , I ) → 𝔸 ♯ × 𝔹 ♯ , ( q ′ , p ′ ) : ( 𝔼 ′ , I ′ ) → 𝔹 ♯ × 𝔸 ♯ b e the marke d (1 , 0) -biﬁbr ation and marke d (0 , 1) -biﬁbr ation c orr esp onding to Φ , r esp e ctively. (i) F or a functor G : 𝔹 → ℂ at ∞ with c orr esp onding 1-ﬁbr e d (0 , 1) -ﬁbr ation ℚ → 𝔹 , the functor Nat 𝔹 , ℂ at ∞ (Φ , G ) : 𝔸 → ℂ at ∞ is classiﬁe d by the 1-ﬁbr e d (0 , 1) - ﬁbr ation p ∗ q ∗ ℚ over 𝔸 . (ii) F or a functor H : 𝔸 op → ℂ at ∞ with c orr esp onding 1-ﬁbr e d (1 , 0) -ﬁbr ation ℙ → 𝔸 , the functor Nat 𝔸 op , ℂ at ∞ (Φ , G ) : 𝔹 op → ℂ at ∞ is classiﬁe d by the 1- ﬁbr e d (1 , 0) -ﬁbr ation q ′ ∗ p ′∗ ℙ over 𝔹 . Pr o of. W e pro ve the ﬁrst case. Let Φ ′ b e the functor 𝔸 op → 𝟙𝔽 ib (0 , 1) / 𝔹 asso ciated to Φ . Then we hav e a natural equiv alence Nat 𝔹 , ℂ at ∞ (Φ , G ) ≃ 𝟙𝔽 ib (0 , 1) / 𝔹 (Φ ′ , ℚ ) ≃ 𝔻ℂ at ( ∞ , 2) / 𝔹 ♯♯ (Φ ′ ♮ , ℚ ♮ ) . Giv en W : 𝔸 → ℂ at ∞ , we get a natural pullback square Nat 𝔸 , ℂ at ∞ ( W , Nat 𝔹 , Cat ∞ (Φ , G )) Nat 𝔸 , ℂ at ∞ ( W , DFun (Φ ′ ♮ , ℚ ♮ )) ∗ Nat 𝔸 , ℂ at ∞ ( W , DFun (Φ ′ ♮ , 𝔹 ♯♯ )) , where we regard Φ ′ as a functor to 𝔻ℂ at ( ∞ , 2) . This allo ws us to identify the top left ∞ -category as DF un / 𝔹 ♯♯ (colim W 𝔸 op Φ ′ ♮ , ℚ ♮ ) with the weigh ted colimit taken in 𝔻ℂ at ( ∞ , 2) . Here p : ( 𝔼 , I ) ♯ → 𝔸 ♯♯ is the dec- orated (1 , 0) -ﬁbration for Φ ′ , so if π : 𝕎 → 𝔸 is the (0 , 1) -ﬁbration for W , then Corollary 5.3.17 identiﬁes this weigh ted colimit as the lo calization of ( 𝔼 , I ) ♯ × 𝔸 ♯♯ 𝕎 ♮ where we inv ert ▶ the 1-morphisms that pro ject to a p -cartesian morphism in 𝔼 and a π -co cartesian morphism in 𝕎 , ▶ the 2-morphisms that pro ject to a p -co cartesian 2-morphism in 𝔼 (as all 2- morphisms in 𝕎 are π -cartesian). 6 In fact, we exp ect this to b e the same as the marked (1 , 0) -biﬁbration for Φ , just with the order of 𝔸 and 𝔹 rev ersed; this w ould follow from an ( ∞ , 2) -categorical v ersion of the uniqueness of biv ariant straigh tening prov ed in [ HHLN23 ], but w e will not pursue this here. 116 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI W e claim that in our case we can ignore this lo calization: given a comm utative triangle ( 𝔼 , I ) ♯ × 𝔸 ♯♯ 𝕎 ♮ ℚ ♮ 𝔹 ♯♯ α π w e see that any 1-morphism in the source that pro jects to a p -cartesian morphism in 𝔼 is decorated, and so is sent to a π -co cartesian morphism in ℚ ♮ — but it also pro jects to an equiv alence in 𝔹 via q , so its image is π -co cartesian ov er an equiv alence and so m ust itself b e an equiv alence. Similarly , the image of any 2- morphism that pro jects to a p -co cartesian morphism in 𝔼 must b e in v ertible in ℚ . It follows that we hav e natural equiv alences Nat 𝔸 , ℂ at ∞ ( W , Nat 𝔹 , Cat ∞ (Φ , G )) ≃ DF un / 𝔹 ♯♯ (( 𝔼 , I ) ♯ × 𝔸 ♯♯ 𝕎 ♮ , ℚ ♮ ) ≃ 𝟙𝔽 ib (0 , 1) / 𝔹 ( q ! p ∗ 𝕎 ♮ , ℚ ♮ ) ≃ 𝟙𝔽 ib (0 , 1) / 𝔹 ( 𝕎 ♮ , p ∗ q ∗ ℚ ♮ ) , as required. □ As a sp ecial case, we can identify the ﬁbration for a functor of the form Fun ( F ( – ) , C ) : Corollary 5.8.3. Consider a functor F : 𝔹 → ℂ at ∞ and let ▶ p : 𝔼 → 𝔹 b e the (0 , 1) -ﬁbr ation for F , ▶ q : 𝔼 ′ → 𝔹 op b e the (1 , 0) -ﬁbr ation for F . Then for any ∞ -c ate gory C , the functor F un ( F ( – ) , C ) : 𝔹 op → ℂ at ∞ is classiﬁe d by ▶ the (1 , 0) -ﬁbr ation p ∗ ( C × 𝔼 ) , ▶ the (0 , 1) -ﬁbr ation q ∗ ( C × 𝔼 ′ ) . □ Com bined with our work on right Kan extensions, w e now obtain the desired functorial version of the Y oneda Lemma: Notation 5.8.4. F or an ( ∞ , 2) -category ℂ , Let h ℂ : ℂ → ℙ𝕊 h ( ℂ ) be the functor obtained by straightening (ev 0 , ev 1 ) : 𝔸 r oplax ( ℂ ) → ℂ × ℂ in the second v ariable to a functor ℂ → 𝟙𝔽 ib (1 , 0) / ℂ ≃ ℙ𝕊 h ( ℂ ) . Corollary 5.8.5. W e have a natur al e quivalenc e Nat ℂ op , ℂ at ∞ ( h ℂ , – ) ≃ id of functors ℙ𝕊 h ( ℂ ) → ℙ𝕊 h ( ℂ ) . Pr o of. F rom Prop osition 5.8.2 , we know that Nat ℂ op , ℂ at ∞ ( h ℂ , – ) can be identiﬁed in terms of ﬁbrations as ev 0 , ∗ ev ∗ 1 : 𝟙𝔽 ib (1 , 0) / ℂ → 𝟙𝔽 ib (1 , 0) / ℂ . But Theorem 4.6.4 identiﬁes the latter as the right adjoin t to pullback along id ℂ . □ FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 117 Remark 5.8.6. A similarly functorial version of the Y oneda lemma for ( ∞ , ∞ ) - categories app ears as [ Lou24 , Theorem 4.2.1.20]. Corollary 5.8.7 (“Co-Y oneda Lemma”) . F or an ( ∞ , 2) -c ate gory ℂ we have a nat- ur al e quivalenc e colim Φ ℂ h ℂ ≃ Φ for Φ ∈ ℙ𝕊 h ( ℂ ) . Pr o of. By deﬁnition, this weigh ted colimit should satisfy ℙ𝕊 h ( ℂ )(colim Φ ℂ h ℂ , Ψ) ≃ Nat ℂ op , ℂ at ∞ (Φ , Nat ℂ op , ℂ at ∞ ( h ℂ , Ψ)) , and from Corollary 5.8.5 the righ t-hand side is naturally equiv alent to the ∞ - category Nat ℂ op , ℂ at ∞ (Φ , Ψ) . □ With these results in hand, we can now turn to the free co completion of a small ( ∞ , 2) -category. Prop osition 5.8.8. Supp ose ℂ is a smal l ( ∞ , 2) -c ate gory and 𝔻 is a c o c omplete ( ∞ , 2) -c ate gory. (i) F or any functor F : ℂ → 𝔻 , the left Kan extension h ℂ , ! F : ℙ𝕊 h ( ℂ ) → 𝔻 exists and is c o c ontinuous. (ii) F or any c o c ontinuous functor G : ℙ𝕊 h ( ℂ ) → 𝔻 , the c ounit map h ℂ , ! h ∗ ℂ G → G is an e quivalenc e. Pr o of. By Corollary 5.6.6 , the Kan extension h ℂ , ! F exists if 𝔻 admits colimits w eighted by ℙ𝕊 h ( ℂ )( h ℂ ( – ) , ϕ ) for all ϕ in ℙ𝕊 h ( ℂ ) ; b y Corollary 5.8.5 this presheaf is naturally equiv alen t to ϕ , so this exists by our assumption that ℂ is small and 𝔻 is co complete. T o see that h ℂ , ! F is co contin uous, we consider Φ : 𝕁 → ℙ𝕊 h ( ℂ ) and W ∈ ℙ𝕊 h ( 𝕁 ) and use Prop osition 5.2.13 to compute h ℂ , ! F (colim W 𝕁 Φ) ≃ colim colim W 𝕁 Φ ℂ F ≃ colim W 𝕁 colim Φ ℂ F ≃ colim W 𝕁 h ℂ , ! F (Φ) . Finally , if G preserves colimits, then the counit map h ℂ , ! h ∗ ℂ G → G at ϕ ∈ PSh( ℂ ) is the canonical map colim φ ℂ G ( h ℂ ) → G (colim φ ℂ h ℂ ) , whic h is an equiv alence since G is co contin uous. □ Corollary 5.8.9. If ℂ is a smal l ( ∞ , 2) -c ate gory and 𝔻 is a c o c omplete ( ∞ , 2) - c ate gory, then the r estriction functor h ∗ ℂ : 𝔽 un ( ℙ𝕊 h ( ℂ ) , 𝔻 ) → 𝔽 un ( ℂ , 𝔻 ) has a ful ly faithful left adjoint h ℂ , ! with image the c