A synthetic construction of universal cocartesian fibrations

A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS CHRISTIAN SA TTLER AND D A VID W ¨ ARN Abstract. W e give a mo del-indep enden t construction of directed univ alent cocartesian ﬁbrations of ( ∞ , 1)-categories, and prov e a straightening equiv alence against suc h ﬁbrations. The key step is sho wing that co cartesian ﬁbrations descend along lo calisations, which we accomplish b y analysing mapping spaces of lo calisations. Along the w ay w e in tro duce a directed v ersion of the join con- struction, giving a sequen tial colimit description of the full image of any functor. 1. Intr oduction In ordinary category theory , one often gets a wa y with constructing categories and functors b e- t w een them in an ad-ho c manner, b y declaring what ob jects and morphisms are, and how functors act on them. In higher category theory how ev er, this would inv olv e an inﬁnite tow er of coherences and quic kly b ecomes infeasible. A more principled approac h is needed. In Lurie’s foundational w ork, this approach mak es use of set-based mo dels of homotopical structures, particularly quasi- categories. In con trast, more recen t work on the foundations and applications of higher category theory increasingly uses the resulting high-level language of ( ∞ , 1)-categories, which is model- indep enden t and homotopy-in v ariant. In this approac h, whic h one might call formal or synthetic , ( ∞ -categorical) universal prop erties pla y a cen tral role. F or example: in ordinary category theory , one can deﬁne the functor category F un( A, B ) by declaring that its ob jects are functors and its morphisms natural transformations. In a syn thetic setting, w e forget about the deﬁnition and instead rely on the univ ersal prop ert y , whic h in this case sa ys that functors X → F un( A, B ) corresp ond to functors X × A → B . Of fundamental importance are the (large) ( ∞ , 1)-categories S of ∞ -group oids and Cat ∞ of ( ∞ , 1)-categories, and these to o enjoy universal prop erties whic h go by the name of str aightening– unstr aightening . Namely , S is the target of a left ﬁbr ation p : S • → S . Ev ery functor F : X → S th us determines a left ﬁbration ov er X , given b y the pullbac k F ∗ p : X × S S • → X ; this is the unstr aightening of F . The univ ersal prop erty of S sa ys that every left ﬁbration o v er X with small ﬁbres arises in this w ay from a functor F : X → S (its straightening) which is unique up to con tractible c hoice. Similarly , Cat ∞ is the target of a c o c artesian ﬁbration q : Cat ∞• → Cat ∞ , and its univ ersal prop erty expresses that functors X → Cat ∞ corresp ond to co cartesian ﬁbrations o ver X with small ﬁbres, under the same op eration of pulling back q along a functor F : X → Cat ∞ . 1 F or short, w e ma y say that S • → S is the universal left ﬁbr ation , and Cat ∞• → Cat ∞ is the univ ersal co cartesian ﬁbration. The existence of a universal co cartesian ﬁbration is a foundational tec hnical result of higher category theory . Since Lurie’s ﬁrst proof [7], simpler treatmen ts ha v e b een giv en [3, 5], and the result has been generalised to categories in ternal to ∞ -topoi [8] and prov en in the framew ork of simplicial t ype theory [4]. But the question remains whether a model-indep endent pro of is possible. On the surface this seems unlikely: ho w could one build the ( ∞ , 1)-category of ( ∞ , 1)-categories without kno wing precisely what an ( ∞ , 1)-category is? 1 In the setting of ordinary categories, (co)cartesian ﬁbrations are known as Gr othendie ck (op)ﬁbr ations , and the op eration of unstraightening a functor X → Cat is known as the Gr othendie ck c onstruction . 1 2 CHRISTIAN SA TTLER AND DA VID W ¨ ARN In the present work, we present a construction of Cat ∞ and pro ve its universal prop erty using mo del-indep enden t means, assuming little more than straightening–unstraigh tening for left ﬁbra- tions. The resulting argument is sim ultaneously simple and general: w e exp ect it to apply also in the setting of categories internal to an ∞ -top os and in constructive settings. This can be view ed as a con tin uation of the program of synthetic homotop y theory; indeed w e use ideas from homotopy t yp e theory . 1.1. Ov erview. F or a co cartesian ﬁbration p : E → B , ev ery morphism f : B ( x, y ) in its base induces a tr ansp ort functor f ! : E ( x ) → E ( y ) on ﬁbres. W e say that p is dir e cte d univalent if the corresp onding map B ( x, y ) → Map( E ( x ) , E ( y )) is an equiv alence of spaces. W e say that a category 2 F is classiﬁe d b y p if there is an equiv alence F ≃ E ( x ) for some x : B . Our ﬁrst main result is that there are enough directed univ alen t co cartesian ﬁbrations, in the follo wing sense. Theorem 7.7. Given a functor q : Y → X , ther e is a dir e cte d univalent c o c artesian ﬁbr ation p : E → B , such that p classiﬁes every ﬁbr e of q . This is a reﬁnemen t of the naiv e idea that there is a ‘category of al l categories’. It expresses that for an y collection of categories whic h is small enough that it can be indexed by a category , there is a category of those categories. Our second main result is a straigh tening theorem against an y directed univ alent cocartesian ﬁbration. Theorem 8.1. Supp ose p : E → B is a dir e cte d univalent c o c artesian ﬁbr ation and C a c ate gory. Given a c o c artesian ﬁbr ation q : D → C such that p classiﬁes every ﬁbr e of q , ther e is a functor f : C → B and an e quivalenc e D ≃ C × B E of c ate gories over C . Mor e over, given two functors f , g : C → B , the c anonic al map fr om Map( f , g ) to the sp ac e of c o c artesian functors f ∗ p → g ∗ p is invertible. The second claim ab ov e implies that the straightening f : C → B of q is unique in an appropriate sense. The map it refers to is deﬁned in Section 2.14. The k ey ingredien t for b oth theorems is a proof that cocartesian ﬁbrations descend along cocom- mas, sequen tial colimits, and lo calisations (cf. Lemmas 6.1 and 6.2 and Theorem 5.7). Lo calisations are the hardest to deal with by far. What we show is that if i : C → D exhibits D as the lo calisa- tion of C at some collection W ⊆ Ar( C ) of morphisms, then pulling back along i induces a fully faithful functor Co cartFib( D ) Co cartFib( C ) i ∗ whose image consists of those co cartesian ﬁbrations p : E → C with the prop ert y that for any morphism f ∈ W , the transp ort functor f ! is inv ertible. Luc kily , it is not so hard to sa y what the in v erse to i ∗ at such a co cartesian ﬁbration p : E → C ough t to b e: it ough t to be the lo calisation E [ V − 1 ] at the collection V of morphisms that are co cartesian lifts of morphisms b elonging to W . Most of the w ork go es in to sho wing that E [ V − 1 ] → D is a co cartesian ﬁbration and that its pullbac k along i is p : E → C (cf. Lemma 5.5). This reduces to fairly concrete claims ab out mapping spaces of E [ V − 1 ] (Lemma 5.1), whic h we pro ve using a general, functorial description of mapping spaces of lo calisations (cf. Example 3.6). The other ingredien t is a mo del-indep endent wa y of decomp osing an arbitrary category into simpler pieces. Sp eciﬁcally , giv en an arbitrary functor f : X → Y , w e explain ho w to build the full image of f as the colimit of a sequence X → X 1 → X 2 · · · where each X n is a localisation of a co comma inv olving X and X n − 1 (cf. Lemma 7.2). This is a directed analogue of Rijk e’s join 2 F rom no w on we take c ate gory to mean ( ∞ , 1)-category . A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 3 construction [9]. Applied to the core inclusion C ≃ → C , this lets us deduce Theorem 8.1 from descen t. Giv en a co cartesian ﬁbration p : E → X , its directed univ alen t completion ought to b e the full image factorisation of its straightening X → Cat ∞ . In our setting, we do not w an t to assume that Cat ∞ exists, since w e are trying to construct it, but w e can still turn this idea into a c onstruction of the directed univ alent completion, b y playing out the directed join construction (cf. Lemma 7.6). In the undirected setting, this idea was recently used b y Uemura to deﬁne univ alent completion of families [10]. Ac kno wledgemen ts. W e thank Dani ¨ el Apol for helpful discussions. 2. Preliminaries In this section w e describ e some of the basic notions and results p ertaining to spaces and categories used in the rest of the pap er. F or a more thorough dev elopment of synthetic category theory we refer to work in progress of Cisinski, Cnossen, Nguy en, and W alde [2]. W e allow ourselves to w ork at a high lev el of abstraction and usually neglect, for example, to build witnesses that certain diagrams commute, with the hop e that the reader can ﬁll in the details in whatever precise setting 3 they are interested in. W e take as primitiv e the notion of space and the space Cat ≃ of categories. F or categories C and D we write Map( C, D ) for the space of functors C → D . W e assume that functors can be comp osed and that functor comp osition is asso ciativ e up to homotopy . Th us categories form what is sometimes called a wild category . 4 W e assume that there is a terminal category 1. W e assume that w e ma y form pullbac ks, pushouts, and functor categories. W e write the latter as F un( C , D ). W e denote Map(1 , C ) by Ob( C ) and refer to it as the sp ac e of obje cts of C . W e take c : C to mean c : Ob( C ). W e write I for the category that is freely generated b y ob jects 0 , 1 : I and a morphism 0 → 1. W e write Ar( C ) for the functor category F un( I , C ). W e write mapping spaces in C as C ( x, y ). Morphisms in functor categories are called natural transformations and are depicted as 2-cells. 2.1. Comma and slice categories. Given functors F : B → A and G : C → A with common co domain, the comma category F ↓ G is giv en b y the pullback of categories F ↓ G Ar( A ) B × C A × A . ⌟ ( dom , cod ) F × G Equiv alently , the comma category is cofreely generated b y the following lax square. F ↓ G C B A . G F In the comma notation, w e use categories as a stand-in for the iden tit y functor on them. F or example, A ↓ G and F ↓ A denote commas with the identit y functor A → A . W e also use ob jects as a stand-in for the functor from the terminal category selecting them. F or example, a ↓ G and 3 E.g., quasicategories, categories in ternal to an ( ∞ , 1)-top os, or simplicial t yp e theory . 4 Here w e use the w ord ‘wild’ to emphasize that w e nev er need inﬁnitely many coherences; w e ha ve not in vestigated ho w muc h coherence is needed. 4 CHRISTIAN SA TTLER AND DA VID W ¨ ARN F ↓ a denote commas with the functor a : 1 → A . Giv en F : C → A and a : A , we may also write a ↓ C for a ↓ F when F is clear from con text. W e write the slice A ↓ a also as A/a . 2.2. The fundamen tal theorem of category theory. W e assume the fundamental the or em of c ate gory the ory , sa ying that I detects equiv alences of categories. Equiv alen tly , a functor F : C → D is in v ertible if and only if it is • surje ctive : for ev ery d : D , there is c : C with F ( c ) ≃ d , • ful ly faithful : for x, y : C , the action C ( x, y ) → D ( F x, F y ) of F on morphisms is in v ertible. 2.3. F ull and wide sub categories. W e assume that w e may form sub categories and that they enjo y the exp ected mapping-in univ ersal properties. That is, if C is a category and P is a predicate on the ob jects of C (i.e., a monomorphism P  → Ob( C )), then there is a category C P with a functor i : C P → C suc h that for any category X , the map Map( X , C p ) Map( X , C ) is a monic, and its image is spanned by those functors F : X → C such that for ev ery ob ject x : X , F ( x ) lies in P (in other words, Ob( F ) : Ob( X ) → Ob( C ) factors through P  → Ob( C )). W e call C P the ful l sub c ate gory of C sp anne d by P . W e similarly assume that we may form wide sub categories. That is, let C b e a category and let P b e a predicate on morphisms of C whic h is closed under comp osition. 5 Then there is a category C P with a functor i : C P → C suc h that for any category X , the map Map( X , C p ) Map( X , C ) is monic, and its image is spanned b y those functors F : X → C such that for every morphism f in X , the map F ( f ) lies in P . 2.4. Adjunctions. Giv en categories C and D with functors L : C → D and R : D → C , an adjunction L ⊣ R consists of an equiv alence of categories e : C ↓ R ≃ L ↓ D ov er C × D . W e sa y that e exhibits L as the left adjoint of R . W e ha v e that left adjoin ts are unique when they exist, and that R has a left adjoint if and only if the category c ↓ R has an initial ob ject for every c : C . An equiv alen t wa y of exhibiting L as left adjoint to R is b y giving a natural transformation η : id C → RL suc h that η c : c → RLc is initial in c ↓ R for ev ery c . W e also mak e use of adjunctions in the wild setting. 2.5. Spaces, group oids, and extensivity. Given a category C , we think of the mapping space Map(1 , C ) as the space of obje cts of C and denote it Ob( C ). The wild functor Ob has a fully faithful left adjoin t, whic h sends a space X to the group oid D X = F X 1 giv en b y the copro duct 6 of X -many copies of the terminal category (equiv alen tly , the tensor of 1 with X ). W e will usually lea v e this inclusion from spaces to categories implicit. 7 This induces an equiv alence betw een spaces and groupoids, i.e., categories in which every morphism is inv ertible. W e ask that the copro duct deﬁning D X is is extensive , i.e., satisﬁes descent. Explicitly , this means that the (wild) functor Cat /D X → Cat X whic h computes the ﬁbres of a category o v er D X is in vertible (and so its inv erse is the left adjoint F X ). This means that a category C o v er a giv en group oid is determined b y its family of ﬁbres, and similarly for a functor of categories o v er a groupoid. 5 This includes nul lary comp osition, i.e., we assume that P contains all isomorphisms of C . 6 In more traditional con texts, the term c opr o duct is reserved for the case when X is a set. 7 W e are not a ware of any established notation for it. A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 5 2.6. Left and righ t ﬁbrations. W e say that a functor D → C is a left (resp. righ t) ﬁbration if it is orthogonal against the left endp oint inclusion 0 : 1 → I (resp. right endpoint inclusion 1 : 1 → I ), in the sense that the square of spaces Map( I , D ) Map( I , C ) Ob( D ) Ob( C ) dom dom is cartesian. If that is the case, one can sho w that the square of c ate gories Ar( D ) Ar( C ) C D dom dom is also cartesian (b y sho wing that the cartesian gap map is surjectiv e and fully faithful). W e denote the (wild) category of left (resp. right) ﬁbrations o v er C b y LFib( C ) (resp. RFib( C )). 2.7. Coﬁnal functors. W e sa y a functor F : C → D is left (resp. right) c oﬁnal if it is orthogonal against left (resp. righ t) ﬁbrations. W e assume that every functor F : C → D factors as a left coﬁnal functor C → E follo w ed b y a left ﬁbration E → D . It follows that this factorisation is unique. Dually , ev ery functor factors uniquely as a righ t coﬁnal functor follow ed by a righ t ﬁbration. 2.8. Left Kan extension. F or a left ﬁbration p : E → C and an arbitrary functor F : C → D , w e may factor the comp osite F p : E → D as a left coﬁnal map E → E follow ed by a left ﬁbration E → D . This deﬁnes a left adjoint F ! : LFib( C ) → LFib( D ) to the base-change functor F ∗ : LFib( D ) → LFib( C ) (whic h sends a left ﬁbration ov er D to its pullbac k along F ). Left Kan extension admits a useful ﬁbrewise description. F or E → C a left ﬁbration and d : D an arbitrary ob ject, w e ha ve a functor E ↓ d → E d to the ﬁbre of E → D o ver d . The relev ant assertion is that this functor exhibits E d as the free group oid on E ↓ d , i.e., the lo calisation at all its arro ws. Dually , if E → C is a right ﬁbration then w e hav e a functor d ↓ E → E d whic h exhibits E d as the free group oid on d ↓ E . Abstractly , w e ha v e the lax square of categories E ↓ d 1 C D . cod dom d F This induces a lax square LFib( D ) LFib( C ) LFib(1) LFib( E ↓ d ), F ∗ d ∗ dom ∗ cod ∗ and the relev ant assertion is that its mate co d ! dom ∗ → d ∗ F ! is inv ertible. More generally , the same prop ert y holds for the lax square coming from any comma category: for functors F : A → B and G : C → B , we ha ve G ∗ F ! ≃ co d ! dom ∗ , where dom : F ↓ G → A and co d : F ↓ G → C . Dually , for restriction of left Kan extension of right ﬁbrations, w e ha ve G ∗ F ! ≃ dom ! co d ∗ where co d : G ↓ F → A and dom : G ↓ F → C . 6 CHRISTIAN SA TTLER AND DA VID W ¨ ARN 2.9. The Y oneda lemma. Let C b e a category with an initial ob ject c . W e assume the Y oneda lemma in the following form: the functor c : 1 → C is left coﬁnal. In particular, for a category C with an arbitrary ob ject c , we can form the coslice category c ↓ C and observe that it has an initial ob ject giv en b y the iden tity on c . So the corresp onding functor 1 → c ↓ C is left coﬁnal. Since co d : c ↓ C → C is a left ﬁbration, this means that it giv es the unique (left coﬁnal, left ﬁbration) factorisation of c : 1 → C . In particular, supp ose F : C → D is some functor. Since 1 → c ↓ C and 1 → F c ↓ D are b oth left coﬁnal, b y 2-out-of-3 the induced functor c ↓ C → F c ↓ D is also left coﬁnal. This exhibits F c ↓ D as the left Kan extension of c ↓ C along F : C → D . Dually , the inclusion of a terminal ob ject is right coﬁnal, and the left Kan extension of C /c is D/F c . Since the ﬁbre of D ↓ F c ov er d : D is the mapping space D ( d, F c ), this lets us express mapping spaces of D in terms of left Kan extension F ! : RFib( C ) → RFib( D ) applied to represen table righ t ﬁbrations. 2.10. Descen t for left ﬁbrations. W e assume that left ﬁbrations descend along pushouts. This means that for an y pushout square of categories A B C D , g f l k ⌜ the induced square LFib( D ) LFib( C ) LFib( B ) LFib( A ) k ∗ l ∗ f ∗ g ∗ is a pullback square of categories. This is a consequence of straightening–unstraigh tening for left ﬁbrations. 2.11. Limits and colimits in functor categories. Giv en categories C and D , we assume that the family of ev aluation functors ev c : F un( C , D ) → D for c : C creates (co)limits. This means that (co)limits are given lev elwise when they exist at every lev el. Such lev elwise (co)limits are preserv ed under restriction f ∗ : F un( C , D ) → F un( C ′ , D ) along an y functor f : C ′ → C . In practice, we apply this only to describ e pushouts, pullbacks, sequen tial colimits, and initial ob jects. W e apply it also in LFib( C ), i.e., F un( C , S ). The relev an t claim is then that for f : C ′ → C , restriction f ∗ : LFib( C ) → LFib( C ′ ) preserv es pushouts and sequen tial colimits. 2.12. Mapping spaces of sequential colimits. A se quenc e in a (wild) category C consists of a family of ob jects a : N → Ob( C ) with morphisms f n : a n → a n +1 for n : N . A sequen tial colimit of suc h a sequence a 0 f 0 − → x 1 f 1 − → · · · is an initial co cone under it. Sequen tial colimits are ﬁltered colimits, so comm ute with pullbac ks in spaces, and hence also in functor categories into spaces suc h as LFib( C ) and RFib( C ). Consider a sequence of categories A 0 F 0 − → A 1 F 1 − → · · · with colimit A ∞ with copro jections i n : A n → A ∞ . Since sequential colimits are ﬁltered, mapping spaces of A ∞ admit a simple description: for ob jects x, y : A 0 , w e hav e a co cone under A 0 ( x, y ) A 1 ( F 0 x, F 0 y ) · · · A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 7 with p oin t A ∞ ( i 0 x, i 0 y ) given by the actions of i n on morphisms, and this exhibits the space A ∞ ( i 0 x, i 0 y ) as a sequen tial colimit. One can use this fact to show that Ob and Map( I , − ) comm ute with sequen tial colimits. 2.13. Co cartesian ﬁbrations. Given a functor p : E → B and a morphism f : x → y in E , w e sa y that f is p -c o c artesian if the square of spaces E ( y , z ) E ( x, z ) B ( py , pz ) B ( px, pz ) −◦ f −◦ pf is cartesian for all z : E . Let Ar cocart ( p ) ⊆ Ar( E ) denote the full sub category spanned by p - co cartesian maps. One can show that the cartesian gap map of the square of categories Ar cocart ( p ) E Ar( B ) B dom p dom is fully faithful. W e sa y p is a c o c artesian ﬁbr ation if that gap map is inv ertible. Equiv alently , the gap map is surjectiv e, i.e., for ob jects x : E and y : B with f : B ( px, y ), there is a lift y ′ : E of y with a p -co cartesian lift f ′ : E ( x, y ′ ) of f . Co cartesian ﬁbrations are stable under base c hange. That is, for p : E → B a cocartesian ﬁbration and x : C → B an arbitrary functor, the base change x ∗ p : E × B C → C of p is also a cocartesian ﬁbration. W e migh t denote E × B C as x ∗ E in this con text. A map in x ∗ E is x ∗ p -co cartesian if and only if its image in E is p -co cartesian. F or co cartesian ﬁbrations p 0 : E 0 → B and p 1 : E 1 → B with base B , we sa y that a functor F : E 0 → E 1 of categories ov er B is c o c artesian if it sends p 0 -co cartesian maps to p 1 -co cartesian maps. W e denote b y CocartFib( B ) the resulting (wild) category of co cartesian ﬁbrations o ver B . If a co cartesian functor F : E 0 → E 1 o v er B is ﬁbr ewise fully faithful, then it is fully faithful. This means that inv ertibilit y of co cartesian functors can b e detected ﬁbrewise. 2.14. Co cartesian transp ort. Given a cocartesian ﬁbration p : E → B and a category C , w e ha v e a corresp ondence betw een functors C → Ar cocart ( p ) and functors C → Ar( B ) × B E . The data of a functor C → Ar cocart ( p ) is equiv alen tly the data of t w o functors F, G : C → E with a natural transformation α : F → G all of whose comp onen ts are p -cocartesian. The data of a functor C → Ar( B ) × B E is the data of functors F : C → E and G 0 : C → B and a natural transformation α 0 : pF → G 0 . So giv en the latter data, w e ha v e a unique lift G : C → E of G 0 together with a lift α : F → G of α 0 , suc h that the comp onen ts of α are p -co cartesian. In particular, pulling back a co cartesian ﬁbration is co v ariantly functorial. That is, giv en a category C (e.g., the terminal category) and functors x, y : C → B with a natural transformation f : x → y , w e obtain a functor f ! : x ∗ E → y ∗ E of co cartesian ﬁbrations o v er C . The image of f ! in E has co cartesian comp onen ts and lifts f in the appropriate sense. Thus, the assignmen t x 7→ x ∗ E deﬁnes a (wild) functor F un( C , B ) → Co cartFib( C ). 