o c ontinuous functors. □ 118 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI 5.9. Presen table ( ∞ , 2) -categories. In this section we consider a rather simple- minded deﬁnition of presentable ( ∞ , 2) -categories, and sho w that this admits a n umber of other characterizations generalizing those for presentable ∞ -categories: Deﬁnition 5.9.1. An ( ∞ , 2) -category ℂ is pr esentable if ℂ is co complete and ℂ ≤ 1 is presentable, or equiv alen tly accessible. W e state the comparison ﬁrst and then in tro duce the terminology needed to unpac k it: Theorem 5.9.2. The fol lowing ar e e quivalent for an ( ∞ , 2) -c ate gory ℂ : (1) ℂ is pr esentable. (2) ℂ is c o c omplete and lo c al ly smal l, and ther e is a r e gular c ar dinal κ and a smal l ful l sub- ( ∞ , 2) -c ate gory ℂ 0 ⊆ ℂ c onsisting of 2- κ -c omp act obje cts, such that every obje ct in ℂ is the c onic al c olimit of a diagr am in ℂ indexe d by a κ -ﬁlter e d ∞ -c ate gory. (3) Ther e is a smal l ( ∞ , 2) -c ate gory 𝕁 and a ful ly faithful functor ℂ  → ℙ𝕊 h ( 𝕁 ) that admits a left adjoint and whose image is close d under c onic al c olimits over κ -ﬁlter e d ∞ -c ate gories for some r e gular c ar dinal κ . (4) Ther e is a smal l ( ∞ , 2) -c ate gory 𝕁 , a smal l set S of 1-morphisms in ℙ𝕊 h ( 𝕁 ) and an e quivalenc e ℂ ≃ Lo c S ( ℙ𝕊 h ( 𝕁 )) . Deﬁnition 5.9.3. An ob ject c ∈ ℂ is 2- κ -c omp act for a regular cardinal κ if ℂ ( c, – ) preserv es conical colimits ov er κ -ﬁltered ∞ -categories. Observ ation 5.9.4. If ℂ admits conical colimits ov er κ -ﬁltered ∞ -categories and tensors by [1] , then an ob ject c ∈ ℂ is 2- κ -compact if and only if c and [1] ⊠ c are b oth κ -compact in ℂ ≤ 1 . (In fact, it suﬃces that [1] boxtimesc is 2- κ -compact as c is a retract of this.) Deﬁnition 5.9.5. F or s : x → y and c in an ( ∞ , 2) -category ℂ , w e sa y that c is 2-lo c al with resp ect to s if the functor s ∗ : ℂ ( y , c ) → ℂ ( x, c ) is an equiv alence. If S is a set of 1-morphisms in ℂ , w e write Lo c S ( ℂ ) for the full sub- ( ∞ , 2) -category of ob jects that are 2-lo cal with resp ect to all elements of S . Observ ation 5.9.6. Suppose ℂ is an ( ∞ , 2) -category that admits tensors by [1] . Then an ob ject c is 2-lo cal with resp ect to s if and only if c is lo cal in ℂ ≤ 1 with resp ect to both s and s ⊠ [1] . Similarly , if ℂ admits cotensors by [1] then c is 2-lo cal with resp ect to s if and only if c and c [1] are lo cal with resp ect to s in ℂ ≤ 1 . No w we consider some easy consequences of standard results on presentable ∞ - categories. Observ ation 5.9.7. Suppose ℂ is a presentable ( ∞ , 2) -category. Then ℂ is also cotensored o ver Cat ∞ , as the presheav es Map( K , ℂ ( – , c )) preserve limits and so are represen table. Since ℂ ≤ 1 is complete, it follo ws that ℂ is also a complete ( ∞ , 2) - category. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 119 Observ ation 5.9.8. If ℂ is a presentable ( ∞ , 2) -category, then for c, c ′ ∈ ℂ we ha ve that Map( K , ℂ ( c, c ′ )) ≃ ℂ ( K ⊠ c, c ′ ) ≃ is a small ∞ -groupoid for all small ∞ -categories K . It follows that ℂ ( c, c ′ ) is a small ∞ -category, and so ℂ is lo cally small. Com bining the adjoint functor theorem of [ Lur09a , Theorem 5.5.2.9] with Prop o- sition 2.1.5 , we get: Prop osition 5.9.9. Supp ose ℂ and 𝔻 ar e lo c al ly smal l ( ∞ , 2) -c ate gories with ℂ pr esentable and c onsider a functor F : ℂ → 𝔻 . (1) F is a left adjoint if and only if it is c o c ontinuous. (2) If 𝔻 is also pr esentable, then F is a right adjoint if and only if it is c ontinuous and F ≤ 1 is ac c essible. □ Corollary 5.9.10. Supp ose ℂ is a pr esentable ( ∞ , 2) -c ate gory. (1) A pr eshe af F : ℂ op → ℂ at ∞ is r epr esentable if and only if F is c ontinuous. (2) A c opr eshe af F : ℂ → ℂ at ∞ is c or epr esentable if and only if F is c ontinuous and F ≤ 1 is ac c essible. Pr o of. In the second case, the condition is equiv alen t to F having a left adjoint L : ℂ at ∞ → ℂ , which is the unique co contin uous functor taking the p oint to x = L ( ∗ ) . Then x corepresents F as we get ℂ ( x, – ) ≃ F un ( ∗ , F ( – )) ≃ F ( – ) . The ﬁrst case is similar, using that F op has a right adjoint. □ Lemma 5.9.11. Supp ose ℂ is a pr esentable ( ∞ , 2) -c ate gory. Then so is 𝔽 un ( 𝕂 , ℂ ) for any smal l ( ∞ , 2) -c ate gory 𝕂 . In p articular, ℙ𝕊 h ( 𝕂 ) := 𝔽 un ( 𝕂 op , ℂ at ∞ ) is al- ways pr esentable. Pr o of. First suppose 𝕂 is an ∞ -category. Then 𝔽 un ( 𝕂 , ℂ ) ≤ 1 ≃ F un ( 𝕂 , ℂ ≤ 1 ) , which is presentable. The general case then follo ws from Corollary 5.7.2 together with [ Lur09a , Corollary 4.2.3.7] or [ HM25 , Corollary 6.8]. □ W e next mak e some simple observ ations ab out 2-compact and 2-lo cal ob jects that will lead us to the pro of of Theorem 5.9.2 . Lemma 5.9.12. Supp ose ℂ is a pr esentable ( ∞ , 2) -c ate gory. Then ther e exists a r e gular c ar dinal κ such that ℂ ≤ 1 is κ -ac c essible and an obje ct c ∈ ℂ is 2- κ -c omp act if and only if it is a κ -c omp act obje ct of ℂ ≤ 1 . Pr o of. By Observ ation 5.9.4 it is enough to show there exists a κ such that [1] ⊠ – : ℂ ≤ 1 → ℂ ≤ 1 preserv es κ -compact ob jects and ℂ ≤ 1 is κ -accessible. But this is a co contin uous functor b etw een presen table ∞ -categories, so such a κ m ust exist: by the adjoin t functor theorem [ Lur09a , Theorem 5.5.2.9] this functor has a right adjoint, which m ust be κ -accessible for some κ such that ℂ ≤ 1 is κ -accessible by (the proof of ) 120 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI [ Lur09a , Lemma 5.4.7.7]; the left adjoint then preserves κ -compact ob jects, as required. □ Lemma 5.9.13. F or any set of maps S in an ( ∞ , 2) -c ate gory, the ful l sub- ( ∞ , 2) - c ate gory Lo c S ( ℂ ) is close d under weighte d limits in ℂ . Pr o of. Giv en W : 𝕂 → ℂ at ∞ and F : 𝕂 → Lo c S ℂ , consider the commutativ e square ℂ ( y , lim W 𝕂 F ) ℂ ( x, lim W 𝕂 F ) Nat 𝕂 , ℂ at ∞ ( W , ℂ ( y , F )) Nat 𝕂 , ℂ at ∞ ( W , ℂ ( x, F )) . s ∗ ∼ ∼ ℂ ( s,F ) ∗ Here the b ottom horizontal map is an equiv alence