2.15. Descen t in spaces and left ﬁbrations. An excellen t prop erty of the category S of spaces is that it has all colimits and these satisfy descent. Abstractly , this means that giv en some diagram of spaces X : J → S , the functor S / colim J X lim j : J S /X j 8 CHRISTIAN SA TTLER AND DA VID W ¨ ARN giv en informally by base-c hange is an equiv alence of categories. Since this functor has a left adjoin t (giv en informally b y colim J ), this sa ys that the unit and counit of the adjunction are in v ertible. In v ertibility of the counit means that colimits are universal , i.e., stable under base c hange. In vertibilit y of the unit means that the wide sub category of Ar( S ) spanned b y c artesian squar es is closed under colimits. Since colimits and pullbac k in LFib( C ) are computed p oint wise, colimits in LFib( C ) also satisfy descen t. W e use descen t only for pushouts and sequential colimits. 2.16. Mapping spaces of co commas. The c o c omma of a span of categories B f ← − A g − → C is the pushout of categories A ⊔ A B ⊔ C A × I f ↑ g . ( id , [0 , 1]) f ⊔ g [ k,l ] More usefully , this means that the cocomma is freely generated by the lax square A C B f ↑ g . g f l k By descen t for left ﬁbrations, the lax square LFib( f ↑ g ) LFib( B ) LFib( C ) LFib( A ) k ∗ l ∗ f ∗ g ∗ exhibits LFib( f ↑ g ) as a comma category . There is a general construction for ﬁnding left adjoints to the pro jections from a comma category , using in this case that g ∗ has a left adjoin t and that LFib( B ) has an initial ob ject preserv ed b y f ∗ . In this case, w e get that the left adjoin ts k ! and l ! resp ectiv ely classify the lax squares LFib( B ) LFib( C ) LFib( B ) LFib( A ) g ! f ∗ η f ∗ g ∗ f ∗ LFib( C ) LFib( C ) LFib( B ) LFib( A ) 0 ! g ∗ f ∗ In particular, the units id LFib( B ) → k ! k ∗ and id LFib( C ) → l ! l ∗ are in v ertible, meaning that the inclusions k and l in to the co comma are fully faithful. Moreo v er, the mate g ! f ∗ → l ∗ k ! is in v ertible, and k ∗ l ! : LFib( C ) → LFib( B ) is the initial functor. 2.17. Lo calisation and pullbac k. The inclusion from spaces to categories admits a left adjoin t |−| , which w e refer to as lo c alisation . This for example shows up when computing left Kan extensions. Lo calization preserv es ﬁnite pro ducts. T o see this the product of categories B and C , one uses that − × C admits a righ t adjoin t F un( C , − ) which preserv es group oids. A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 9 Corollary 2.1. Consider a c artesian squar e of c ate gories D C B A wher e A is a gr oup oid. Then the squar e r emains c artesian after lo c alisation. Pr o of. By extensivit y , w e ma y work ﬁbrewise ov er a : A . The claim then reduces to the fact lo calization preserv es the pro duct of B a and C a . □ 2.18. Left adjoints from arrow categories. The following lemma gives a simple description of left adjoin ts from arrow categories. Lemma 2.2. Consider a functor m : C → Ar( D ) , c orr esp onding to a natur al tr ansformation f ∗ → g ∗ of functors f ∗ , g ∗ : C → D . Supp ose f ∗ and g ∗ have left adjoints f ! , g ! : D → C and that C has pushouts. Then m : C → Ar( D ) has a left adjoint given by pushout g ! dom g ! co d f ! dom m . ⌜ Pr o of. Given an ob ject x → y of Ar( D ) and c : C , we ha v e to sho w that the space of dashed square completions x f ∗ x y g ∗ c is naturally equiv alen t to the space of dashed square completions g 1 x g ! y f ! x c . This is straightforw ard. □ 2.19. 2-limits and 2-colimits of categories. Limits and colimits of categories enjoy ( ∞ , 2)- categorical univ ersal properties. Consider for example the copro duct of categories C and D . A priori, it has a ( ∞ , 1)-categorical universal property , which expresses that the mapping space Map( C ⊔ D , X ) is equiv alen t to the pro duct of mapping spaces Map( C , X ) × Map( D , X ). But more is true: w e ha ve an equiv alence of c ate gories F un( C ⊔ D , X ) ≃ F un( C , X ) × F un( D , X ). T o see this, note that Map( Y , F un( C ⊔ D , X )) and Map( Y , F un( C, X ) × F un( D , X )) are b oth equiv alent to Map( C , F un( Y , X )) × Map( D , F un( Y , X )). Similarly , w e ha v e that F un( X , C × D ) ≃ F un( X , C ) × F un( X, D ). W e will make implicit use of this t yp e of result for constructions lik e comma categories and lo calisations of categories. 10 CHRISTIAN SA TTLER AND DA VID W ¨ ARN 2.20. Co cartesian ﬁbrations and the Conduc h´ e condition. W e sa y that a functor p : E → B is a Conduch´ e ﬁbration 8 if for ob jects x, z : E and b : B , the square of spaces | x ↓ ﬁb p ( b ) ↓ z | E ( x, z ) B ( px, b ) × B ( b, pz ) B ( px, pz ) is cartesian. Here x ↓ ﬁb p ( b ) ↓ z denotes the iterated comma category whose ob jects are giv en b y a lift y : E of b with morphisms x → y and y → z . Lemma 2.3. Every c o c artesian ﬁbr ation is a Conduch´ e ﬁbr ation. Pr o of. Consider a cocartesian ﬁbration p : E → B . The comma category x ↓ ﬁb p ( b ) has a core- ﬂectiv e sub category spanned b y p -co cartesian maps x → y . This subcategory is equiv alen t to the space B ( px, b ). In this w a y one shows that the localisation | x ↓ ﬁb p ( b ) ↓ z | is equiv alen t to B ( px, b ) ↓ z , where the functor B ( px, b ) → E sends a map f : px → b to f ! x . The claim follo ws from taking ﬁbres o ver B ( px, b ) and using that x → f ! x is co cartesian. □ Lemma 2.4. Consider a c o c artesian ﬁbr ation p : E → B with a morphism f : B ( b, c ) such that the c o c artesian ﬁbr e tr ansp ort functor f ! : E b → E c is inverible. Then f has al l p -c artesian lift, and a lift of f is p -c artesian if and only if it is p -c o c artesian. Pr o of. Supp ose f : y → z is a p -co cartesian lift of f ; w e claim that f is also p -cartesian. Th us supp ose x : E lying o v er a : B . W e w an t to sho w that E ( x, y ) → E ( x, z ) × B ( a,c ) B ( a, b ) is in vertible. Let g : B ( a, b ); w e need to show that the map of spaces from the ﬁbre of E ( x, y ) → B ( a, b ) o v er g to the ﬁbre of E ( x, z ) → B ( a, c ) o v er f g is inv ertible. The ﬁrst space is equiv alent to the mapping space ﬁb q ( b )( g ! x, y ) and the second is equiv alen t to ﬁb q ( c )(( f g ) ! x, z ) ≃ ﬁb q ( c )( f ! g ! x, f ! y ). The map is equiv alent to the action of f ! , whic h is inv erible by assumption. Thus f is p -cartesian. Since f ! is surjective, every lift of c is the target of some co cartesian lift of f . This lift is then also cartesian, so f has all cartesian lifts. Uniqueness of cartesian lifts means that any cartesian lift is co cartesian. □ 3. Pulling ba ck a reflective subca tegor y In this short section w e describ e a simple metho d, due to Kelly in the setting of ordinary categories [6], for pulling back a reﬂective sub category C  → A along a right adjoin t f ∗ : B → A . That is, we describ e a formula for a left adjoin t to the inclusion C × A B  → B . Note that C × A B is the full sub category of B spanned by ob jects b suc h that f ∗ b lies in the image of C  → A . W e are particularly in terested in pulling back the reﬂective sub category RFib( W )  → Ar(RFib( W )), spanned b y isomorphisms, along a functor m ∗ : RFib( C ) → Ar(RFib( W )); this will allow us to compute mapping spaces in lo calisations of C . Construction 3.1. Consider categories A, B , C with functors f ∗ : B → A and g ∗ : C → A . Sup- p ose that f ∗ and g ∗ ha v e resp ective left adjoin ts f ! and g ! , and that B has pushouts. W e obtain an endofunctor S : B → B with a p ointing s : id B → S b y the follo wing pushout square: f ! f ∗ id B f ! g ∗ g ! f ∗ S . ε f f ! η g f ∗ ⌜ 8 It is true but not direct that a functor p is a Conduc h´ e ﬁbration if and only if it is exp onen tiable, in the sense that base c hange along p admits a right adjoint [1]. W e do not use this. A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 11 Note that the ab o ve square transp oses under the adjunction f ! ⊣ f ∗ to the square (1) f ∗ f ∗ g ∗ g ! f ∗ f ∗ S . id η g f ∗ f ∗ s Lemma 3.2. Consider a c osp an B f ∗ − → A g ∗ ← − C as in Construction 3.1. Given x, y : B with f ∗ y lying in C , the map s x : x → S x is ortho gonal against y , in the sense that the map B ( S x, y ) → B ( x, y ) of sp ac es is invertible. Pr o of. Since s x is a cobase c hange of f ! η g f ∗ x , it suﬃces to show that f ! η g f ∗ x is orthogonal against y . T ransp osing under the adjunction f ! ⊣ f ∗ , it suﬃces to show that η g f ∗ x is orthogonal against f ∗ y . This is immediate since f ∗ y lies in C . □ Construction 3.3. Consider a cospan of categories B f ∗ − → A g ∗ ← − C as in Construction 3.1. Supp ose that B has sequen tial colimits. Deﬁne S ∞ : B → B as the sequentia l colimit of the sequence id B S S S · · · . s sS sS S W e consider S ∞ to be p ointed b y the sequen tial comp osition id B → S ∞ . Lemma 3.4. Consider a c osp an of c ate gories B f ∗ − → A g ∗ ← − C as in Construction 3.1. Supp ose that B has se quential c olimits, that f ∗ pr eserves them, and that C  → A is close d under se quential c olimits. Then S ∞ takes values in C × A B  → B . Pr o of. Let b : B . W e ha v e to show that f ∗ S ∞ b lies in C . Since f ∗ preserv es sequential colimits, this is the colimit of the sequence f ∗ b f ∗ S b f ∗ S S b · · · . f ∗ s b f ∗ s S b f ∗ s S S b By the square (1), eac h map in this sequence factors through an ob ject of C . The colimit is th us equiv alently a sequential colimit of ob jects of C , so it lies in C since C is closed under sequential colimits. □ Corollary 3.5. Consider a c osp an of c ate gories B f ∗ − → A g ∗ ← − C with adjoints f ! ⊣ f ∗ and g ! ⊣ g ∗ . Assume that g ∗ is ful ly faithful, B has pushouts and se quential c olimits, f ∗ pr eserves se quential c olimits, and C  → A is close d under se quential c olimits. Then the natur al tr ansformation id B → S ∞ fr om Construction 3.3 exhibits S ∞ as a r eﬂe ctor onto the sub c ate gory C × A B  → B . Pr o of. Combine Lemmas 3.2 and 3.4. □ Example 3.6. Let C b e a category and m : W → Ar( C ) an arbitrary functor. Consider the pushout square of categories W × I W C C [ W − 1 ]. fst m j i ⌜ Then i : C → C [ W − 1 ] is the lo calisation of C at the collection of morphisms mw for w : W . By descen t for right ﬁbrations, the induced square of categories RFib( C [ W − 1 ]) RFib( C ) RFib( W ) Ar(RFib( W )) i ∗ j ∗ m ∗ ∆ 12 CHRISTIAN SA TTLER AND DA VID W ¨ ARN is cartesian. The reﬂector of ∆ is co d : Ar(RFib( W )) → RFib( W ). All the conditions of Corol- lary 3.5 are satisﬁed, so w e get a description of the reﬂector i ! : RFib( C ) → RFib( C [ W − 1 ]) as the sequen tial colimit of a sequence id s − → S sS − → S S → · · · . W riting m as a natural transformation k → l of functors k , l : W → C , restriction m ∗ corresp onds to the map l ∗ → k ∗ , and so we can use Lemma 2.2 to obtain a more explicit description of m ! and hence of the endofunctor S . Namely , we hav e m ! ∆ ≃ l ! , and w e hav e the following pushout squares of endofunctors on RFib( C ): k ! l ∗ k ! k ∗ l ! l ∗ m ! m ∗ , ⌜ m ! m ∗ id RFib( C ) l ! k ∗ S . ⌜ Giv en a presheaf P : RFib( C ), we think of S P as the result of freely non-recursiv ely adding inverses to the transp ort maps P ( lw ) → P ( k w ) for w : W . Indeed an ob ject of ( l ! k ∗ P )( c ) is, roughly sp eaking, giv en by an ob ject w : W together with a map f : c → l w and an element p : P ( k w ); we think of this as the “inv erse” to P ( lw ) → P ( k w ) at p follo wed by restriction along f . The ob jects k ! l ∗ , k ! k ∗ , and l ! l ∗ in the pushout square deﬁning m ! m ∗ describ e w a ys in whic h one can build an ob ject of ( l ! k ∗ P )( c ) that ough t to liv e already in P ( c ), by some in v erse la w. An ob ject of ( S n P )( c ) thus corresponds, roughly sp eaking, to an ob ject of P at some p oint x : C , together with a zigzag connecting x with c , con taining at most n ‘in v erse’ zags, all of the form k w → l w . In particular the example ab ov e gives a description of mapping spaces (i.e., representable preshea v es) of the lo calisation C [ W − 1 ]. W e are particularly in terested in the functoriality of this description. T o this end, we employ the organisational device of gluing . 