since the natural transformation ℂ ( s, F ) is p oint wise inv ertible and so an equiv alence in 𝔽 un ( 𝕂 , ℂ at ∞ ) . □ Lemma 5.9.14. Supp ose ℂ is a pr esentable ( ∞ , 2) -c ate gory and ℂ ′  → ℂ is a ful l sub- ( ∞ , 2) -c ate gory that is close d under c onic al c olimits over κ -ﬁlter e d ∞ -c ate gories for some κ . If the inclusion has a left adjoint, then ℂ ′ is a pr esentable ( ∞ , 2) - c ate gory. Pr o of. It is clear that ℂ ′ is cocomplete, since colimits can be computed by applying the left adjoin t of the inclusion to the colimit in ℂ . Moreov er, ℂ ′≤ 1 is presentable by [ Lur09a , Prop osition 5.5.4.2] since it is a reﬂective sub category of the presentable ∞ -category ℂ ≤ 1 , with the inclusion preserving κ -ﬁltered colimits. □ Lemma 5.9.15. Supp ose ℂ is a pr esentable ( ∞ , 2) -c ate gory and ℂ 0 is a ful l sub- ( ∞ , 2) -c ate gory. Then the fol lowing ar e e quivalent: (1) ℂ 0 is close d under c onic al c olimits over κ -ﬁlter e d ∞ -c ate gories for some r e gular c ar dinal κ and the inclusion ℂ 0  → ℂ has a left adjoint. (2) Ther e exists a smal l set S of morphisms in ℂ such that ℂ 0 ≃ Loc S ℂ . Pr o of. Giv en (1), we know from [ Lur09a , Prop osition 5.5.4.2] that there exists a small set S of morphisms in ℂ such that ℂ ≤ 1 0 ≃ Lo c S ( ℂ ≤ 1 ) . But w e also know that ℂ 0 is closed under cotensors in ℂ (since the inclusion is a right adjoint and so preserv es limits by Prop osition 5.4.1 ), so if x ∈ ℂ ≤ 1 is S -lo cal then so is x [1] , hence x is also S -2-lo cal and so ℂ 0 con tains precisely the S -2-lo cal ob jects, as required. Con versely , given (2) w e know from Observ ation 5.9.6 that ℂ ≤ 1 0 ≃ Loc S ′ ℂ ≤ 1 where S ′ consists of S and S ⊠ [1] . Hence the inclusion ℂ ≤ 1 0 → ℂ ≤ 1 has a left adjoint b y [ Lur09a , Prop osition 5.5.4.15] and therefore also a left adjoint at the ( ∞ , 2) -lev el b y Prop osition 2.1.5 since Lo c S ℂ is closed under cotensors with [1] by Lemma 5.9.13 . □ Pr o of of The or em 5.9.2 . Suppose ﬁrst that ℂ is presentable. Then ℂ is lo cally small b y Observ ation 5.9.8 , and the rest of condition (2) follows from [ Lur09a , Proposition 5.4.2.2] together with Lemma 5.9.12 . Conv ersely , condition (2) implies that ℂ is presen table b y applying [ Lur09a , Prop osition 5.4.2.2] again. The equiv alence of the last tw o conditions is a sp ecial case of Lemma 5.9.15 , while condition (3) implies that ℂ is presentable by Lemma 5.9.14 . It remains to sho w that if ℂ is presentable then condition (3) holds. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 121 Cho ose κ as in Lemma 5.9.12 and consider the inclusion i : ℂ κ  → ℂ . Since ℂ κ is small and ℂ is co complete, we ha ve a Y oneda extension i ′ = h ℂ κ , ! i : ℙ𝕊 h ( ℂ κ ) → ℂ , whic h is co contin uous, and so has a right adjoint R : ℂ → ℙ𝕊 h ( ℂ κ ) b y Prop osition 2.1.5 . W e claim that R preserv es conical colimits ov er κ -ﬁltered ∞ -categories and is fully faithful, which will complete the pro of. T o see the former, w e note that colimits in