4. Gluing a functor Giv en a functor p : A → B , the gluing Gl( p ) is the comma category B ↓ p . 9 In other w ords, Gl( p ) is cofreely generated by functors π 1 : Gl( p ) → B and π 2 : Gl( p ) → A with a natural transformation α : π 1 → pπ 2 . In this section, we recall some basic results on this construction. Lemma 4.1. L et p : A → B and supp ose A and B have c olimits of shap e J . Then so do es Gl( p ) , and π 1 , π 2 pr eserve c olimits of shap e J . Pr o of. Direct. □ Construction 4.2. Gluing is functorial in lax squares: giv en a lax square of categories A ′ C B D , p q w e obtain a functor Gl( p ) → Gl( q ) which classiﬁes the pasting of the lax square with the lax triangle α : π 1 → pπ 2 . Lemma 4.3. Consider a c ommutative squar e of c ate gories A C B D u q p v 9 This construction is also kno wn as Artin gluing . A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 13 with left adjoints f ⊣ u and g ⊣ v . The induc e d functor U : Gl( p ) → Gl( q ) has a left adjoint F : Gl( q ) → Gl( p ) , which is induc e d by the mate of the squar e ab ove. In particular, the comm utative squares Gl( p ) B Gl( q ) D , π 1 U v π 1 Gl( p ) A Gl( q ) C . π 2 U u π 2 ha v e in v ertible mates, i.e., the mates g π 1 → π 1 F and f π 2 → π 2 F are inv ertible. Pr o of. Consider an ob ject x : b → pa of Gl( p ) and an ob ject y : d → q c of Gl( q ). Maps y → U x are giv en by the dashed data in the square d q c v b v pa q ua . q (?) ∼ Maps F y → x are giv en b y the dashed data in the square g d g q c pg c b pa . p (?) These are naturally equiv alent. □ 4.1. Cartesian maps. Given a functor p : A → B , w e ma y understand the pro jection π 2 : Gl( p ) → A via the restriction of co cartesian ﬁbrations Gl( p ) Ar( B ) A B . π 2 ⌟ cod p A morphism x → Y of Ar( B ) is said to b e c artesian if it is π 2 -cartesian. Equiv alently , its image in Ar( B ) is cartesian, meaning that the follo wing square in B is cartesian: π 1 x π 1 y q π 2 x q π 2 y . α x α y Lemma 4.4. Given a functor p : A → B b e a functor, the c ol le ction of c artesian maps of Gl( p ) the fol lowing closur e pr op erties. (1) Any isomorphism is c artesian. Given maps f : x → y and g : y → z in Gl( p ) with g c artesian, f is c artesian if and only if g f is c artesian. (2) Given a c ommutative squar e v p ≃ q u wher e v pr eserves pul lb acks, the induc e d functor Gl( p ) → Gl( q ) pr eserves c artesian maps. (3) Supp ose A and B have pushouts, that p pr eserves them, and that pushouts in B satisfy desc ent. Then for any sp an y f ← − x g − → z in Gl( p ) with f and g c artesian, the maps y → y ⊔ x z , z → y ⊔ x z ar e also c artesian. 14 CHRISTIAN SA TTLER AND DA VID W ¨ ARN (4) Supp ose A and B have pushouts, that p pr eserves them, and that pushouts in B ar e stable under b ase change. Then for any squar e in Gl( p ) with al l four maps c artesian, the c o gap map is also c artesian. (5) Supp ose A and B have pushouts, that p pr eserves them, and that pushouts in B satisfy desc ent. Then for any map of sp ans in Gl( p ) with al l seven maps involve d c artesian, the induc e d map on pushouts is also c artesian. (6) Supp ose A and B have se quential c olimits, that p pr eserves them, and that se quential c ol- imits in B satisfy desc ent. Then c artesian maps in Gl( p ) ar e close d under tr ansﬁnite c omp osition. (7) Supp ose A and B have se quential c olimits, that p pr eserves them, and that se quential c ol- imits c ommute with pul lb ack in B . Given a se quenc e of maps in Gl( p ) x 0 x 1 · · · x ∞ y 0 y 1 · · · y ∞ , f 0 f 1 f ∞ if the vertic al maps f n ar e al l c artesian, so is the induc e d map f ∞ on se quential c olimits. Pr o of. (1) This is pasting of cartesian morphisms for the functor π 2 . (2) Say q : C → D . In the square of categories Gl( p ) Ar( B ) Gl( q ) Ar( D ), Ar( v ) the horizon tal functors create cartesian morphisms. By assumption, the righ t functor preserv es cartesian morphisms, hence so does the left functor. (3) Since p preserv es pushouts, w e ma y as w ell w ork in Ar( B ) rather than Gl( p ). Here, the statemen t follo ws directly from descen t. (4) Same as ab o v e. (5) Consider a diagram in Gl( p ) b a c y x z with all maps cartesian. By part (3), the map y → y ⊔ x z is cartesian, and so by part (1), so is b → y ⊔ x z , and b y symmetry so is c → y ⊔ x z . W e are now done b y part (4). (6) As ab ov e, this reduces to the claim in Ar( B ), where it is simply the assumption that sequen tial colimits in B satisfy descent. (7) This reduces to the claim in Ar( B ), where it is simply the assumption that sequen tial colimits comm ute with pullback. □ A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 15 4.2. Pulling bac k and gluing reﬂectiv e sub categories. Consider a map of cospans of cate- gories, (2) B u A u C u B d A d C d f ∗ u g ∗ u f ∗ d q p g ∗ d r where each cospan (ro w) satisﬁes the assumptions of Corollary 3.5. By Lemmas 4.1 and 4.3, w e obtain a cospan Gl( q ) Gl( p ) Gl( r ) f ∗ g ∗ whic h again satisﬁes the assumptions of Corollary 3.5. Denote the pullbac ks B u × A u C u and B d × A d C d b y D u and D d , respectively , and the induced functor D d → D u b y t . Note that Gl( t ) — the full sub category of Gl( q ) spanned b y ob jects whose comp onen ts lie in D u and D d — is equiv alently the pullbac k Gl( q ) × Gl( p ) Gl( r ). Moreo ver, the reﬂector S ∞ : Gl( q ) → Gl( t ) is computed comp onent wise, by Lemma 4.3. W e w ould lik e to kno w when it happ ens that, for an ob ject x : Gl( q ), the unit map x → S ∞ x is cartesian. Since this decomp oses as a transﬁnite comp osition x S x S S x · · · , it is natural to look for conditions that ensure that eac h map in this sequence is cartesian. By deﬁnition, s x : x → S x is the cobase change of f ! η g f ∗ x : f ! f ∗ x → f ! g ∗ g ! f ∗ x along ε f x : f ! f ∗ x → x . Sa y that x : Gl( q ) is go o d if η g f ∗ x and ε f x are b oth cartesian. W e are thus led to ask if it happ ens that S x is go o d as so on as x is goo d. Lemma 4.5. Supp ose we ar e given a diagr am of c ate gories as in (2) , wher e b oth c osp ans satisfy the assumptions of Cor ol lary 3.5. Supp ose mor e over that: (1) Se quential c olimits and pushouts in B u and A u satisfy desc ent. (2) The functors p , q , f ∗ , g ∗ pr eserve se quential c olimits and pushouts. 10 (3) g ! : Gl( p ) → Gl( r ) pr eserves c artesian maps. (4) f ! : Gl( p ) → Gl( q ) pr eserves every c artesian map x → y such that η g y : y → g ∗ g ! y is c artesian. (5) F or c : C , the obje ct f ! g ∗ c of Gl( q ) is go o d. Then for any x : Gl( q ) which is go o d, we have that S x is also go o d, and the unit map x → S ∞ x is c artesian. Explicitly , given ob jects x u : B u and x d : B d with a map m : x u → px d , determining a go o d ob ject x : Gl( q ), the statement that the unit map x → S ∞ x is cartesian means that the square in B u x u h u ! x u q x d q h d ! x d η m ⌟ q η is cartesian where h u ! and h d ! denote the reﬂectors B u → D u and B d → D d , respectively . Pr o of. Supp ose x : Gl( q ) is go o d; w e w ant to show that S x is also go o d. That is, w e wan t to sho w that η g f ∗ S x and ε f S x are cartesian. Since b oth the domains and co domains of η g f ∗ and ε f preserv e pushouts, in particular the pushout deﬁning S x , these maps are induced by the following 10 Note that if f ∗ u , f ∗ d , g ∗ u , g ∗ d all preserv e colimits, then so do f ∗ and g ∗ . 16 CHRISTIAN SA TTLER AND DA VID W ¨ ARN maps of spans and we are in a p osition to apply part (5) of Lemma 4.4. W e ha v e that all lab elled maps in the diagrams f ∗ f ! g ∗ g ! f ∗ x f ∗ f ! f ∗ x f ∗ x g ∗ g ! f ∗ f ! g ∗ g ! f ∗ x g ∗ g ! f ∗ f ! f ∗ x g ∗ g ! f ∗ x η g f ∗ f ! g ∗ g ! f ∗ x f ∗ ε f x η g f ∗ x g ∗ g ! f ∗ f ! η g f ∗ x g ∗ g ! f ∗ ε f x and f ! f ∗ f ! g ∗ g ! f ∗ x f ! f ∗ f ! f ∗ x f ! f ∗ x f ! g ∗ g ! f ∗ x f ! f ∗ x g ∗ g ! f ∗ x ε f f ! g ∗ g ! f ∗ x f ! f ∗ f ! η g f ∗ x f ∗ ε f x ε f x f ! η g f ∗ x ε f x are cartesian; it then follo ws b y part (1) of Lemma 4.4 that the unlab elled maps are cartesian, so that S x is go o d. By assumption, x is go o d, i.e., ε f x and η g f ∗ x are cartesian. W e hav e that f ∗ and g ∗ preserv e cartesian maps b y part (2) of Lemma 4.4, and g ! preserv es them by assumption, so f ∗ ε f x and g ∗ g ! f ∗ ε f are cartesian. T o v erify that f ! preserv es the cartesian map η g f ∗ x , we apply condition (4), noting that η g g ∗ g ! f ∗ x is in vertible and so in particular cartesian. W e also w ant to kno w that f ! preserv es the cartesian map f ∗ ε f x ; indeed η g f ∗ x is cartesian by assumption. The remaining tw o maps, η g f ∗ f ! g ∗ g ! f ∗ x and ε f f ! g ∗ g ! f ∗ x , are b oth cartesian b y condition (5) with c = g ! f ∗ x . This ﬁnishes the pro of that if x is go o d then so is S x . If x is go o d, then s x : x → S x is cartesian b y part (3) of Lemma 4.4. Since S x is also go o d, every map in the sequence x s x − → S x s S x − − → S S x → · · · deﬁning S ∞ x is go o d. By part (6) of Lemma 4.4, the transﬁnite comp osition x → S ∞ x is also go o d, as needed. □ 5. Localising a cocar tesian fibra tion Let C b e a category and W a collection of morphisms in C . In this section w e pro v e descen t for co cartesian ﬁbrations along the lo calisation functor i : C → C [ W − 1 ]. Giv en a co cartesian ﬁbration q : E → C , say that q inverts W if for ev ery morphism f : x → y in W , the transport functor f ! : E x → E y on ﬁbres of q is inv ertible. Any co cartesian ﬁbration o v er C [ W − 1 ] pulls bac k along i to giv e a co cartesian ﬁbration ov er C that inv erts W . W e w ould lik e to kno w that every co cartesian ﬁbration whic h inv erts W arises in this wa y . Giv en a co cartesian ﬁbration q : E → C which in v erts W , let W u b e the collection of q -co cartesian lifts of morphisms in W . The comp osite E → C → C [ W − 1 ] extends along i u : E → E [ W − 1 u ] to a functor p : E [ W − 1 u ] → C [ W − 1 ]. Most of this section is devoted to pro ving that q ′ is a co cartesian ﬁbration, and that q is the pullbac k of q ′ . Both of these statemen ts concern mapping spaces of the lo calisations E [ W − 1 u ] and C [ W − 1 ]. W e will see that they can b e reduced to the following more concrete statemen ts. Lemma 5.1. L et q : E → C b e a c o c artesian ﬁbr ation which inverts W and let W u b e the c ol le ction of q -c o c artesian lifts of morphisms in W . (1) F or obje cts x, y : E , the fol lowing squar e of sp ac es is c artesian: E ( x, y ) E [ W − 1 u ]( i u x, i u y ) C ( q x, q y ) C [ W − 1 ]( iq x, iq y ) . ⌟ A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 17 (2) F or obje cts x, y : E and c : C , the fol lowing squar e is c artesian: | x ↓ E c ↓ i u y | E [ W − 1 u ]( i u x, i u y ) C ( q x, c ) × C [ W − 1 ]( ic, iq y ) C [ W − 1 ]( iq x, iq y ) . ⌟ Her e x ↓ E c ↓ i u y denotes the iter ate d c omma c ate gory whose obje cts c onsist of an obje ct z in the ﬁbr e of p over c to gether with maps x → z and i u z → i u y . W e will prov e this by appealing to Lemma 4.5. In fact, we establish a sligh tly more general result. Say a Conduch ´ e ﬁbration q : E → C in v erts W if morphisms in W hav e all q -co cartesian and q -cartesian lifts and these coincide. 11 Lemma 5.2. L et p : E → C b e a Conduch ´ e ﬁbr ation which inverts W and let W u b e the c ol le ction of q -c o c artesian (e quivalently, q -c artesian) lifts of morphisms in W . Then p satisﬁes the c onclusion of L emma 5.1. W e defer the pro of of Lemma 5.2 to the following subsection, but let us ﬁrst remark that it is a direct generalisation of Lemma 5.1. Pr o of of L emma 5.1 fr om L emma 5.2. By Lemmas 2.3 and 2.4, any co cartesian ﬁbration that in- v erts W is also a Conduc h ´ e ﬁbration that in verts W . □ 5.1. Setting up the pro of of the main lemma. Throughout this subsection, w e work in the con text of Lemma 5.2 and w ork tow ard its pro of. W e start by establishing more systematic notation, whic h will be ﬁxed until the lemma is pro v ed. Let us write A d and A u for C and E ; w e systematically use subscripts d and u to denote things taking place at the lev el of A d and A u , resp ectively . So w e write W d instead of W . W e also iden tify W d with its sp ac e of morphisms. Thus W d is a space with a functor W d → Ar( A d ). W e denote the localisation functor A d → A d [ W − 1 d ] b y i d . W e denote b y W u the full subcategory of W d × Ar( A d ) Ar( A u ) spanned by ob jects whose underlying arrow in A u is q -co cartesian (equiv alen tly , q -cartesian). As b efore, w e denote the functor A u [ W − 1 u ] → A d [ W − 1 d ] by q ′ . W e denote the functor W u → W d b y p , and the functors W d → Ar( A d ) and W u → Ar( A u ) b y m d and m u , resp ectively . W e denote the comp osites of m d with dom and co d b y s d , t d : W d → A d , resp ectiv ely , and similarly deﬁne s u , t u : W u → A u . W e emphasize the asymmetry b etw een W d and W u : it is imp ortant that w e consider W d as a group oid and W u as a category . Since W u consists precisely of q -co cartesian (equiv alently , q - cartesian) lifts of W d , both of the follo wing squares of categories are cartesian: W u A u W d A d , s u p ⌟ q s d W u A u W d A d . t u p ⌟ q t d In the lemma b elo w, q ∗ is understo o d to refer to restriction RFib( A d ) → RFib( A u ) of righ t ﬁbrations along q : A u → A d . Given an ob ject a u : A u , w e denote b y ょ Gl a the ob ject of Gl( q ∗ ) giv en b y the unit ょ a u → q ∗ q ! ょ a u where ょ a u denotes the representable righ t ﬁbration dom : A u /a u → A u . Note that q ∗ q ! ょ a u is the righ t ﬁbration q ↓ q a u → A u , and the unit ょ a u → q ∗ q ! ょ a u is given ﬁbrewise b y the action A u ( b, a u ) → A d ( q b, q a u ) of q on morphisms. 11 This can b e read as saying that for a morphism x → y in W , the profunctor E x − 7 − → E y is representable and in vertible. 18 CHRISTIAN SA TTLER AND DA VID W ¨ ARN Lemma 5.3. L et A d , A u , q , W d , W u , p, t d , t u b e as ab ove. Supp ose we ar e given a c artesian squar e of c ate gories as b elow, wher e C d is a gr oup oid: C u A u C d A d . k u r ⌟ q k d Then: (1) The functor k ! : Gl( r ∗ ) → Gl( q ∗ ) pr eserves c artesian maps. (2) F or any a u : A u , the c ounit k ! k ∗ ょ Gl a → ょ Gl a is c artesian. (3) F or any x : Gl( p ) , the c ounit k ! k ∗ t ! x → t ! x is c artesian. Pr o of. (1) Let x → y b e a cartesian map in Gl( r ∗ ). W e denote b y x u → C u the righ t ﬁbration π 1 x , and similarly deﬁne x d , y u , y d . Th us the fact that x → y is cartesian means that x u ≃ x d × y d y u . Using the p oint wise formula for left Kan extension, we hav e to sho w that the follo wing square of spaces is cartesian for a u : A u with a d : = q a u : | a u ↓ x u | | a u ↓ y u | | a d ↓ x d | | a d ↓ y d | . Since C d is a group oid, so is y d , and hence so is a d ↓ y d . So b y Corollary 2.1, it suﬃces to sho w that the square ab ov e is a cartesian square of categories b efore lo calisation. This follo ws from pullback pasting. (2) Given a u , b u : A u with a d : = q a u and b d : = q b u , we ha v e to show that the following square of spaces is cartesian: | b u ↓ C u ↓ a u | A u ( b u , a u ) | b d ↓ C d ↓ a d | A d ( b d , a d ). W orking ﬁbrewise ov er C d and using that C u ≃ C d × A d A u , this amounts to a Conduc h ´ e condition. (3) T o describe k ! k ∗ t ! x , w e use that k ∗ t ! can be rewritten as restriction follo w ed b y left Kan extension (see Section 2.8). W e thus ha v e to show that for a u : A u with a d : = q a u , the follo wing square of spaces is cartesian: | a u ↓ C u ↓ x u | | a u ↓ x u | a d ↓ C d ↓ x d a d ↓ x d . Using Corollary 2.1 and w orking ﬁbrewise o ver C d , x d , this reduces to a Conduch ´ e condition. □ A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 19 Lemma 5.4. With notation and assumptions as b efor e, the diagr am of c ate gories b elow satisﬁes al l the assumptions of L emma 4.5. Mor e over, for any a u : A u , the obje ct ょ Gl a : Gl( q ∗ ) is go o d. RFib( A u ) Ar(RFib( W u )) RFib( W u ) RFib( A d ) Ar(RFib( W d )) RFib( W d ) m ∗ u ∆ m ∗ d q ∗ Ar( p ∗ ) ∆ p ∗ Pr o of. It is immediate that b oth cospans (ro ws) satisfy the assumptions of Corollary 3.5, and that conditions (1) and (2) of Lemma 4.5 are satisﬁed. W e ha v e Gl(Ar( p ∗ )) ≃ Ar(Gl( p ∗ )), with the left adjoin t ∆ ! : Gl(Ar( p ∗ )) → Ar( p ∗ ) to ∆ ∗ cor- resp onding to cod : Ar(Gl( p ∗ )) → Gl( p ∗ ). A morphism in Gl(Ar( p ∗ )) is cartesian if it is sent to cartesian morphisms in Gl( p ∗ ) by b oth dom and co d . It follows that ∆ ! preserv es cartesian mor- phisms, i.e. condition (3) of Lemma 4.5 is satisﬁed. W e also ha ve that, for x : Gl(Ar( p ∗ )), the unit η ∆ x : x → ∆ ∗ ∆ ! x is cartesian if and only if x is cartesian as a morphism in Gl( p ∗ ). The functor m ∗ : Gl( q ∗ ) → Gl(Ar( p ∗ )) corresp onds to the morphism t ∗ → s ∗ of functors Gl( q ∗ ) → Gl( p ∗ ). By Lemma 2.2, the left adjoin t m ! : Gl(Ar( p ∗ )) → Gl( q ∗ ) is giv en by the follo wing pushout: s ! dom s ! co d t ! dom m ! . ⌜ W e claim that the map s ! x → t ! x is cartesian for an y x : Gl( p ∗ ). This amoun ts to showing that the follo wing square of spaces below is cartesian: | a u ↓ s x u | | a u ↓ t x u | a d ↓ s x d a u ↓ t x u . This follo ws from Corollary 2.1 and the fact that the comp onents of s u → t u are q -cartesian. W e no w argue that m ! preserv es ev ery cartesian map x → y in Gl(Ar( p ∗ )) suc h that y determines a cartesian map of Gl( p ∗ ). W rite x 0 → x 1 , y 0 → y 1 for the maps in Gl( p ∗ ) determined b y x, y . The fact that x → y is cartesian means that x 0 → y 0 and x 1 → y 1 are cartesian; by comp osition and cancellation, so is x 0 → x 1 . Now m ! x → m ! y is the map on pushouts induced by the map of spans below: t ! x 0 s ! x 0 s ! x 1 t ! y 0 s ! y 0 s ! y 1 Ev ery map in the ab ov e diagram is cartesian either b y part (1) of Lemma 5.3 or by the fact that s ! → t ! is alw a ys cartesian, so m ! x → m ! y is cartesian b y part (5) of Lemma 4.4. This establishes condition (4) of Lemma 4.5. It remains to v erify that ょ Gl a is go o d for all a u : A u and that m ! ∆ ∗ y is go o d for all y : Gl( p ∗ ). Recall that an ob ject x : Gl( q ∗ ) is go o d if ε m x and t ∗ x → s ∗ x are b oth cartesian. Note that m ! ∆ ∗ y 20 CHRISTIAN SA TTLER AND DA VID W ¨ ARN is simply t ! y . W e ha v e that ε m x is the cogap map of the follo wing square. s ! t ∗ x s ! s ∗ x t ! t ∗ x x , The map s ! t ∗ x → t ! t ∗ x is alwa ys cartesian, as sho wn ab ov e. By parts (1) and (4) of Lemma 4.4, ε m x is cartesian if b oth counits t ! t ∗ x → x and s ! s ∗ x → x are cartesian. In the cases at hand, where x is either ょ Gl a or t ! c , this follows from parts (2) and (3) of Lemma 5.3 The map t ∗ ょ Gl a → s ∗ ょ Gl a b eing cartesian is a direct consequence of the fact that the comp onents of s u → t u are q -co cartesian. The map t ∗ t ! y → s ∗ t ! y b eing cartesian is the statemen t that the square t u w u ↓ t y u s u w u ↓ t y u t d w d ↓ t y d s d w d ↓ t y d is cartesian, which again follo ws from s u w u → t u w u b eing q -co cartesian, together with Corol- lary 2.1. □ W e are no w ﬁnally ready to return to the main lemma. Recall that we c hanged notation after stating the lemma, so that E and C in the statemen t of the lemma correspond to A u and A d , resp ectiv ely . Pr o of of L emma 5.2. W e need to show b oth parts of Lemma 5.1. The square in part (1) is precisely the square underlying the unit map ょ Gl y → S ∞ ょ Gl y ev aluated at an ob ject x : A u . This is cartesian b y Lemmas 5.4 and 4.5. F or part (2), supp ose y : A u and c : A d and consider the cartesian square of categories 12 ﬁb q ( c ) A u { c } A d . k u r ⌟ q k d Sa y that an ob ject x : Gl( q ∗ ) is nic e if the counit k ! k ∗ x → x is cartesian. The desired statement is precisely the statemen t that S ∞ ょ Gl y is nice. Since k ! k ∗ and q ∗ preserv e sequen tial colimits and sequen tial colimits in RFib( A u ) comm ute with pullbacks, any sequential colimit of nice ob jects is nice. W e ha v e that ょ Gl y is nice b y part (2) of Lemma 5.3. W e claim that if x : Gl( q ∗ ) is nice and go o d , then S x is also nice. It then follo ws that S n ょ Gl y is nice and go o d for ev ery n (by Lemmas 5.4 and 4.5), so that S ∞ ょ Gl y is nice. Since k ! k ∗ preserv es pushouts, in particular the pushout deﬁning S x , it suﬃces b y part (5) of Lemma 4.4 to sho w that ev ery lab elled map in the b elow diagram is cartesian: k ! k ∗ m ! ∆ ∗ ∆ ! m ∗ x k ! k ∗ m ! m ∗ x k ! k ∗ x m ! ∆ ∗ ∆ ! m ∗ x m ! m ∗ x x . ε k ··· ϵ k ··· k ! k ∗ ϵ m x ε k x m ! η ∆ m ∗ x ε m x 12 Here, { c } is the terminal category , and the functor k d : { c } → A d selects the ob ject c . A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 21 The b ottom maps are cartesian since x is assumed to b e goo d. The right map is cartesian since x is assumed to b e nice. T o see that the left map is cartesian, note that m ! ∆ ∗ ∆ ! m ∗ x ≃ t ! s ∗ x and apply part (3) of Lemma 5.3. T o see that k ! k ∗ ε m x is cartesian we apply part (1) of Lemma 5.3. □ 5.2. Descen t along lo calisations. At this p oint, we treat Lemma 5.1 as a blac k b o x and mak e no further use of gluings or of the sequential-colimit description of mapping spaces of lo calisations. Let us switch back to writing C , E , W in place of A d , A u , W d , respectively . W e record t w o easy observ ations ab out the lo calisation i : C → C [ W − 1 ]. First, it is surjectiv e; indeed the inclusion of the essential image of i into C [ W − 1 ] has a section by the universal property of C [ W − 1 ]. Second, given ob jects x, y : C and a map f : ix → iy in the lo calisation, there exists a sequence of ob jects x = x 0 , x 1 , . . . , x n = y and maps f j : ix j → ix j +1 whose comp osite is f and suc h that each f j is either of the form ig with g : x j → x j +1 , or of the form ( ig ) − 1 with g in W . This can b e sho wn by using that i ∗ i ! C ( − , y ) is the initial presheaf under C ( − , y ) with the prop erty of in v erting W . Lemma 5.5. L et q : E → C b e a c o c artesian ﬁbr ation which inverts W and let W u b e the c ol le ction of q -c o c artesian lifts of morphisms in W . L et q ′ b e the induc e d functor E [ W − 1 u ] → C [ W − 1 ] . Then q ′ is a c o c artesian ﬁbr ation and the fol lowing squar e is c artesian: E E [ W − 1 u ] C C [ W − 1 ] . i u q ⌟ q ′ i Pr o of. Let us denote the lo calisations C [ W − 1 ] and E [ W − 1 u ] by C ′ and E ′ , resp ectively . By part (1) of Lemma 5.1, we hav e that the gap map E → E ′ × C ′ C is fully faithful. In particular, this means that for c : C , the functor ﬁb q ( c ) → ﬁb q ′ ( ic ) is fully faithful. W e can reformulate part (2) of Lemma 5.1 using the fact that q is a co cartesian ﬁbration. Let x, y : E and c : C . Note that for x : E , the comma category x ↓ E c has a coreﬂectiv e subcategory spanned by q -co cartesian maps. This means that for f : q x → c , the ﬁbre of | x ↓ E c ↓ i u y | o v er f is the mapping space E ′ ( i u f ! x, i u y ). In this w a y w e get that the follo wing square of spaces is cartesian: E ′ ( i u f ! x, i u y ) E ′ ( i u x, i u y ) C ′ ( ic, iq y ) C ′ ( iq x, iq y ). ⌟ Since every ob ject of E ′ is equiv alen t to one of the form i u y , this means that i u g : i u x → i u f ! x is q ′ -co cartesian, where g denotes the q -co cartesian map x → f ! x . W e no w claim that given a, b : C , f : C ′ ( ia, ib ), and x : ﬁb q ′ ( ia ) suc h that x lies in the image of ﬁb q ( a )  → ﬁb q ′ ( ia ), there is y : ﬁb q ′ ( ib ) and a q ′ -co cartesian lift x → y of f , and moreo v er, y lies in the image of ﬁb q ( b )  → ﬁb q ′ ( ib ). Since q ′ -co cartesian maps are closed under comp osition, it suﬃces to consider the cases where f is either of the form ig , or of the form ( ig ) − 1 with g in W . The ﬁrst case w as dealt with ab ov e, so w e consider the case of f = ( ig ) − 1 . Since f is in v ertible in this case, there is trivially a q ′ -co cartesian lift, given b y an isomorphism x ≃ y . The fact that y lies in the image of ﬁb q ( b )  → ﬁb q ′ ( ib ) can b e seen b y considering the square ﬁb q ( b ) ﬁb q ′ ( ib ) ﬁb q ( a ) ﬁb q ′ ( ia ), g ! ( ig ) ! 22 CHRISTIAN SA TTLER AND DA VID W ¨ ARN in whic h the vertical maps are in vertible, and x : ﬁb q ′ ( ia ) is the image of y under ( ig ) ! . No w let a : C and x : ﬁb q ′ ( ia ) b e arbitrary . Since i u : E → E ′ is surjectiv e, there is y : E with g : i u y ≃ x . In particular, q ′ g gives us an isomorphism f : iq y ≃ ia , and g is precisely the q ′ -co cartesian lift of f . By the claim ab o v e, this means that x lies in the image of ﬁb q ( a )  → ﬁb q ′ ( ia ), so this functor is surjective. This means that the gap map E → E ′ × C ′ C is surjectiv e and hence in v ertible. This also means that the claim ab ov e, together with surjectivity of i : C → C ′ , establishes that q ′ is a co cartesian ﬁbration. □ Remark 5.6. Conduc h ´ e ﬁbrations enjoy a straigh tening–unstraigh tening corresp ondence which generalises that for (co)cartesian ﬁbrations [1], and it is reasonable to ask if our methods apply also to Conduc h ´ e ﬁbrations. Indeed, the main ingredien t, Lemma 5.2, applies to Conduch ´ e ﬁbrations. Ho w ever, straightening of Conduch ´ e ﬁbrations is inherently more complicated than straightening of co cartesian ﬁbrations, since Conduc h´ e ﬁbrations are classiﬁed b y ﬂagge d functors (i.e., not just functors in the ordinary sense) into the ﬂagged category of categories and profunctors b etw een them. Essen tially this is b ecause not ev ery in v ertible profunctor is representable. A concrete manifestation of this is that Lemma 5.5 does not apply to Conduc h´ e ﬁbrations. Indeed, let F : A → B b e some functor which deﬁnes an in v ertible profunctor but not an equiv alence of categories. F or example A could b e the w alking idemp otent and B the walking section-retraction pair. W e build a Conduch ´ e ﬁbration q : E → ∆ 3 as follo ws: the ﬁbres ov er the ob jects 0 , 1 , 2 , 3 of ∆ 3 are A , B , A , B , resp ectively , and the profunctors b et w een ﬁbres induced b y the generating morphisms 0 → 1, 1 → 2, 2 → 3 of ∆ 3 are given by F , F − 1 , and F , resp ectively . It can b e seen that q in v erts the morphisms 0 → 2 and 1 → 3. The lo calisation of ∆ 3 at these tw o morphisms is the terminal category 1. Since the ﬁbres A and B o v er 0 , 1 : ∆ 3 are not equiv alen t, it cannot b e that q : E → ∆ 3 is the pullback of a Conduch ´ e ﬁbration E ′ → 1. Theorem 5.7. L et C b e a c ate gory W a c ol le ction of morphisms, and let i : C → C [ W − 1 ] b e the c orr esp onding lo c alisation. Then the (wild) functor i ∗ : Co cartFib( C [ W − 1 ]) → Co cartFib( C ) given by b ase change along i is ful ly faithful, and its image is sp anne d by c o c artesian ﬁbr ations that invert W . Pr o of. Since i is surjectiv e, i ∗ is conserv ative. No w let q : E → C b e a cocartesian ﬁbration that in v erts W . By Lemma 5.5, we obtain a cocartesian ﬁbration q ′ : E [ W − 1 u ] → C [ W − 1 ] together with an isomorphism η : q → i ∗ q ′ . W e claim that ( q ′ , η ) is initial in q ↓ i ∗ . This means that for an y other co cartesian ﬁbration p : Y → C [ W − 1 ], the space of functors of co cartesian ﬁbrations q ′ → p is equiv alent to the space of functors of co cartesian ﬁbrations q → i ∗ p , b y the map F 7→ i ∗ ( F ) η . By construction, the space Map C [ W − 1 ] ( q ′ , p ) of al l functors of categories ov er C [ W − 1 ], not neces- sarily preserving co cartesian maps, is equiv alent to the subspace of Map C ( q , i ∗ p ) spanned b y those functors F : E → i ∗ Y such that the comp osite E → Y inv erts ev ery morphism of W u . If a functor F : E → i ∗ Y ov er C preserves co cartesian maps, then so do es E → Y , so morphisms in W u are sen t to co cartesian lifts of isomorphisms in Y , and so E → Y in verts W u . It remains only to show that the induced functor F ′ : E [ W − 1 u ] → Y also preserv es co cartesian maps. This follows from the fact that ev ery morphism in C [ W − 1 ] can b e written as a comp osite of morphisms in the image of i (whose co cartesian lifts are preserv ed by F ′ since F preserves cocartesian maps) and isomorphisms (whose cocartesian lifts are in v ertible and so preserved b y ev ery functor). Th us, q 7→ q ′ determines a left adjoint to the functor from Co cartFib( C [ W − 1 ]) to the full sub category of Co cartFib( C ) spanned by co cartesian ﬁbrations that in v ert W . Since the unit of this adjunction is in v ertible and the right adjoin t is conserv ativ e, this is an equiv alence of (wild) categories. □ A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 23 6. Other descent st a tements In this section we prov e descen t statements for co cartesian ﬁbrations o v er co commas, sequential colimits, and group oids. Lemma 6.1. Given a c o c omma lax squar e of c ate gories A C B D , g f k h the c orr esp onding lax squar e of (wild) c ate gories of c o c artesian ﬁbr ations Co cartFib( D ) Co cartFib( B ) Co cartFib( C ) Co cartFib( A ) h ∗ k ∗ f ∗ g ∗ is a c omma lax squar e. Pr o of. Since h and k are join tly surjective, h ∗ and k ∗ are jointly conserv ative, and so the functor R : Co cartFib( D ) → f ∗ ↓ g ∗ is conserv ative. W e claim that it has a left adjoint and that the unit is in v ertible; it then follo ws that it is an equiv alence of (wild) categories. An ob ject of f ∗ ↓ g ∗ is given b y co cartesian ﬁbrations p : Y → B and q : Z → C together with a cocartesian functor f ∗ Y → g ∗ Z o v er A . This data is describ ed b y the diagram Y f ∗ Y Z B A C . p u v ⌞ q f g Let W denote the co comma of Y u ← − f ∗ Y v − → Z , and denote the inclusions in to W b y s : Y  → W and t : Z  → W . W e claim that the induced functor r : W → D is a co cartesian ﬁbration. There are three kinds of morphisms in D : those coming from B , those coming from C , and “heteromorphisms”, of the form hb → k c . The base changes of r along h and k are p and q , resp ectiv ely , essentially b ecause h and k are fully faithful. This sho ws that r has co cartesian lifts of morphisms coming from C , with t : Z  → W sending q -co cartesian maps to r -co cartesian maps. It also sho ws that r has lo c al ly c o c artesian lifts of morphisms coming from B . It remains to show that it has co cartesian lifts of heteromorphisms hc → k c , and that these are stable under pre- and p ostcomp osition with morphisms coming from B or C . Consider ob jects y : Y and z : Z . Then the mapping space W ( sy , tz ) is the lo calisation of y ↓ f ∗ Y ↓ z , and the mapping space D ( hpy , k q z ) is the lo calisation of pq ↓ A ↓ q z . Consider the coreﬂectiv e sub category y ↓ cocart f ∗ Y ↓ Z of y ↓ f ∗ Y ↓ z spanned by ob jects whose underlying morphism in Y is p -co cartesian. Then w e can equiv alently compute D ( hpy , k q z ) as the lo calisation of y ↓ cocart f ∗ Y ↓ Z . No w consider the functor F : ( y ↓ cocart f ∗ Y ↓ Z ) → ( pq ↓ A ↓ q z ). W e claim that F is a Kan ﬁbr ation , i.e., that it is both a left ﬁbration and a righ t ﬁbration. The ﬁbre of F o v er an ob ject giv en b y a : A with m : pq → f a and n : g a → q z is the mapping space Z q z ( n ! v ( m ! y , a ) , z ). Showing that F is a Kan ﬁbration b oils do wn to the fact that, given a, a ′ : A with maps m : pq → f a and l : a → a ′ and n : g a ′ → q z , w e hav e ( ng ( l )) ! v ( m ! y , a ) ≃ n ! v (( f ( l ) m ) ! y , a ′ ), since v sends the co cartesian map ( m ! y , a ) → (( f ( l ) m ) ! y , a ′ ) to a q -co cartesian lift of g ( l ). 24 CHRISTIAN SA TTLER AND DA VID W ¨ ARN Admitting that F is a Kan ﬁbration, descen t implies that the square of categories y ↓ cocart f ∗ Y ↓ z W ( sy , tz ) py ↓ A ↓ q z D ( hpy , k q z ). is cartesian. Since the b ottom map is surjectiv e (indeed, a lo calisation), this means that ev ery ﬁbre of W ( sy , tz ) → D ( hpy , k q z ) is of the form Z q z ( n ! v ( m ! y , a ) , z ). In particular, heteromorphisms ha v e co cartesian lifts: these corresp ond to isomorphisms n ! v ( m ! y , a ) ≃ n ! v ( m ! y , a ). F rom this description, it is clear that they are stable under precomp osition with lo cally cocartesian lifts of morphisms from B . This ﬁnishes the pro of that r is a co cartesian ﬁbration. The inclusions in to the cocomma W deﬁne a map from Y u ← − f ∗ Y v − → Z to Rr in f ∗ ↓ g ∗ . Moreo v er, this map is inv ertible, since Y ≃ h ∗ W and Z ≃ k ∗ W . Moreov er, this deﬁnes left adjoint to R , essen tially by the univ ersal prop erty of W as a co comma. Since R is conserv ativ e and has a left adjoin t with inv ertible unit, R is in v ertible. □ Lemma 6.2. Given a se quential c olimit diagr am of c ate gories C 0 C 1 · · · C ∞ , f 0 i 0 f 1 i 1 the c orr esp onding diagr am of (wild) c ate gories of c o c artesian ﬁbr ations Co cartFib( C ∞ ) · · · Co cartFib( C 1 ) Co cartFib( C 0 ) i ∗ 1 i ∗ 0 is a se quential limit diagr am. Pr o of. Since I detects equiv alences of categories and Map( I , − ) commutes with sequen tial colimits, sequen tial colimits of categories satisfy descent, i.e., Cat /C ∞ is the sequen tial limit of Cat /C 0 f ∗ 0 ← − Cat /C 1 · · · . Again using compactness of I and the fact that sequential colimits commute with pullbac k in spaces, one can chec k that a category ov er C ∞ is a co cartesian ﬁbration as soon as its pullback to C n is a co cartesian ﬁbration for every n , and similarly with functors of cocartesian ﬁbrations o v er C ∞ . □ Lemma 6.3. Given a gr oup oid X , the (wild) functor Co cartFib( X ) → Cat X which c omputes the ﬁbr es of a c o c artesian ﬁbr ation is an e quivalenc e of (wild) c ate gories. Pr o of. Every category ov er X is a cocartesian ﬁbration, with cocartesian maps given by the iso- morphisms. Therefore, Co cartFib( X ) is just the (wild) category of categories ov er X . The claim th us reduces to extensivity of the (wild) category of categories. □ 7. A directed join constr uction In this section, w e describe a model-dep endent wa y of decomposing a general category in to simpler pieces. More precisely , for an y functor f : A → B , we give a sequential colimit form ula for the full image of f . This is a directed generalisation of the join construction [9], which deals with A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 25 the case where A and B are group oids. W e are primarily interested in the case where f is a c or e inclusion C ≃ → C . Construction 7.1. Consider categories A 0 , A 1 , B with functors f 0 : A 0 → B and f 1 : A 1 → B . W e deﬁne A 0  ∗ B A 1 to b e the category freely generated by functors i 0 : A 0 → A 0  ∗ B A 1 and i 1 : A 1 → A 0  ∗ B A 1 together with a natural transformation α : i 0 dom → i 1 co d of functors f 0 ↓ f 1 → A 0  ∗ B A 1 , sub ject to the condition that the restriction of α along A 0 × B A 1  → f 0 ↓ f 1 is in v ertible. This category comes with a functor A 0  ∗ B A 1 → B , whic h we denote b y i 0  ∗ i 1 : A 0 × B A 1 f 0 ↓ f 1 A 1 A 0 A 0  ∗ B A 1 B . cod dom α i 1 i 0 i 0  ∗ i 1 Explicitly , w e can build A 0  ∗ B A 1 as the lo calisation of the co comma of A 0 dom ← − − f 0 ↓ f 1 cod − − → A 1 at the collection of morphisms given b y applying the generating 2-cell to an ob ject of f 0 ↓ f 1 whose underlying morphism in B is in vertible. Lemma 7.2. L et A , B , X 0 b e c ate gories and let f : A → B and g 0 : X 0 → B b e functors. F or every n : N , a c ate gory g n : X n → B over B is given by taking X n +1 : = A  ∗ B X n , g n +1 : = f  ∗ g n . L et X ∞ b e the c olimit of the se quenc e X 0 i 1 − → X 1 i 1 − → · · · . Then assuming the image of g 0 is c ontaine d in the image of f , we have that X ∞ g ∞ − → B is ful ly faithful, and exhibits X ∞ as the ful l image of f . Pr o of. F or 0 ≤ m ≤ n ≤ ∞ , denote the functor X m → X n b y ι n m . F or n : N , denote the functor A → X n b y j n . Since the functors j n : A → X n +1 and ι n +1 n : X n → X n +1 are join tly surjective, the image of g n +1 is the union of the images of f and g n . It follo ws that the image of g n is con tained in the image of f for ev ery n . Since the functors ι ∞ n : X n → X ∞ are jointly surjectiv e, the image of g ∞ coincides with the image of f . Th us it remains to show that g ∞ is fully faithful. W e wan t to show that, for ev ery pair of ob jects x ′ , y : X ∞ , the map X ∞ ( x ′ , y ) → B ( g ∞ x ′ , g ∞ y ) is an equiv alence of spaces. Since the functors ι ∞ n : X n → X ∞ are join tly surjectiv e, w e ma y assume that x ′ is of the form ι ∞ n x with x : X n . Without loss of generalit y , we may assume n = 0. W e ma y also assume we are giv en a : A and an isomorphism e : f a ≃ g 0 x in B . Since X 1 is deﬁned as A  ∗ B X 0 , w e hav e a lax square f ↓ g 0 X 0 A X 1 . cod dom α ι 1 0 j 0 This induces a functor f ↓ g 0 → j 0 ↓ ι 1 0 . Restricting along a : 1 → A , w e get a functor H : f a ↓ g 0 → j 0 a ↓ ι 1 0 of left ﬁbrations o ver X 0 . By construction of X 1 , w e ha v e that H sends e : f a ∼ − → g 0 x to an isomorphism j 0 a ≃ ι 1 0 x . W e also hav e a functor G : j 0 a ↓ ι 1 0 → g 1 j 0 a ↓ g 1 ι 1 0 , giv en essen tially b y the action of g 1 on morphism. The comp osite GH : f a ↓ g 0 → g 1 j 0 a ↓ g 1 ι 1 0 is the equiv alence induced b y f ≃ g 1 j 0 and g 0 ≃ g 1 ι 1 0 . W e can no w build the commutativ e diagram in LFib( X 0 ): (3) f a ↓ g 0 j 0 a ↓ ι 1 0 g 1 j 0 a ↓ g 1 ι 1 0 g 0 x ↓ g 0 ι 1 0 x ↓ ι 1 0 g 1 ι 1 0 x ↓ g 1 ι 1 0 . H ∼ G ∼ ∼ H ′ G ′ 26 CHRISTIAN SA TTLER AND DA VID W ¨ ARN The v ertical equiv alences are induced b y e : f a ≃ g 0 x and H e : j 0 a ≃ ι 1 0 x . The functor G ′ is giv en b y the action of g 1 on morphisms, and the composite G ′ H ′ b y the isomorphism g 0 ≃ g 1 ι 1 0 . W e can no w forget ab out a and the top ro w of the diagram. Since G ′ is the base c hange of the functor ι 1 0 x ↓ X 1 → g 1 ι 1 0 x ↓ g 1 along ι 1 0 , w e hav e also the follo wing square: x ↓ X 0 ι 1 0 x ↓ X 1 g 0 x ↓ g 0 g 1 ι 1 0 x ↓ g 1 . All sides admit a simple description not in v olving a or e : the v ertical functors are given by the actions of g 0 and g 1 on morphisms, the top map is induced b y the action of ι 1 0 on morphisms, and the b ottom map is induced b y the isomorphism g 0 ≃ g 1 ι 1 0 . The indicated diagonal functor is induced b y H ′ . The bottom right triangle commutes since it is induced by G ′ H ′ , whic h we discussed abov e, and the top left triangle can b e seen to comm ute b y the Y oneda lemma. Rep eating this construction for ev ery n : N , we obtain the following commutativ e diagram: x ↓ X 0 ι 1 0 x ↓ X 1 ι 2 0 x ↓ X 2 · · · g 0 x ↓ g 0 g 1 ι 1 0 x ↓ g 1 g 2 ι 2 0 x ↓ g 2 · · · . The colimit of the top ro w is ι ∞ 0 x ↓ X ∞ . The colimit of the b ottom ro w is g ∞ ι ∞ 0 x ↓ g ∞ . The diagonal maps tell us that the induced functor F : ι ∞ 0 x ↓ X ∞ → g ∞ ι ∞ 0 x ↓ g ∞ is in v ertible. It remains to show that F is isomorphic to the functor given b y the action of g ∞ on morphisms; since F is inv ertible it follows from this that g ∞ is fully faithful. By the Y oneda lemma, it suﬃces to show that F deﬁnes a morphism of categories o v er X ∞ , and that it sends the identit y on ι ∞ 0 x to the identit y on g ∞ ι ∞ 0 x . The former amounts to the fact that the witnesses of commutativit y in (3) witness commutativit y in LFib( X 0 ) and not just in Cat . The latter follo ws from the fact that the functor x ↓ X 0 → g 0 x ↓ g 0 sends the iden tity on x to the iden tit y on g 0 x . □ The directed join construction gives the follo wing induction principle, whic h allo ws us to prov e a result for a general category “by induction” on how the category is built from simpler pieces. Lemma 7.3. L et P b e a class of c ate gories such that: (1) Every gr oup oid is in P . (2) If B f ← − A g − → C is a sp an of c ate gories such that A , B , C ar e al l in P , then f ↑ g is also in P . (3) If C 0 → C 1 → · · · is a se quenc e of c ate gories that ar e al l in P , then their c olimit is also in P . (4) If C is in P and D is a lo c alisation of C at some c ol le ction of morphisms, then D is in P . Then every c ate gory is in P . Pr o of. Let C be a category; w e w ant to sho w that C is in P . Consider Lemma 7.2 with A = X 0 = C ≃ and B = C . Since the full image of the core inclusion C ≃ → C is all of C , this exhibits C as the sequential colimit of a sequence of categories X 0 → X 1 → · · · . It thus suﬃces to sho w that X n is in P for ev ery n . Let Q b e the class of categories Y such that for every left ﬁbration Z → Y , the category Z lies in P . W e will show, by induction on n , that X n lies in Q ; it follo ws that X n lies in P since the iden tit y X n → X n is a left ﬁbration. A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 27 Note that if Y is a group oid, then Y lies in Q , since for Z → Y a left ﬁbration, Z is also a group oid. W e claim that if Y lies in Q and Y ′ is a lo calisation of Y , then Y ′ also lies in Q . Indeed, if Z → Y ′ is a left ﬁbration, then Z × Y ′ Y lies in P since Y lies in Q , and Z is a lo calisation of Z × Y ′ Y , so it also lies in P . Next we claim that for any span of categories B f ← − A g − → C where B is a group oid, g is a left ﬁbration, and C lies in Q , the co comma category f ↑ g also lies in Q . Supp ose W → f ↑ g is a left ﬁbration. Denote by Y → B and Z → C its base changes along Y → f ↑ g and Z → f ↑ g , resp ectiv ely . Then W is the co comma of Y ← f ∗ Y → Z . Y lies in P since it is a group oid. Z lies in P since Z → C is a left ﬁbration, and f ∗ Y lies in P since f ∗ Y → A and g : A → C are left ﬁbrations. Th us W lies in P , as needed. It is now direct b y induction that X n lies in Q for ev ery n . □ 7.1. A virtual directed join construction. T o construct directed univ alen t co cartesian ﬁbra- tions, we no w wan t to repla y the directed join construction for cocartesian ﬁbrations E → C (though t of as functors C → Cat ). The analogue of comma categories turns out to b e a bit subtle. Lemma 7.4. L et p 0 : E 0 → C 0 and p 1 : E 1 → C 1 b e two c o c artesian ﬁbr ations. Assuming that C 0 is a gr oup oid, then ther e is a c ate gory F un cocart ( p 0 , p 1 ) which is c ofr e ely gener ate d by functors u 0 : F un cocart ( p 0 , p 1 ) → C 0 , u 1 : F un cocart ( p 0 , p 1 ) → C 1 with a functor u 01 : u ∗ 0 E 0 → u ∗ 1 E 1 of c o c artesian ﬁbr ations over F . In the end, such a category can b e shown to exist ev en if C 0 is not a group oid (e.g., by taking the comma category of the straightenings of p 0 and p 1 ), but for no w we will mak e do with the restricted statemen t. Pr o of. W e assume that C 0 is the terminal category; the general case follo ws b y extensivit y ov er C 0 . Deﬁne a category F ′ b y the pullback square F ′ F un( E 0 , E 1 ) C 1 F un( E 0 , C 1 ) ⌟ ∆ Then F ′ is cofreely generated by functors u 0 : F ′ → C 0 (this data is trivial) and u 1 : F ′ → C 1 together with a functor u 01 : u ∗ 0 E 0 → u ∗ 1 E 1 of categories o ver F ′ . In particular, an ob ject of F ′ is giv en b y an ob ject c : C 1 together with a functor a : E 0 → ﬁb p 1 ( c ). A morphism from ( c, a ) to ( d, b ) is giv en b y a morphism f : c → d together with a natural transformation f ! a → b of functors E 0 → ﬁb p 1 ( d ). Let F denote the wide sub category of F ′ spanned by morphisms where the natural transformation f ! a → b is in v ertible. W e claim that a functor h : D → F ′ factors through F if and only if the functor h ∗ u 01 : h ∗ u ∗ 0 E 0 → h ∗ u ∗ 1 E 1 of categories ov er D preserv es cocartesian maps; it then follo ws that F has the desired universal prop ert y . Since C 0 is terminal, h ∗ u ∗ 0 E 0 is simply E 0 × D , and a morphism is co cartesian for the pro jection E 0 × D → D if and only if it is of the form ( id e , g ) : ( e, x ) → ( e, y ) for g : D ( x, y ). The functor h : D → F ′ assigns to x and y resp ective functors a : E 0 → ﬁb p 1 ( u 1 hx ) and b : E 0 → ﬁb p 1 ( u 1 hy ), and to g a natural transformation β : ( u 1 hg ) ! a → b . The functor h ∗ u 01 sends ( id e , g ) to a morphism in E 1 dgiv en b y the comp osite ae cocart − − − → ( u 1 g ) ! ae β e − → be . This is p 1 -co cartesian if and only if β e is in v ertible. Th us h ∗ u 01 preserv es ev ery co cartesian morphism of the form ( id e , f ) if and only if hf lies in F ′ . □ 28 CHRISTIAN SA TTLER AND DA VID W ¨ ARN The follo wing construction should b e though t of as a virtual analogue of Construction 7.1. Construction 7.5. Giv en co cartesian ﬁbrations p 0 : E 0 → C 0 and p 1 : E 1 → C 1 where C 0 is a group oid, w e denote by C 0  ∗ p 0 ,p 1 C 1 the localisation of the co comma of C 0 u 0 ← − F un cocart ( p 0 , p 1 ) u 1 − → C 1 at the collection of morphisms corresp onding to ob jects x of F un cocart ( p 0 , p 1 ) whose underlying functor ﬁb p 0 ( u 0 x ) → ﬁb p 1 ( u 1 x ) is inv ertible. Denote the inclusions from C 0 , C 1 in to C 0  ∗ p 0 ,p 1 C 1 b y i 0 , i 1 . Denote the natural transformation i 0 u 0 → i 1 u 1 b y α . By Theorem 5.7 and Lemma 6.1, we obtain a co cartesian ﬁbration p 0  ∗ p 1 o v er C 0  ∗ p 0 ,p 1 C 1 with witnesses that its base c hanges along i 0 , i 1 are p 0 , p 1 , as well as a witness that the cocartesian functor u 01 : u ∗ 0 p 0 → u ∗ 1 p 1 is induced by α : i 0 u 0 → i 1 u 1 . Recall that a functor p : E → B is said to classify a category C if there exists an ob ject b : B together with an equiv alence of categories C ≃ ﬁb p ( b ). A co cartesian ﬁbration p : E → B is said to b e dir e cte d univalent if for every x, y : B , the map B ( x, y ) → Map(ﬁb p ( x ) , ﬁb p ( y )) given b y f 7→ f ! is an equiv alence of spaces. The follo wing lemma can b e though t of as a virtual analogue of Lemma 7.2, or as a dir e cte d analogue of Uemura’s construction of univ alen t completion [10, Prop osition 4.7]. Lemma 7.6. Consider c o c artesian ﬁbr ations p : B → A and q 0 : Y 0 → X 0 wher e A is a gr oup oid. Deﬁne inductively over n : N a c o c artesian ﬁbr ation q n : Y n → X n by taking q n +1 to b e the c o c arte- sian ﬁbr ation p  ∗ q n of Construction 7.5. L et X ∞ denote the c olimit of the se quenc e X 0 i 1 − → X 1 i 1 − → X 2 → · · · . By L emma 6.2, we obtain a c o c artesian ﬁbr ation q ∞ : Y ∞ → X ∞ whose b ase change along X n → X ∞ is q n . Supp ose that every c ate gory classiﬁe d by q 0 is also classiﬁe d by p . Then q ∞ is dir e cte d univalent, and a c ate gory is classiﬁe d by q ∞ if and only if it is classiﬁe d by p . Pr o of. W e follo w the pro of of Lemma 7.2 m utatis m utandis. Denote the functor A → X n +1 b y j n , and the functor X m → X n for 0 ≤ m ≤ n ≤ ∞ by ι n m . W e ha v e that j n and ι n +1 n are join tly surjectiv e, so every category classiﬁed by q n is also classiﬁed by p . Since the functors ι ∞ n are join tly surjectiv e, a category is classiﬁed b y q ∞ if and only if it is classiﬁed b y p . It remains to show that q ∞ is directed univ alent, i.e., that for all x ′ , y : X ∞ , the map X ∞ ( x ′ , y ) → Map(ﬁb q ∞ ( x ′ ) , ﬁb q ∞ ( y )) is in v ertible. Since the functors ι ∞ n are join tly surjective, it suﬃces to consider x ′ of the form ι ∞ n x with x : X n . Without loss of generality w e may assume n = 0. W e ma y also assume that we are giv en a : A and an equiv alence e : ﬁb p ( a ) ≃ ﬁb q 0 ( x ). Since X 1 is deﬁned as A  ∗ p,q 0 X 0 , w e hav e a lax square F un cocart ( p, q 0 ) X 0 A X 1 . u 1 u 0 α ι 1 0 j 0 Restricting along a : 1 → A , this giv es a functor H : F un cocart ( a ∗ p, q 0 ) → j 0 a ↓ ι 1 0 of left ﬁbrations o v er X 0 . By construction of X 1 , we ha ve that H sends e to an isomorphism j 0 a ≃ ι 1 0 x . W e also ha ve a functor G : j 0 a ↓ ι 1 0 → F un cocart (( j 0 a ) ∗ q 1 , ι 1 ∗ 0 q 1 ) of left ﬁbrations o v er X 0 , which sends a morphism f : j 0 a → ι 1 0 y to the transport functor f ! : ﬁb q 1 ( j 0 a ) → ﬁb q 1 ( ι 1 0 y ). The composite GH : F un cocart ( a ∗ p, q 0 ) → F un cocart (( j 0 a ) ∗ q 1 , ι 1 ∗ 0 q 1 ) is the equiv alence induced by p ≃ j ∗ 0 q 1 and A SYNTHETIC CONSTR UCTION OF UNIVERSAL COCAR TESIAN FIBRA TIONS 29 q 0 ≃ ι 1 ∗ 0 q 1 . W e can no w build the following comm utative diagram in LFib( X 0 ): F un cocart ( a ∗ p, q 0 ) j 0 a ↓ ι 1 0 F un cocart (( j 0 a ) ∗ q 1 , ι 1 ∗ 0 q 1 ) F un cocart ( x ∗ q 0 , q 0 ) ι 1 0 x ↓ ι 1 0 F un cocart (( ι 1 0 x ) ∗ q 1 , ι 1 ∗ 0 q 1 ). H ∼ G ∼ ∼ H ′ G ′ The vertical equiv alences are induced by e : a ∗ p ≃ x ∗ q 0 and H e : j 0 a ≃ ι 1 0 x . The functor G ′ sends a morphism f to the transp ort functor f ! . The comp osite G ′ H ′ is induced b y the equiv alence q 0 ≃ ι 1 0 q 1 . W e can no w forget ab out a and the top row of the diagram ab o ve. Since G ′ is the base change of ι 1 0 x ↓ X 1 → F un cocart (( ι 1 0 x ) ∗ q 1 , q 1 ) along ι 1 0 : X 0 → X 1 , w e also ha v e the follo wing square: (4) x ↓ X 0 ι 1 0 x ↓ X 1 F un cocart ( x ∗ q 0 , q 0 ) F un cocart (( ι 1 0 x ) ∗ q 1 , q 1 ). All sides admit a simple description not in v olving a or e . The indicated diagonal functor is induced b y H ′ . The b ottom right triangle comm utes since it is induced b y G ′ H ′ , which w e discussed abov e. T op left triangle can b e seen to commute by the Y oneda lemma. Rep eating this construction for ev ery n : N , we obtain the following commutativ e diagram: x ↓ X 0 ι 1 0 x ↓ X 1 ι 2 0 x ↓ X 2 · · · F un cocart ( x ∗ q 0 , q 0 ) F un cocart (( ι 1 0 x ) ∗ q 1 , q 1 ) F un cocart (( ι 2 0 x ) ∗ q 2 , q 2 ) · · · . The colimit of the top row is ι ∞ 0 x ↓ X ∞ . The colimit of the b ottom row is F un cocart (( ι ∞ 0 x ) ∗ q ∞ , q ∞ ). The diagonal maps tell us that the induced map F : ι ∞ 0 x ↓ X ∞ → F un cocart (( ι ∞ 0 x ) ∗ q ∞ , q ∞ ) is in v ertible. It remains to show that F is isomorphic to the functor f 7→ f ! ; since F is inv ertible it follo ws from this that q ∞ is directed univ alen t. By the Y oneda lemma, it suﬃces to sho w that F deﬁnes a morphism of categories o ver X ∞ , and that it sends the iden tit y on ι ∞ 0 x to the identit y functor on ( ι ∞ 0 x ) ∗ q ∞ . The former amounts to the fact that the witnesses of commutativit y in (4) witness commutativit y in LFib( X 0 ) and not just in Cat . The latter follows from the fact that the functor ( x ↓ X 0 ) → F un cocart ( x ∗ q 0 , q 0 ) sends the iden tit y on x to the iden tit y functor on x ∗ q 0 . □ W e can no w easily prov e the existence of enough directed univ alent co cartesian ﬁbrations. Theorem 7.7. Given a functor q : Y → X , ther e is a dir e cte d univalent c o c artesian ﬁbr ation p : E → B , such that p classiﬁes every ﬁbr e of q . Pr o of. Let q : Y → X b e an arbitrary functor, and let P b e the collection of categories classiﬁed b y q . Equiv alen tly , P is the collection of categories classiﬁed b y q ′ : Y × X X ≃ → X ≃ . By Lemma 7.6, w e obtain a directed univ alent co cartesian ﬁbration whic h classiﬁes the same categories. □ 8. Straightening of cocar tesian fibra tions Finally w e prov e a straightening theorem against any directed univ alent ﬁbration. Theorem 8.1. Supp ose p : E → B is a dir e cte d univalent c o c artesian ﬁbr ation and C a c ate gory. Given a c o c artesian ﬁbr ation q : D → C such that p classiﬁes every ﬁbr e of q , ther e is a functor f : C → B and an e quivalenc e D ≃ C × B E of c ate gories over C . Mor e over, given two functors 30 CHRISTIAN SA TTLER AND DA VID W ¨ ARN f , g : C → B , the c anonic al map fr om Map( f , g ) to the sp ac e of c o c artesian functors f ∗ p → g ∗ p is invertible. A direct proof of the result abov e is p ossible if one assumes that co cartesian ﬁbrations are exp onen tiable [2]. W e instead deduce it from descen t. Pr o of. F or a category C , denote the full sub category of CocartFib( C ) spanned b y co cartesian ﬁbrations all of whose ﬁbres are classiﬁed by p b y Co cartFib p ( C ). Sa y a category C is go o d if the (wild) functor ( − ) ∗ C p : F un( C , B ) → Co cartFib p ( C ) is inv ertible. W e hav e to sho w that every category is go o d. By Lemma 7.3, it suﬃces to show that every group oid is go o d, and that go o d categories are closed under co commas, lo calisations, and sequen tial colimits. Since p is directed univ alen t, in particular the map B ≃ → Cat ≃ whic h computes ﬁbres of p is a monomorphism. T ogether with Lemma 6.3, this means that every group oid is go o d. Supp ose we are given a span of categories Y f ← − X g − → Z where X , Y , and Z are go o d, and let W denote the co comma category f ↑ g . W e then obtain the following cub e-shaped diagram with lax top and b ottom faces: F un( W, B ) F un( Z , B ) F un( Y , B ) F un( X , B ) Co cartFib p ( W ) Co cartFib p ( Z ) Co cartFib p ( Y ) Co cartFib p ( X ). ( − ) ∗ W p ( − ) ∗ Z p ( − ) ∗ X p ( − ) ∗ Y p The top square is a comma square since W is a co comma. The b ottom square is a comma square b y Lemma 6.1. The three vertical maps ( − ) ∗ X p , ( − ) ∗ Y p , and ( − ) ∗ Z p are in v ertible by assumption. The fourth vertical map ( − ) ∗ W p is thus inv ertible b y uniqueness of comma categories. Thus go o d categories are closed under co comma. The facts that go o d categories are closed under lo calisation and sequential colimits similarly follo w from Theorem 5.7 and Lemma 6.2. □ Remark 8.2. The problem of deﬁning a fully coheren t (as opp osed to wild) unstraigh tening functor is in some sense orthogonal to the problem of showing that the unstraigh tening functor is inv ertible. Indeed, by the fundamental theorem of category theory , the statement that a functor is in vertible in v olves only its action on ob jects and morphisms. So far w e hav e only considered Co cartFib( C ) as a wild category , and so hav e no c hoice but to consider ( − ) ∗ C p : F un( C, B ) → Co cartFib( C ) as a wild functor. Ho wev er, given a univ ersal co cartesian ﬁbration Cat • → Cat , it b ecomes p ossible to realise Co cartFib( C ) as a fully coherent category and w e exp ect it to then also b e p ossible to realise the straightening functor as a fully coheren t functor (cf. [2, Section 7.7]). References [1] David Ayala and John F rancis. “Fibrations of ∞ -categories”. In: Higher Structur es 4.1 (F eb. 2020), pp. 168–265. doi : 10.21136/HS.2020.05 . [2] Denis-Charles Cisinski, Bastiaan Cnossen, Kim Nguy en, and T ashi W alde. Synthetic Cate gory The ory . 2026. REFERENCES 31 [3] Denis-Charles Cisinski and Hoang Kim Nguyen. The universal c oCartesian ﬁbr ation . 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Co c artesian ﬁbr ations and str aightening internal to an ∞ -top os . May 2022. doi : 10.48550/arXiv.2204.00295 . arXiv: 2204.00295 [math] . [9] Egb ert Rijke. The join c onstruction . Jan. 2017. doi : 10 . 48550 / arXiv . 1701 . 07538 . arXiv: 1701.07538 [math] . [10] T aichi Uem ura. Colimits in the ∞ -c ate gory of ∞ -top oi and ´ etale morphisms . June 2025. doi : 10.48550/arXiv.2506.10431 . arXiv: 2506.10431 [math] .

A synthetic construction of universal cocartesian fibrations

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