presheav es are computed p oint wise, so given a diagram ϕ : J → ℂ with J a κ -ﬁltered ∞ -category it suﬃces to chec k that for x ∈ ℂ κ , colim J ℙ𝕊 h ( ℂ κ )( h ℂ κ x, R ( ϕ )) ≃ ℙ𝕊 h ( ℂ κ )( h ℂ κ x, colim J R ( ϕ )) → ℙ𝕊 h ( ℂ κ )( h ℂ κ x, R (colim J ϕ )) is an equiv alence. Via the adjunction and the equiv alence b et ween i ′ ◦ h ℂ κ and the inclusion i , we can identify this as colim J ℂ ( x, ϕ ) → ℂ ( x, colim J ϕ ) . This is indeed an equiv alence since x is 2- κ -compact. T o see that R is fully faithful w e can equiv alen tly prov e that the counit map i ′ R ( c ) → c is an equiv alence for all c ∈ ℂ . Since R and i ′ preserv e κ -ﬁltered colimits and any c is the κ -ﬁltered colimit of a diagram in ℂ κ , we may reduce to the case where c is 2- κ -compact. In this case w e know c ≃ i ′ h ℂ κ ( c ) , so applying the 2-of-3 property to the triangle equiv alence for i ′ w e see that it is enough to show that the unit map h ℂ κ c → Rc is an equiv alence, or equiv alen tly that the map ℙ𝕊 h ( ℂ κ )( ϕ, h ℂ κ ( c )) → ℂ ( i ′ ϕ, i ( c )) is an equiv alence for all presheav es ϕ ; w e can detect this on representable presheav es, where we are asking for the map ℂ κ ( x, c ) → ℂ ( i ( x ) , i ( c )) to b e an equiv alence, which is clear. □ References [Abe23] F ernando Ab ellán. Comparing lax functors of ( ∞ , 2) -categories, 2023, arXiv:2311.12746 . [AG22] F ernando Abellán García. Marked colimits and higher coﬁnality . J. Homotopy R elat. Struct. , 17(1):1–22, 2022. [AGH25] F ernando Ab ellán, Andrea Gagna, and Rune Haugseng. Straightening for lax trans- formations and adjunctions of ( ∞ , 2) -categories. Selecta Math. (N.S.) , 31(4):Paper No. 85, 81, 2025, . [AM24] F ernando Ab ellán and Louis Martini. ( ∞ , 2) -topoi and descent, 2024, arXiv:2410.02014 . [AMGR17] David A yala, Aaron Mazel-Gee, and Nick Rozenblyum. F actorization homology of enriched ∞ -categories, 2017, . [AMGR24] David A yala, Aaron Mazel-Gee, and Nick Rozenblyum. Stratiﬁed noncommutativ e geometry . Mem. Amer. Math. Soc. , 297(1485):iii+260, 2024, . 122 FERNANDO ABELLÁN, RUNE HA UGSENG, AND LOUIS MAR TINI [AS23a] F ernand o Ab ellán and W alker H. Stern. 2-cartesian ﬁbrations I I: A Grothendieck construction for ∞ -bicategories. J. Inst. Math. Jussieu (to app e ar) , 2023, arXiv:2201.09589 . [AS23b] F ernand o Ab ellán and W alker H. Stern. On coﬁnal functors of ∞ -bicategories, 2023, arXiv:2304.07028 . [Ber24] John D. Berman. On lax limits in ∞ -categories. Pr oc. Amer. Math. So c. , 152(12):5055–5066, 2024, . [CM23] Timothy Campion and Y uki Maehara. A mo del-independent gray tensor pro duct for ( ∞ , 2) -categories, 2023, . [DDS18] M.E. Descotte, E.J. Dubuc, and M. Szyld. Sigma limits in 2-categories and ﬂat pseudofunctors. A dv. Math. , 333:266–313, 2018. [GH15] David Gepner and Rune Haugseng. Enriched ∞ -categories via non-symmetric ∞ - operads. Adv. Math. , 279:575–716, 2015. [GHL21] Andrea Gagna, Y onatan Harpaz, and Edoardo Lanari. Gray tensor products and lax functors of ( ∞ , 2) -categories. A dv. Math. , 391:Paper No. 107986, 32, 2021, arXiv:2006.14495 . [GHL24] Andrea Gagna, Y onatan Harpaz, and Edoardo Lanari. Cartesian ﬁbrations of ( ∞ , 2) - categories. Algebr. Ge om. T op ol. , 24(9):4731–4778, 2024, . [GHL25] Andrea Gagna, Y onatan Harpaz, and Edoardo Lanari. Marked limits in ( ∞ , 2) - categories. Do c. Math. , published online, 2025. [GHN17] Da vid Gepner, Rune Haugseng, and Thomas Nikolaus. Lax colimits and free ﬁbra- tions in ∞ -categories. Do c. Math. , 22:225–1266, 2017. [Hau15] Rune Haugseng. Rectiﬁcation of enric hed ∞ -categories. A lgebr. Ge om. T opol. , 15(4):1931–1982, 2015. [Hei23] Hadrian Heine. An equiv alence between enriched ∞ -categories and ∞ -categories with weak action. A dv. Math. , 417:Paper No. 108941, 140, 2023. [Hei24] Hadrian Heine. The higher algebra of weigh ted colimits, 2024, . [HHLN23] Rune Haugseng, F abian Heb estreit, Sil Linskens, and Jo ost Nuiten. Lax monoidal adjunctions, tw o-var iable ﬁbrations and the calculus of mates. Pr o c. Lond. Math. So c. (3) , 127(4):889–957, 2023, . [Hin16] Vladimir Hinic h. Dwy er-Kan lo calization revisited. Homolo gy Homotopy Appl. , 18(1):27–48, 2016. [Hin20] Vladimir Hinic h. Y oneda lemma for enric hed ∞ -categories. A dv. Math. , 367:107129, 119, 2020. [HM25] Simon Henry and Nicholas J. Meadows. Higher theories and monads. High. Struct. , 9(1):227–268, 2025. [Lam17] Michael Lam bert. Computing weigh ted colimits, 2017, . [LMGR + 24] Y u Leon Liu, Aaron Mazel-Gee, Da vid Reutter, Catharina Stropp el, and Paul W edrich. A braided monoidal ( ∞ , 2) -category of So ergel bimodules, 2024, arXiv:2401.02956 . [Lou24] Félix Loubaton. Categorical theory of ( ∞ , ω ) -categories, 2024, . [Lur09a] Jacob Lurie. Higher top os the ory , v olume 170 of Annals of Mathematics Studies . Princeton Universit y Press, Princeton, NJ, 2009. [Lur09b] Jacob Lurie. ( ∞ , 2) -categories and the Go odwillie calculus I, 2009, . [Lur17] Jacob Lurie. Higher algebra, 2017. https://www.math.ias.edu/~lurie/papers/HA. pdf . [MGS24] Aaron Mazel-Gee and Reub en Stern. A universal characterization of noncommutativ e motives and secondary algebraic K-theory . Ann. K-The ory , 9:369–445, 2024. [NS18] Thomas Nikolaus and Peter Scholze. On top ological cyclic homology . A cta Math. , 221(2):203–409, 2018. [Nui24] Joost Nuiten. On straightening for Segal spaces. Comp os. Math. , 160(3):586–656, 2024. FREE FIBRA TIONS, LAX COLIMITS AND KAN EXTENSIONS FOR ( ∞ , 2) -CA TEGORIES 123 [R V22] Emily Riehl and Dominic V erity . Elements of ∞ -c ate gory the ory , volume 194 of Cam- bridge Studies in A dvance d Mathematics . Cambridge Universit y Press, Cambridge, 2022. [Sha21] Jay Shah. P arametrized higher category theory II: Univ ersal constructions, 2021, arXiv:2109.11954 . [Ste20] Germán Stefanic h. Presen table ( ∞ , n ) -categories, 2020, . [Str87] Ross Street. The algebra of oriented simplexes. J. Pur e Appl. A lgebr a , 49(3):283–335, 1987.

Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment