Framed bicategories and monoidal fibrations

FRAMED BICA TEGORIES AND MONOID AL FIBRA TIONS MICHAEL SHULMAN Abstract. In some bicategories, the 1-cells are ‘morphisms’ b et we en the 0- cells, suc h as funct ors b et we en categories, but in o thers they are ‘ob jects’ o ver the 0-cells, such as bimo dules, spans, di stributors, or parametrized sp ec- tra. Many bi catego rical notions do not work well in these cases, b ecause the ‘morphisms b et wee n 0-cells’, such as ring homomorphisms, ar e missing. W e can include them by using a pseudo double category , but usuall y these mor- phisms also induc e base c hange functors act ing on the 1-ce ll s. W e av oid compli- cated coheren ce problems b y describing base chang e ‘nonalgebraically’, using categorical ﬁbrations. The resulting ‘framed bicategories’ assemb le i n to 2- categories, with attend ant notions of equiv alence, adjunction, and so on whic h are more appropriate for our examples than are the usual bicategorical ones. W e then describ e tw o wa ys to construct framed bicategories. One is an analogue of rings and bimo dules which starts from one fr amed bicategory and builds another. The other s tarts f rom a ‘m onoidal ﬁbrati on’, meaning a parametrized family of monoidal categories, and pro duces an analogue of the f ramed bicategory of spans. Combining the tw o, we obtain a construction whic h includes b oth enri c hed and internal categories as special cases. Contents 1. In tro duction 2 2. Double ca tegories 4 3. Review of the theory o f ﬁbrations 9 4. F ramed bicategor ies 13 5. Dualit y theor y 19 6. The 2-ca tegory of fr a med bicategor ies 23 7. F ramed equiv ale nce s 30 8. F ramed adjunctions 33 9. Monoidal fra med bicategor ies 35 10. In volutions 38 11. Monoids and mo dules 40 12. Monoidal ﬁbratio ns 44 13. Closed monoidal ﬁbrations 49 14. F rom ﬁbrations to fra med bicategor ies 53 15. Monoids in monoidal ﬁbrations and examples 57 16. Tw o technical lemmas 61 17. Pro ofs of Theorems 14.4 a nd 14.11 64 Appendix A. Connection pairs 71 Appendix B. Biequiv alences, biadjunctions, and monoidal bicatego r ies 73 Appendix C. Equipmen ts 75 Appendix D. Epilogue: framed bica tegories versus bicategories 77 References 78 1 2 MICHAEL SHULMAN 1. Introduction W e b egin with the obse r v ation that there are really t wo sor ts of bicatego ries (or 2-catego ries). This fact is well a ppreciated in 2-catego rical circles , but not as widely known as it ought to b e. (In fact, there ar e other sorts of bicategory , but we will only b e concerned with tw o.) The ﬁrst sort is exempliﬁed by the 2-ca tegory C at of categ ories, functors, and natural transformations. Her e, the 0 -cells are ‘ob jects’, the 1 - cells are maps betw een them, and the 2-cells are ‘maps b et ween ma ps .’ This sort of bicatego ry is well- describ ed by the slog an “a bicategory is a catego r y e nr ich ed over ca teg ories.” The sec ond so rt is exempliﬁed by the bicateg o ry M o d of rings, bimodules, and bimo dule homomorphisms. Here, the 1-cells are themselves ‘o b ject s’, the 2-cells are maps b etw een them, and the 0 - cells are a diﬀeren t sort of ‘ob ject’ which play a ‘b o okkeeping’ role in organizing the rela tionships b etw een the 1-cells. This so rt of bicategor y is well-describ ed by the sloga n “a bicategor y is a monoidal categor y with many ob jects.” Many notions in bicateg o ry theor y work as well for one sor t as for the other . F or example, the notion of 2-functor (including lax 2-functors a s well as pse udo ones ) is w ell-suited to describ e morphisms of either so r t of bicategor y . Other notions, such a s that o f internal adjunction (or ‘dual pair’), are useful in bo th situations , but their meaning in the tw o cases is very diﬀerent. How e ver, so me bicategor ical ideas make more sense for one so rt of bicategory than for the o ther , and frequently it is the seco nd sort that gets slig hted. A prime example is the notion o f equiv alence o f 0-c e lls in a bicategory . This specia lizes in C at to equiv alence of categ ories, which is unquestio na bly the fundamen tal notion of ‘sameness’ for categ ories. But in M o d it s pecia lizes to Morita eq uiv alence of ring s, which , while very interesting, is not the most fundamen tal so rt of ‘sameness’ for rings; isomor phism is. This may not seem lik e s uch a big dea l, since if we wan t to talk ab out when t wo rings a re isomorphic, w e can use the catego ry o f rings instead of the bicategor y M o d . How e ver, it b ecomes mo r e acute when we consider the no tio n of biequiv a - lence of bicateg ories, which inv olves pseudo 2- functors F and G , a nd equiv a lence s X ≃ GF X and Y ≃ F GY . This is ﬁne for C at -lik e bicategor ies, but for M o d - like bicategories, the rig h t notion of equiv a lence ough t to include something corresp ond- ing to ring isomorphisms instead. This problem ar ose in [MS06, 19.3.5], where t wo M o d -like bica teg ories were c lea rly ‘equiv a le nt’, yet the languag e did not exist to describ e what s ort of equiv alence was mean t. Similar pro blems arise in many other situations, such a s the following. (i) C at is a monoidal bicategory in the usual sense, which entails (among other things) natura l equiv a lences ( C × D ) × E ≃ C × ( D × E ). But although M o d is ‘mora lly monoidal’ under tensor pr o duct of rings, the asso ciativity constraint is really a r ing is omorphism ( R ⊗ S ) ⊗ T ∼ = R ⊗ ( S ⊗ T ), not an in vertible bimodule (althoug h it can be ma de into one). (ii) F or C at -like bicategor ies , the notions o f pseudona tural transformation and mo diﬁcation, making bicatego ries in to a tricategory , a r e natural and useful. But for M o d -like bicategories , it is signiﬁcantly less c le a r what the right sort o f higher morphisms ar e. (iii) The notion o f ‘biadjunction’ is w ell-suited to adjunctions b etw een C at - like bicategories, but fails badly for M o d -like bicategories. Attempts to FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 3 solve this problem have r esulted in some work, such as [V er92, CK W91, CKVW98], which is closely r elated to o urs. These problems a ll stem from ess e n tially the same source: the bica tegory struc- ture do es not include the correct ‘maps b etw een 0-cells’, since the 1 - cells of the bicategory a r e b eing used for something else. In this pap er, we show how to use an abstract s tructure to dea l with this sort of situation b y incorp or ating the maps of 0-cells separa tely fro m the 1-cells. This structure forms a pseudo double catego r y with extra prop erties, which we call a fr ame d bic ate gory . The ﬁr st part of this pap er is devoted to framed bicateg o ries. In §§ 2–5 we review basic notions ab out double catego r ies and ﬁbrations, deﬁne framed bicategories, and prov e some basic facts ab out them. Then in §§ 6– 10 we apply framed bicate- gories to resolve the problems ment ioned a b ove. W e deﬁne lax, oplax, and strong framed functors and framed transforma tions, and ther eb y o btain thr e e 2 -categor ies of framed bicatego r ies. W e then a pply general 2-ca teg ory theory to o btain useful notions of framed equiv alence, framed a djunction, and monoidal framed bicategor y . The seco nd part o f the pap er , consisting of §§ 11–1 7 , deals with t wo imp or ta n t wa ys of constructing f ramed bicategories. The ﬁrst, which we describ e in § 1 1, starts with a framed bica tegory D and constructs a new framed bicateg ory M o d ( D ) of mo noids and mo dules in D . The s econd star ts with a diﬀerent ‘par ametrized monoidal structure’ ca lled a monoidal ﬁbr ation , and is essentially the same a s the construction of the bica tegory of par ametrized sp ectra in [MS06]. In §§ 12 –13 we in tro duce monoidal ﬁbrations, and in § 14 w e explain the c onnection to framed bicategories. Then in § 15, we com bine these tw o constructio ns and thereby obtain a natural theory of ‘categories which ar e b oth int ernal and enriched’. §§ 16– § 1 7 are devoted to the pro ofs of the main theor ems in § 14. Finally , in the app endices we tre a t the relationship of framed bicatego ries to other work. This includes the theory of c onne ction p airs and foldings in double categories , v arious par ts of pure bicategory theo ry , and the bicategorica l theor y o f e quipments . Our co nclusion is that they ar e a ll, in suitable sens es, eq uiv alent , but each ha s adv an tages a nd disadv antages, and we b elieve that framed bicateg ories are a better choice for many purpo ses. There are tw o impo r tant themes running throughout this pa per. O ne is a pre- o ccupation with deﬁning 2-catego ries and mak ing constructions 2-functorial. As- semblin g ob jects into 2-catego ries allows us to apply the theory of adjunctions, equiv alences, monads, and so on, internal to these 2-categories. Thu s, without any extra work, we obtain notions such as framed adjunctions and fr a med mon- ads, which b ehave muc h like o rdinary adjunctions and mo nads. Mak ing v arious constructions 2-functoria l makes it easy to obtain fra med adjunctions and monads from more ordinar y ones. W e do not use very m uch 2 -categor y theory in this pa p er, so a pass ing acquain- tance with it should suﬃce. Since we ar e not writing primarily for categor y theo- rists, we hav e a ttempted to av o id o r explain the more esoteric categorical concepts which arise. A class ic reference for 2 -categor y theory is [KS74]; a mo re mo dern and comprehensive o ne (going far b eyond what we will need) is [Lac0 7]. The second impo rtant theme of this pap er is the mixture of ‘alge br aic’ and ‘non- algebraic’ structures. A monoidal categor y is an a lgebraic s tructure: the pro duct is a sp eciﬁed op eration on ob jects. On the o ther hand, a catego ry with c artesian pro ducts is a nonalg ebraic structure: the pro ducts are characterized by a univ ersal 4 MICHAEL SHULMAN prop erty , and merely as s umed to exist. W e can always make a choice o f pro ducts to make a category with pro ducts into a mo noidal categor y , but there ar e many po ssible choices, all is o morphic. There a re man y tec hnical adv ant ages to working with nonalgebraic structures . F or example, no coherence axioms ar e re q uired of a category with pro ducts, whereas a monoidal ca tegory requires several. This a dv antage b ecomes mo re signiﬁcant as the coherence ax ioms multiply . On the other ha nd, when doing concr ete work, one often wan ts to make a sp eciﬁc choice of the structure and work with it algebra ically . Moreov er, not all alge br aic str uctur e satisﬁes an obvious universal prop erty , and while it c an usua lly b e to rtured into do ing so, frequently it is easier in these c a ses to stick with the algebr aic version. F ramed bicategories a re a mixture o f algebraic and nonalg ebraic notions ; the comp osition of 1-cells is algebraic, while the base change op era tions are g iven no n- algebraica lly , using a ‘categ orical ﬁbra tion’. Our exper ience sho ws that this mixture is very technically convenien t, and we hop e to co nvince the reader of this too . In particular, the pro of of Theorem 14.4 is muc h simpler than it would be if we used fully alg ebraic deﬁnitions. This is to b e cont rasted with the similar structures w e will consider in app endices A and C, which are purely a lgebraic. Our inten t in this pap er is not to pr e s en t any o ne par ticular result, but rather to argue for the genera l pro po sition tha t framed bicategories, and rela ted structures , provide a useful framework for many diﬀerent kinds o f mathematics. Despite the length of this paper, w e hav e only had space in it to lay down the most basic deﬁnitions and ideas, and muc h re ma ins to b e said. The theory of framed bicategories was largely mo tiv ated by the desire to ﬁnd a go o d categorical structure for the theor y o f parametrized sp ectra in [MS06]. The reader familiar with [MS06] should ﬁnd the idea of a framed bicatego ry natural; it was realized clea r ly in [MS06] that existing categor ical structures were inadequate to describ e the com bination of a bicateg ory with base c hange o per ations which arose naturally in that context. Another motiv ation for this work came from the bicategorica l ‘shadows’ of [Pon07], a nd a des ir e to explain in what way they ar e actually the same a s the ho rizontal comp osition in the bicategory; w e will do this in the forthcoming [P S07]. I would lik e to thank my advisor, Peter May , as well as Kate Pon to, for many useful discussions a bo ut these s tructures; T o m Fiore, for the idea of using double categories ; and J oachim Ko ck and Stephan Stolz for po in ting out problems with the orig inal version of E x ample 2.7. The term ‘framed bicategory’ was suggested b y Peter May . 2. Double ca tegories As mentioned in the introduction, most of the problems with M o d -lik e bicat- egories c a n be traced to the fa c t that the ‘morphisms’ of the 0 -cells a re missing. Thu s, a na tural r eplacement which sug gests itself is a double c ate gory , a structure which is like a 2 -categor y , except that it has tw o types of 1-cells, called ‘vertical’ and ‘horizo n tal’, and its 2- c e lls are sha ped like squares . Double catego ries go back originally to E hresmann in [Ehr 6 3]; a brief int ro duction can b e found in [KS7 4]. Other refer ences include [BE 74, GP99, GP04, Gar 06]. In this section, we intro duce basic notions of double ca tegories. O ur terminolo g y and notation will sometimes b e diﬀeren t fr o m that commonly used. F o r example, FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 5 usually the term ‘double catego r y’ refers to a strict o b ject, a nd the weak v ersion is called a ‘pseudo double categ ory’. Since we are primar ily interested in the weak version, we will use the term double c ate gory for these, a nd add the word ‘strict’ if necessary . Deﬁnition 2.1. A double category D cons is ts of a ‘ca tegory of ob jects’ D 0 and a ‘categ o ry of ar rows’ D 1 , with structure functors U : D 0 → D 1 L, R : D 1 ⇒ D 0 ⊙ : D 1 × D 0 D 1 → D 1 (where the pullback is over D 1 R − → D 0 L ← − D 1 ) such that L ( U A ) = A R ( U A ) = A L ( M ⊙ N ) = LM R ( M ⊙ N ) = RM equipped with natural isomorphisms a : ( M ⊙ N ) ⊙ P ∼ = − → M ⊙ ( N ⊙ P ) l : U A ⊙ M ∼ = − → M r : M ⊙ U B ∼ = − → M such that L ( a ), R ( a ), L ( l ), R ( l ), L ( r ), and R ( r ) are a ll iden tities, and such that the standa rd coherence axioms for a monoidal categ ory or bicategory (such as Mac Lane’s p entagon; see [ML98]) ar e satisﬁed. W e ca n think o f a double categ o ry as an in ternal ca tegory in C at which is suit- ably weak ened, althoug h this is not strictly true beca use C at con tains only small categories while we allow D 0 and D 1 to b e larg e catego ries (but still lo cally small, that is, having only a set of mor phisms b etw een any t wo ob jects). W e call the o b jects o f D 0 ob jects or 0-cells , and we c a ll the morphisms of D 0 v ertical arro ws and write them as f : A → B . W e call th e ob jects o f D 1 horizon tal 1-cells or just 1-cells . If M is a horizontal 1-cell with L ( M ) = A and R ( M ) = B , we write M : A p → B , and say that A is the le ft frame of M and B is the right frame . W e use this termino logy in preference to the mo re usual ‘source’ and ‘tar get’ b ecause of our philosophy that the ho rizontal 1-cells ar e no t ‘morphisms’, but rather ob jects in their own right whic h just happ en to b e ‘lab eled’ b y a pair of ob jects of another type. A mor phism α : M → N o f D 1 with L ( α ) = f and R ( α ) = g is called a 2-cell , written α : M g = ⇒ f N , o r just M α − → N , and drawn as follows: (2.2) A | M / / f   ⇓ α B g   C | N / / D . 6 MICHAEL SHULMAN W e say that M and N are the so urce and target of α , while f and g are its le ft frame and right frame . W e write the co mpos ition of vertical arrows A f − → B g − → C and the vertical comp osition of 2-cells M α − → N β − → P with juxtap osition, g f or β α , but we write the horizo ntal co mpo s ition of horizontal 1 -cells a s M ⊙ N and that of 2-cells as α ⊙ β . W e write horizontal co mpo s ition ‘forwards’ ra ther than backw ards: for M : A p → B and N : B p → C , w e hav e M ⊙ N : A p → C . This is a ls o called ‘dia grammatic o r- der’ and has several adv antages. First, in examples such a s that of rings and bimo dules (Exa mple 2.3), w e can deﬁne a horiz o nt al 1-cell M : A p → B to be an ( A, B )-bimodule, rather than a ( B , A )-bimodule, a nd still preserve the order in the deﬁnition M ⊙ N = M ⊗ B N of horizontal comp osition. It also makes it easier to av oid mistakes in working with 2- cell diagra ms; it is easier to comp ose A | M / / B | N / / C and get A | M ⊙ N / / C than to remember to switch the o rder in which M and N a ppea r e very time hori- zontal 1- cells a re comp osed. Finally , it a llows us to say that an adjunction M ⊣ N in the ho rizontal bicategory is the same as a ‘dual pair’ ( M , N ) (see § 5), with the left adjoint also b eing the left dual. Every ob ject A o f a do uble ca teg ory has a vertical iden tity 1 A and a horizontal unit U A , every ho rizontal 1 -cell M has an identit y 2-cell 1 M , every vertical a r row f has a ho r izontal unit U f , and we hav e 1 U A = U 1 A (b y the functoriality of U ). W e will frequent ly a buse no ta tion by writing A or f instead of U A or U f when the context is clear. The impo r tant p oint to remember is that vertical co mpo sition is strictly asso ciative and unital, while horizo n tal co mposition is as so ciative a nd unital only up to sp eciﬁed coherent isomorphisms. Note that if D 0 is the terminal ca tegory , then the deﬁnition of do uble category just says that D 1 is a monoida l categor y . W e ca ll s uch double categor ies v ertically trivial . W e ca ll D 0 the v ertical category of D . W e say that tw o ob jects a r e isomorphic if they are is omorphic in D 0 , and that tw o horizontal 1-c e lls a re isomorphic if they are isomorphic in D 1 . W e will never refer to a horizontal 1 -cell as an iso morphism. A 2-cell whose left and right frames are identities is called globular . Note that the constraints a , l , r ar e globular isomor phisms, but they ar e na tural with resp ect to all 2-c e lls , not just globular o nes. Every double category D has a horizon tal bicategory D consisting of the ob jects, ho rizontal 1-cells, a nd globular 2- c e lls . If A and B are o b jects of D , we write D ( A, B ) for the set of vertical arr ows from A to B and D ( A, B ) for the c ate gory of ho rizontal 1-cells a nd glo bular 2- c e lls fro m A to B . It is standar d in bicatego ry theory to say tha t something ho lds l o cally when it is true o f a ll hom-catego ries D ( A, B ), and we will extend this usag e to double ca teg ories. W e also wr ite f D g ( M , N ) for the se t of 2- cells α o f the shap e (2.2). If f a nd g ar e ident ities, we write instead D ( M , N ) for the set o f globular 2- cells from M to N . This may b e r egarded a s shorthand for D ( A, B )( M , N ) and is standa rd in bicategory theory . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 7 W e now consider some exa mples. Note that unlike 1-ca tegories, which we gener- ally name b y their ob jects, we genera lly na me double ca tegories by their horizontal 1-cells. Example 2.3 . Let M o d b e the double category deﬁned a s follo ws. Its ob jects are (not necessar ily co mm utative) rings a nd its vertical mo rphisms a re ring homomor - phisms. A 1-ce ll M : A p → B is an ( A, B )-bimodule, and a 2 -cell α : M g = ⇒ f N is an ( f , g ) -biline ar map M → N , i.e. a n a belia n gro up homomorphism α : M → N such that α ( amb ) = f ( a ) α ( m ) g ( b ). This is equiv alent to saying α is a map of ( A, B )- bimo dules M → f N g , where f N g is N regarded as an ( A, B )-bimodule by means of f a nd g . The hor izontal comp osition of bimo dules M : A p → B and N : B p → C is given by their tensor pr o du ct, M ⊙ N = M ⊗ B N . F or 2 -cells A M / / f   α C g   P / / β E h   B N / / D Q / / F we deﬁne α ⊙ β to b e the comp osite M ⊗ C P α ⊗ β / / f N g ⊗ C g Q h / / / / f N ⊗ D Q h ∼ = f ( N ⊗ D Q ) h . This example may be gener alized by replacing Ab with any monoidal catego ry C that ha s co e q ualizers pr eserved b y its tensor pr o duct, giving the double catego ry M o d ( C ) of monoids, monoid homomorphisms, and bimo dules in C . If C = Mo d R is the catego ry of mo dules over a commutativ e ring R , then the res ulting double category M o d ( C ) = M o d ( R ) is made of R -algebras , R -alg e br a homomor phisms, and bimo dules ov er R -a lgebras. Similarly , we deﬁne the double ca tegory CM o d whose o b jects are co mm utative rings, and if C is a s ymmetric monoidal categor y , we hav e CM o d ( C ). Example 2.4. Let C b e a catego r y with pullbacks, and deﬁne a double ca teg ory S pan ( C ) whose vertical ca teg ory is C , whose 1 -cells A p → B are spans A ← C → B in C , and whos e 2-cells a re commuting diagrams: A   C o o   / / B   D F o o / / E in C . Horizontal compo s ition is by pullback. Example 2. 5. Ther e is a double category of par a metrized sp ectra called E x , whose construction is essentially contained in [MS06]. The vertical ca tegory is a category of (nice) top olog ical spa ces, and a 1-cell A p → B is a spe c tr um parametrized ov er A × B (or B × A ; see the note a bove abo ut the o r der of comp osition). In [MS06] this structure is descr ibed only as a bicategory with ‘ba s e change op erations’, but it is p ointed out there that existing ca tegorical structures do not suﬃce to des c r ibe it. W e will see in § 14 how this s o rt of structure gives rise, quite generally , not only to a double categor y , but to a framed bicatego ry , which supplies the missing catego r ical structure. 8 MICHAEL SHULMAN Example 2.6 . Let V b e a complete and co co mplete clo sed symmetric mo noidal category , such as Set , Ab , Cat , o r a conv enient ca r tesian closed sub categor y of topo logical spac e s, and deﬁne a double ca tegory D is t ( V ) as follows. Its o b j ects are (sma ll) catego ries enriched ov er V , or V -ca tegories. Its vertical arrows are V - functors, its 1 -cells are V -distributors , a nd its 2-cells are V -natural transformations. (Goo d r eferences for enr ic hed categ ory theory include [Kel82] a nd [Dub70].) A V - distributor H : B p → A is simply a V - functor H : A op ⊗ B → V . When A a nd B hav e one ob ject, they are just monoids in V , and a distributor b etw e en them is a bimo dule in V ; thus we have a n inclusio n M o d ( V ) ֒ → D ist ( V ). Horizontal comp osition o f distributors is given by the coend construction, also known as ‘tensor pro duct of functors’. In the bicategor ic a l literature, distributors are often called ‘bimo dules’ or just ‘mo dules’, but we prefer to r eserve that term fo r the clas s ical one-ob ject version. The term ‘distribut or ’, due to Benab ou, is intended to sugges t a gener alization of ‘funct or ’, just as in ana lysis a ‘distribut ion ’ is a g e ner alized ‘funct ion ’. The term ‘profunctor’ is also used for these ob jects, but w e prefer to avoid it b ecause a distributor is nothing like a pro-o b j ect in a functor categ ory . Example 2.7 . W e deﬁne a double category n C ob as fo llows. Its vertical ca tegory consists of oriented ( n − 1)-manifolds without bo undary a nd diﬀeomorphisms. A 1-cell M p → N is a (pos sibly thin) n -dimensional co bo rdism from M to N , and a 2-cell is a compatible diﬀeomorphism. Horizontal comp osition is g iven by g luing o f cob ordisms. More for mally , if A a nd B are or ien ted ( n − 1)-manifolds, a hor izontal 1-cell M : B p → A is either a diﬀeomor phism A ∼ = B (regarded as a ‘thin’ co bo r dism from A to B ), or an n -manifold with b oundary M equipp ed with a ‘colla r’ map ( A op ⊔ B ) × [0 , 1) → M which is a diﬀeomorphism o nt o its imag e and r estricts to a diﬀeomor phism A op ⊔ B ∼ = ∂ M . (Here A op means A with the o ppos ite orientation.) The unit is the identit y 1 A , regarded as a thin co bo r dism. Example 2.8. The following double categor y is known as A dj . Its o b jects are categories , and its horizontal 1-cells are functors. Its vertical arrows C → D are adjoint pa irs of functors f ! : C ⇄ D : f ∗ . W e then seem to hav e t wo choices for the 2-cells; a 2-cell with b oundary A h / / f !   C g !   B k / / f ∗ L L D g ∗ L L could be chosen to b e either a natural tra nsformation g ! h → k f ! or a natural transformation hf ∗ → g ∗ k . How ever, it turns out that there is a natural bijec- tion b et ween na tural transformations g ! h → k f ! and hf ∗ → g ∗ k which r esp e cts comp osition, so it do esn’t matter which we pick. P airs of natural tra ns formations corresp onding to ea ch other under this bijection are called mates ; the mate of a FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 9 transformation α : hf ∗ → g ∗ k is given explicitly as the comp osite g ! h g ! hη / / g ! hf ∗ f ! g ! αf ! / / g ! g ∗ k f ! εkf ! / / k f ! where η is the unit o f the adjunction f ! ⊣ f ∗ and ε is the counit o f the a djunction g ! ⊣ g ∗ . The inv e rse construction is dua l. More g enerally , if K is an y (strict) 2 -categor y , we can deﬁne the notion of an ad- junction int ernal to K : it consists of morphisms f : A → B and g : B → A tog ether with 2-cells η : 1 A ⇒ g f and ε : f g ⇒ 1 B satisfying the usual triang le iden tities. W e can then deﬁne a double ca tegory A dj ( K ) formed by ob jects, morphisms, ad- junctions, and mate-pairs internal to K . These double categories hav e a diﬀerent ﬂavor than the others intro duced a bove. W e mention them par tly to po in t out that double categor ies hav e uses other than those we are in terested in, and par tly b ecause we will need the notion of mates later on. More ab out mate-pair s in 2-ca tegories and their r elationship to A dj can be found in [K S74]; one fact w e will need is that if h and k are iden tities, then α is an isomo rphism if and only if its mate is an isomo rphism. 3. Review of the theor y of fibra tions Double ca tegories incorp orate b oth the 1- cells of a M o d -like bica tegory and the ‘morphisms of 0- cells’, but there is something missing. An impor tant feature o f all our examples is that the 1-cells ca n b e ‘base changed’ along the vertical arrows. F or example, in M o d , we can extend and r e s trict scalar s along a ring ho momorphism. An appropria te abstract structure to descr ibe these base change functors is the well-kno wn categor ical notion of a ‘ﬁbration’. In this section w e will review some of the theory of ﬁbrations, and then in § 4 we will a pply it to base change functors in double catego ries. All the materia l in this section is standard. The theory of ﬁbrations is originally due to Grothendiec k and his schoo l; see, for example [Gro03, Exp os´ e VI]. Mo dern references include [Joh0 2 a, B1 .3] and [Bor94, Ch. 8]. More abstract versions can b e found in the 2- c ategorical literature, such as [Str80]. Deﬁnition 3 .1. Let Φ : A → B be a functor, le t f : A → C b e an arr ow in B , and let M b e an ob ject of A with Φ( M ) = C . An arrow φ : f ∗ M → M in A is cartesian o v er f if, ﬁrstly , Φ( φ ) = f : f ∗ M φ / / _   M _   A f / / C 10 MICHAEL SHULMAN and secondly , whenev er ψ : N → M is an arrow in A and g : B → A is an arrow in B such that Φ( ψ ) = f g , there is a unique χ such that ψ = φχ and Φ( χ ) = g : N ψ * * U U U U U U U U U U U U U U U U U U U U U U U _   χ " " D D D D f ∗ M φ / / _   M _   B g " " E E E E E E E E E * * U U U U U U U U U U U U U U U U U U U U U U U A f / / C W e s ay that Φ is a ﬁbration if for every f : A → C and M with Φ( M ) = C , there exists a cartesia n arrow φ f ,M : f ∗ M → M ov er f . If Φ is a ﬁbra tion, a cleav age for Φ is a choice, for every f and M , o f such a φ f ,M . Th e cleav a ge is normal if φ 1 A ,M = 1 M ; it is split if φ g,M φ f ,g ∗ M = φ gf ,M for all comp osa ble f , g . F or an arr ow f : A → B , we think of f ∗ as a ‘base change’ op eration that maps the ﬁb er A B (consisting of all ob jects ov er B and morphisms over 1 B ) to the ﬁb er A A . W e think of A as ‘glued tog ether’ from the ﬁb er categor ies A B as B v a ries, using the ba se change op erations f ∗ . W e think o f the who le ﬁbr ation as ‘a ca tegory parametrized by B ’. Example 3.2. Let R i ng b e the categor y of rings, and Mo d be the category of pairs ( R, M ) wher e R is a ring and M is a n R -mo dule, with mor phisms consisting of a ring ho momorphism f a nd an f -equiv ar iant module map. Then the forgetful functor Mod : Mo d → Ring , which s e nds ( R, M ) to R , is a ﬁbration. If M is a n R -mo dule a nd f : S → R is a ring homomorphism, then if w e denote by f ∗ M the ab elian g roup M re g arded as an S - mo dule via f , the iden tity map of M deﬁnes an f -equiv ar iant map f ∗ M → M , which is a cartesia n arrow over f . Note that the ﬁb er A R is the ordinary catego ry o f R -mo dules. Thus we may say that mo dules form a ca tegory parametrized by rings . Example 3.3. Let C b e a category with pullbacks, let C ↓ denote the categor y of arrows in C (whose mor phisms are commutativ e squa res), and let Ar r C : C ↓ → C take each arr ow to its co do main. Then A rr C is a ﬁbration; a commu tative squar e is a cartesia n arrow in C ↓ precisely when it is a pullback square. This ﬁbr ation is sometimes refer r ed to a s the self-indexing of C . W e recor d some useful facts ab out ﬁbrations . Prop osition 3.4. L et Φ : A → B b e a ﬁbr ation. (i) The c omp osite of c artesian arr ows is c artesian. (ii) If φ : ( f g ) ∗ M → M and ψ : g ∗ M → M ar e c artesian over f g and g r esp e c- tively, and χ : ( f g ) ∗ M → g ∗ M is the unique factorization of ψ thr ough φ lying over f , then χ is c artesian. (iii) If if φ : f ∗ M → M and φ ′ : ( f ∗ M ) ′ → M ar e two c artesian lifts of f , then ther e is a unique isomorphi sm f ∗ M ∼ = ( f ∗ M ) ′ c ommut ing with φ and φ ′ . (iv) Any isomorphism in A is c artesian. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 11 (v) If f is an isomorphism in B , then any c artesian lift of f is an isomor- phism. In Exa mple 3.2 , there is a ‘canonical’ choice o f a cleav age , but this is not true in Example 3.3 , since pullbacks a re o nly deﬁned up to isomorphism. Prop osi- tion 3.4(iii) tells us that mo re generally , cleav ages in a ﬁbration are unique up to canonical is omorphism. Thus, a ﬁbration is a ‘nonalgebraic’ a pproach to deﬁning base change functor s: the op eration f ∗ is characterized b y a universal prope rty , and the deﬁnition mer ely stipulates that an ob ject sa tisfying that pro pe r t y exists, rather than c ho osing a particular such ob ject as pa rt o f the str ucture. In the terminology of [Mak01], they a re virtual op erations. The ‘alg ebraic’ notion corr esp onding to a ﬁbration Φ : A → B is a pseudo- functor P : B op → C at . Given a ﬁbra tion Φ, if we choose a cleav age, then we obtain, for each f : A → B in B , a functor f ∗ : A B → A A . If we deﬁne P ( A ) = A A and P ( f ) = f ∗ , the uniquenes s -up-to-iso of ca rtesian lifts ma kes P in to a pseudo- functor. C o nv er sely , g iven a pseudofunctor P : B op → C at , we can build a ﬁbratio n ov er B whose ﬁb er ov er A is P ( A ). (This is sometimes ca lled the ‘Gro thendiec k construction’.) In or der to s tate the full 2-catego rical s ense in whic h these co nstructions are in verse equiv alences, we need to int ro duce the mor phisms and transforma tions b e- t ween ﬁbrations. Consider a commuti ng s quare of functors (3.5) A F 1 / / Φ   A ′ Φ ′   B F 0 / / B ′ where Φ and Φ ′ are ﬁbrations, and let φ : g ∗ M → M be c a rtesian in A o ver g . Then we hav e F 1 ( φ ) : F 1 ( g ∗ M ) → F 1 M in A ′ ov er F 0 ( g ). But since Φ ′ is a ﬁbration, there is a cartesian arrow ψ : ( F 0 g ) ∗ ( F 1 M ) → F 1 M over F 0 ( g ), s o F 1 ( φ ) fac to rs uniquely through it, g iving a canonical map (3.6) F 1 ( g ∗ M ) − → ( F 0 g ) ∗ ( F 1 M ) which is an isomorphism if and only if F 1 ( φ ) is cartesia n. It should th us be unsurpr ising that any co mm uting squar e (3 .5) gives r ise to an oplax na tural transformatio n b etw een the corresp onding pseudofunctor s . Reca ll that an oplax natu r al tr ansformation b etw een pseudofunctor s P , Q : B op → C at consists of functors φ x : P x → Qx and natural transforma tions P x P g / / φ x       }  φ g P y φ y   Qx Qg / / Qy satisfying appropr iate coher ence conditions. In a lax natur al tr ansformatio n , the 2-cells go the other direction, a nd in a pseudo natur al tr ansformation the 2-cells are inv ertible. Deﬁnition 3. 7. Any co mm uting square of functors (3.5) is called an o plax mor- phism o f ﬁbrations . It is a strong morphism o f ﬁbrations if whenever φ is a 12 MICHAEL SHULMAN cartesian arrow in A ov er g , then F 1 ( φ ) is ca r tesian in A ′ ov er F 0 ( g ). If F 0 is an iden tity B = B ′ , then we say F 1 is a morphism ov er B . A transformation of ﬁbrations b etw een tw o (oplax) morphisms of ﬁbrations is just a pair o f natura l trans formations, one lying ab ov e the other. If the tw o morphisms are ov er B , the transfor mation is ov er B if its downstairs c o mpo nen t is the ident ity . Prop osition 3.8. L et F ib op ℓ, B denote the 2-c ate gory of ﬁbr ations over B , oplax morphisms of ﬁbr ations over B , and tr ansformations over B , and let [ B op , C at ] op ℓ denote the 2 -c ate gory of pseudofunctors B op → C at , oplax natu ra l tr ansforma- tions, and mo diﬁc ations. Then the ab ove c onstructions deﬁne an e quivalenc e of 2-c ate gories F ib op ℓ, B ≃ [ B op , C at ] op ℓ . If we r estrict to the str ong morphisms of ﬁbr ations over B on t he left and t he pseudo natur al t r ansformations on the right, we again have an e quivalenc e F ib B ≃ [ B op , C at ] . Compared to pseudofunctors, ﬁbra tions have the adv ant age that they incorp o- rate all the base change functors f ∗ and all their coherence data automa tically . W e m ust remember, how ever, that the functors f ∗ are not determined uniquely by the ﬁbration, only up to natura l isomor phism. If Φ is a functor such that Φ op : A op → B op is a ﬁbratio n, we say that Φ is an o pﬁbration . (The ter m ‘coﬁbration’ used to b e common, but this carr ie s the wrong intuition for homotopy theorists, since a n opﬁbration is still characterized b y a lifting prop erty .) The car tesian a rrows in A op are called op cartesian arrows in A . A cleav ag e for an opﬁbration consists of op car tesian ar r ows M → f ! M , g iving rise to a functor f ! : A A → A B for each morphism f : A → B in B . F or any opﬁbration, the co llection of functors f ! forms a c ovariant pseudofunctor B → C at , and c o nv er sely , any c ov ar iant pseudofunctor gives rise to an o pﬁbration. A commutativ e squa re (3.5) in which Φ a nd Φ ′ are o pﬁbrations is ca lled a lax morphism of opﬁbrations , and it is strong if F 1 preserves op cartesia n arr ows; these corr esp ond to la x and pse udo natural transforma tions, resp ectiv ely . Prop osition 3.9. A ﬁbr ation Φ : A → B is also an opﬁbr ation pr e cisely when al l the functors f ∗ have left adjoints f ! . Pr o of. By deﬁnition of f ∗ , there is a na tural bijection betw een mor phisms M → N in A lying ov er f : A → B and mo rphisms M → f ∗ N in the ﬁb er A A . But if Φ is also a n opﬁbration, these mor phisms are also bijective to mo rphisms f ! M → N in A B , so we have an adjunction A A ( M , f ∗ N ) ∼ = A B ( f ! M , N ) as desired. The conv erse is stra ight forward.  W e will refer to a functor which is both a ﬁbration and an opﬁbration as a biﬁbration . A square (3.5) in which Φ and Φ ′ are biﬁbra tions is called a lax morphism of biﬁbrations if F 1 preserves cartesian ar r ows, an oplax m orphism of biﬁbrations if it preserves op cartesian ar rows, and a strong morphism of biﬁbrations if it preserves b oth. Examples 3.10. The ﬁbration Mod : Mo d → Ring is in fact a biﬁbra tion; the left adjoint f ! is given by extension of scalar s . F or any categor y C with pullbacks, the ﬁbra tion Arr C : C ↓ → C is a lso a biﬁbration; the left adjoint f ! is given by comp osing with f . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 13 In many ca ses, the functors f ∗ also hav e righ t a djoints, usually written f ∗ . Thes e functors a r e not a s conv eniently descr ib ed by a ﬁbra tional condition, but we will see in § 5 that in a framed bicategor y , they can b e describ ed in terms o f bas e change ob jects and a c lo sed s tructure. W e say that a ﬁbration is a ∗ - ﬁbration if a ll the functors f ∗ hav e right adjoints f ∗ . Similar ly we hav e a ∗ -biﬁbration , in which every mo r phism f g ives rise to a n adjoint string f ! ⊣ f ∗ ⊣ f ∗ . Examples 3. 11. M od is a ∗ -biﬁbration; the right adjoints are given by co extension of s c alars. Arr C is a ∗ - biﬁbr a tion pr ecisely when C is lo c al ly c artesian close d (that is, each slice ca teg ory C /X is cartesia n closed). Often the mere existence of left o r right adjoints is insuﬃcient, and we need to require a commutativit y condition. W e will explore this further in § 16. 4. Framed bica tegories Morally sp eaking, a framed bicategor y is a double catego ry in which the 1-cells can be restr ic ted and extended along the vertical arrows. W e will for ma lize this by saying that L a nd R a re biﬁbrations. Thus, for any f : A → B in D 0 , there will b e t wo diﬀerent functors which should b e called f ∗ , one arising from L and o ne fro m R . W e distinguish b y writing the ﬁrst on the left a nd the sec o nd on the right. In other words, f ∗ M is a horizontal 1-cell equipped with a car tesian 2-cell A | f ∗ M / / f   cart D B | M / / D while M g ∗ is e q uipped with a ca rtesian 2- cell B | M g ∗ / / cart C g   B | M / / D . A genera l cartesian arr ow in D 1 lying over ( f , g ) in D 0 × D 0 can then be written as f ∗ M g ∗ g = ⇒ f M . W e do similarly for o pca rtesian arrows and the cor resp onding functors f ! . W e r e fer to f ∗ as restriction and to f ! as extensi on . If f ∗ also has a rig h t adjoint f ∗ , we refer to it as co extension . It is worth commenting explicitly on what it mea ns for a 2-cell in a double category to b e cartesian or op cartesia n. Supp ose given a ‘niche’ of the for m A f   C g   B | M / / D 14 MICHAEL SHULMAN in a double catego ry D . This cor resp onds to an ob ject M ∈ D 1 and a morphism ( f , g ) : ( A, C ) → ( B , D ) = ( L, R )( M ) in D 0 × D 0 . A cartesia n lifting of this mor- phism is a 2 -cell A | f ∗ M g ∗ / / f   cart C g   B | M / / D such that any 2-cell of the form E | N / / f h   ⇓ F gk   B | M / / D factors uniquely as follows: E | N / / h   ⇓ F k   A f ∗ M g ∗ / / f   cart C g   B | M / / D . In particular , if h = 1 A and k = 1 C , this says that a n y 2-cell A | N / / f   ⇓ C g   B | M / / D can b e represented b y a glo bular 2-cell A | N / / ⇓ B A | f ∗ M g ∗ / / B . Therefore, ‘all the information’ a bo ut the 2-cells in a framed bicateg ory will be carried by the globular 2-cells and the base change functors. In par ticular, we can think of D a s ‘the bicategory D eq uipp ed with ba se change functors’. This can b e made precise; see a ppendix C. The interaction of ﬁbratio nal conditions with the do uble category structure has further implicatio ns . It is reas onable to expect that restriction and ex tension will commut e with horizontal c ompo sition; thus we will hav e f ∗ ( M ⊙ N ) g ∗ ∼ = f ∗ M ⊙ N g ∗ . This implies, how ever, that for any 1- cell M : B p → C and arrow f : A → B , we have f ∗ M ∼ = f ∗ ( U B ⊙ M ) ∼ = f ∗ U B ⊙ M , FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 15 and hence the base change functor f ∗ can b e r epresented by horizo n tal compo sition with the sp ecial o b ject f ∗ U B , which we call a b ase change obje ct . In the case of M o d , this is the s ta ndard fact that restricting along a r ing ho- momorphism f : A → B is the same as tensoring with the ( A, B )-bimodule f B , by which we mean B reg arded as an ( A, B )-bimodule via f on the left. F or this rea s on, we write f B for the base change o b j ect f ∗ U B in a ny do uble categ ory . Similarly , we write B f for U B f ∗ . The exis tence of such base change ob jects, suitably formalized, turns out to be suﬃcient to ensure that al l restrictions exist. This formaliza tion of ba se change ob jects can be given in an essentially diagrammatic wa y , which mo reov er is s e lf- dual. Thus, it it is also equiv alent to the existence of ex tensions . This is the co nten t of the following result. Theorem 4.1. The fol lowing c onditions on a double c ate gory D ar e e quivalent. (i) ( L, R ) : D 1 → D 0 × D 0 is a ﬁbr ation. (ii) ( L, R ) : D 1 → D 0 × D 0 is an opﬁbr ation. (iii) F or every vertic al arr ow f : A → B , ther e exist 1-c el ls f B : A p → B and B f : B p → A to gether with 2-c el ls (4.2) | f B / / f   ⇓ | U B / / , | B f / / ⇓ f   | U B / / , | U A / / f   ⇓ | B f / / , and | U A / / ⇓ f   | f B / / such that the fol lowing e qu ations hold. | U A / / ⇓ f   f B / / f   ⇓ | U B / / = | U A / / f   ⇓ U f f   | U B / / | U A / / f   ⇓ B f / / ⇓ f   | U B / / = | U A / / f   ⇓ U f f   | U B / / (4.3) | f B / / ∼ = | U A / / ⇓ | f B / / f   ⇓ | f B / / | U B / / | f B / / ∼ = = | f B / / | f B / / | B f / / ∼ = | f B / / ⇓ | U A / / f   ⇓ | U B / / | B f / / | B f / / ∼ = = | B f / / | B f / / (4.4) Pr o of. W e ﬁr s t show that (i) ⇒ (iii). As indica ted ab ov e, if ( L, R ) is a ﬁbr a tion we deﬁne f B = f ∗ U B and B f = U B f ∗ , and we let the ﬁrst t wo 2-cells in (4.2) b e the cartesian 2- c e lls characterizing these tw o restrictions . The unique fac to rizations of U f through these t wo 2-cells then gives us the second tw o 2-cells in (4.2) suc h that the equa tions (4.3) a re satisﬁed by deﬁnition. 16 MICHAEL SHULMAN W e show that the ﬁrst equation in (4.4) is satisﬁed. If we c o mpo s e the left side of this equation with the cartesia n 2 -cell deﬁning f B , we obtain | f B / / ∼ = | U A / / ⇓ | f B / / f   ⇓ | f B / / ∼ = | U B / / f B / / f   ⇓ | U B / / = | f B / / ∼ = | U A / / ⇓ | f B / / f   ⇓ f B / / f   ⇓ | U B / / | U B / / ∼ = | U B / / | U B / / = | f B / / ∼ = | U A / / f   ⇓ U f | f B / / f   ⇓ | U B / / ∼ = | U B / / | U B / / = | f B / / f   ⇓ | U B / / , which is once aga in the cartes ia n 2-cell deﬁning f B . How ever, we also have | f B / / | f B / / f   ⇓ | U B / / , = | f B / / f   ⇓ | U B / / , Thu s, the uniqueness o f factor izations through cartesian arrows implies that the given 2-cell is equal to the identit y , a s desired. This shows the ﬁrst equation in (4.4); the seco nd is a na logous. Thus (i) ⇒ (iii). Now assume (iii), a nd let M : B p → D be a 1-cell and f : A → B and g : C → D be vertical arrows; w e claim that the following co mpo site is cartesian: (4.5) | f B / / f   ⇓ | M / / 1 M | D g / / ⇓ g   | U B / / M / / | U D / / ∼ = | M / / T o show this, supp ose that | N / / f h   ⇓ α gk   | M / / FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 17 is a 2-cell; we m ust sho w that it factors uniquely th roug h (4.5). Consider the comp osite (4.6) | N / / ∼ = | U A / / h   ⇓ U h | N / / h   ⇓ α k   | U C / / ⇓ U k k   U A / / ⇓ f   g   U C / / ⇓ | f B / / | M / / | D g / / . Comp osing this with (4 .5 ) and using the equa tions (4.3) on each side, we get α back again. Thus, (4.6) gives a factorization of α through (4.5 ). T o prove uniqueness, suppo se that we had another factorization (4.7) | N / / ⇓ β h   k   | f B / / f   ⇓ M / / 1 M | D g / / ⇓ g   | U B / / M / / | U D / / ∼ = | M / / = | N / / f h   ⇓ α gk   | M / / Then if we substitute the left-hand side of (4 .7) for α in (4.6) and use the equa- tions (4.4) on the left a nd rig h t, we see that everything cancels a nd we just g et β . Hence, β is equa l to (4.6), so the facto rization is unique. This prov es that (4.5 ) is cartesian, so (iii) ⇒ (i). The pro o f that (ii) ⇔ (iii) is e x actly dual.  Deﬁnition 4.8. When the equiv alent co nditions of Theorem 4.1 are satisﬁed, we say that D is a framed bi category . Thu s, a framed bicategory has b oth restrictions and extensions. By the con- struction for (iii) ⇒ (i), we see that in a framed bicategory we hav e (4.9) f ∗ M g ∗ ∼ = f B ⊙ M ⊙ D g . The dual construction for (iii) ⇒ (ii) shows that (4.10) f ! N g ! ∼ = B f ⊙ N ⊙ g D . In particular , taking N = U B , we see tha t addition to (4.11) f B ∼ = f ∗ U B and B f ∼ = U B f ∗ , we have (4.12) f B ∼ = U A f ! and B f ∼ = f ! U A . More sp eciﬁcally , the ﬁr st tw o 2-cells in (4 .2) are always cartesia n and the second t wo are op cartesia n. It thus follows that from the uniqueness o f ca r tesian and op- cartesian arrows that 1-cells f B a nd B f equipped with the data of Theo rem 4.1(iii) 18 MICHAEL SHULMAN are unique up to isomor phism. In fact, if f B and f f B are tw o s uc h 1-cells, the canonical isomo r phism f B ∼ = − → f f B is given explicitly by the co mpos ite | f B / / ∼ = | U A / / ⇓ | f B / / f   ⇓ | f f B / / | U B / / | f f B / / ∼ = The case of B f is s imilar. W e can now prov e the expe c ted compatibility b etw een base c hange and ho rizon- tal comp osition. Corollary 4.13. In a fr ame d bic ate gory, we have f ∗ ( M ⊙ N ) g ∗ ∼ = f ∗ M ⊙ N g ∗ and f ! ( M ⊙ N ) g ! ∼ = f ! M ⊙ N g ! . Pr o of. Use (4.9) and (4 .10), together with the a s so ciativity of ⊙ .  On the other hand, if co extensions exist, we have a canonical morphism (4.14) f ∗ M ⊙ N g ∗ − → f ∗ ( M ⊙ N ) g ∗ given by the adjunct o f the c o mpo site f ∗  f ∗ M ⊙ N g ∗  g ∗ ∼ = f ∗ f ∗ M ⊙ N g ∗ g ∗ − → M ⊙ N , but it is rar ely a n isomorphism. In general, co extensio n is often less well behaved than restriction and extensio n, which partly justiﬁes our choice to use a formalism in which it is less natural. Examples 4.15. All of the double categor ies we introduced in § 2 are actually framed bicatego ries, and many of them have co extensio ns as well. • In M o d , if M is an ( A, B )-bimodule and f : C → A , g : D → B are ring homomorphisms, then the restric tio n f ∗ M g ∗ is M regar ded as a ( C, D )- bimo dule via f and g . Similarly , f ! is given by extension of scalar s and f ∗ b y co extension of sca lars. The base ch ange ob jects f B a nd B f are B regarded as an ( A, B )-bimodule a nd ( B , A )-bimo dule, respec tiv ely , via the map f . • In S pan ( C ), re s trictions ar e given b y pullback and extensions ar e given b y comp osition. The base change ob jects f B a nd B f for a map f : A → B in C ar e the spans A 1 A ← − A f − → B a nd B f ← − A 1 A − → B , r esp ectiv ely . These are often known as the gr aph o f f . Coe x tensions ex ist when C is lo cally cartesian closed. • In E x , the base change functors a re deﬁned in [MS06, § 11.4 a nd § 12.6 ], and the base change ob jects are a version of the sphere sp ectrum describ ed in [MS06, § 17.2 ]. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 19 • In D is t ( V ), restrictions are given by pre c o mpo sition, and extensio ns and co extensions are given by left a nd right Kan extension, r espec tively . F or a V -functor f : A → B , the base change ob jects f B and B f are the dis- tributors B ( − , f − ) a nd B ( f − , − ), resp ectively . • In n C ob , r e s triction, extensio n, and co extension ar e all giv en by compos ing a diﬀeomorphism of ( n − 1 )-manifolds with the given diﬀeomor phism ont o a collar of the b oundary . The ba s e change o b j ects of a diﬀeomor phism f : A ∼ = B ar e f and its inv e r se, regar ded as thin cob ordisms. • In A dj , res triction and extension are g iven by comp osing with suitable adjoints. F or example, given h : B → D and adjunctions f ! : A ⇄ B : f ∗ and g ! : C ⇄ D : g ∗ , then a cartesian 2- cell is g iven by the square A | g ∗ hf ! / / f !       |  ε C g !   B | h / / D where ε is the counit of the a djunction g ! ⊣ g ∗ . The bas e change ob jects for an a djunction f ! ⊣ f ∗ are f ! and f ∗ , resp ectively . Co extensions do no t generally exist. W e als o obser ve that the base change ob jects are pseudofunctorial. This is related to, but distinct from, the pseudofunctoriality of the base change functors . Pseudofunctoriality o f base change functors means that for A f − → B g − → C , we hav e f ∗ ( g ∗ ( M )) ∼ = ( g f ) ∗ M coherently , while pseudofunctor ialit y o f base change ob jects means that we hav e f B ⊙ g C ∼ = gf C coherently . Howev e r , since base change ob jects represent all base change functors, either implies the other. Prop osition 4.1 6. If D is a fr ame d bic ate gory with a chosen cle avage, then the op er ation f 7→ f B deﬁnes a pseudofunctor D 0 → D which is the identity on obje cts. Similarly, the op er ation f 7→ B f deﬁnes a c ont r avariant pseudofunctor D 0 op → D . 5. Duality theor y W e mentioned in Example 2.8 that the notion o f an adjunction can be deﬁned in ternal to any 2 - category . In fact, the deﬁnition can eas ily b e extended to any bicategory: an adjunction in a bicategory B is a pair o f 1 -cells F : A p → B and G : B p → A tog ether with 2 -cells η : U B → G ⊙ F a nd ε : F ⊙ G → U A , s a tisfying the usual tr iangle identities with appro priate ass o ciativit y and unit isomorphisms inserted. An internal adjunction is an exa mple of a formal concept whic h is useful in both t yp es of bicatego ries discussed in the in tro duction, but its me aning is very diﬀerent in the tw o cases. In C at - lik e bicatego ries, a djunctions b ehav e m uch lik e ordinary adjoint pa irs of functors; in fact, we will use them in this w ay in § 8 . In M o d -like bicategories, on the other hand, adjunctions enco de a notion of duality . In par ticular, if C is a monoidal ca tegory , co nsidered as a one-ob ject bicatego ry , an adjunction in C is b etter known as a dual p air in C , and o ne sp eaks of an ob ject 20 MICHAEL SHULMAN Y as b eing left o r right dual to a n o b ject X ; see, for example, [May01 ]. When C is symmetric monoida l, left duals and right duals coincide. Examples 5 .1. When C = M o d R for a comm utative ring R , the dualizable ob jects ar e the ﬁnit ely generated pro jectiv es. When C is the stable homo to p y category , the dualizable ob jects are the ﬁnite CW s pectra . The terminology of dual pairs was extended in [MS06] to adjunctions in M o d -like bicategories, which behave more like dual pair s in monoidal categor ies than they do like adjoint pair s of functors . Of course, now the distinction b etw een left and right matters. Explicitly , we hav e the following. Deﬁnition 5. 2. A dual pair in a bica tegory D is a pair ( M , N ), with M : A p → B , N : B p → A , toge ther with ‘ev aluatio n’ and ‘co ev aluation’ maps N ⊙ M → U B and U A → M ⊙ N satisfying the triangle identities. W e say that N is the right dual of M and that M is righ t dualizable , and dually . The deﬁnition of dual pair g iven in [MS06 , 16.4.1 ] is a ctually reversed from ours , although it do esn’t lo ok it, be cause of our diﬀeren t conv ent ions a bo ut which wa y to write ho rizontal comp osition. But bec a use we also turn around the horizontal 1-cells in all the examples, the terms ‘right dualizable’ and ‘left dualizable’ refer to the same actual ob j ects a s b efor e. O ur conv en tion ha s the adv a n tage that the right dual is also the rig h t adjoint. Because a dual pair is formally the s ame as an adjunction, all formal prop erties of the latter apply as well to the former . One example is the calculus of mates, as deﬁned in Exa mple 2.8: if ( M , N ) and ( P , Q ) are dual pairs , then there is a natural bijection b etw een morphisms M → P and Q → N . W e deﬁne a dual pair in a framed bicategor y D to b e just a dual pair in its underlying horizontal bica teg ory D . In this case, we hav e natura l exa mples coming from the base change o b j ects. Prop osition 5.3. If f : A → B is a vertic al arr ow in a fr ame d bic ate gory D , then ( f B , B f ) is natur al ly a dual p air. Pr o of. Since the base change functor f ! is left adjoint to f ∗ , we hav e equiv alences D ( M ⊙ f B , N ) ≃ D ( M f ! , N ) ≃ D ( M , N f ∗ ) ≃ D ( M , N ⊙ B f ) . By the bicategorica l Y oneda lemma (see, for example, [Str80]), which applies to dual pairs just a s it applies to a djunctions, this implies the desired r esult. Alternately , the unit and counit can be co nstructed dir e ctly from the da ta in Theorem 4.1(iii); the unit is | U A / / ∼ = | U A / / ⇓ | U A / / f   ⇓ | f B / / | B f / / FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 21 and the counit is | B f / / | f B / / f   | U B / / ⇓ | U B / / ⇓ | U B / / ∼ = Equations (4.3) a nd (4.4) ar e then exactly what is needed to prov e the triangle iden tities.  In par ticular, each of the base change o b jects f B and B f determines the other up to isomorphism. Co mb ining these dual pairs with ano ther genera l fact a bo ut adjunctions in a bica tegory , we hav e the following generalization o f [MS06, 1 7.3.3– 17.3.4]. Prop osition 5.4. L et ( M , N ) b e a dual p air in a fr ame d bic ate gory with M : A p → B , N : B p → A , and let f : B → C b e a vertic al arr ow. Then ( M f ! , f ! N ) is also a dual p air. Similarly, for any g : D → A , ( g ∗ M , N g ∗ ) is a dual p air. Pr o of. W e have M f ! ∼ = M ⊙ f B and f ! N ∼ = B f ⊙ N , so the result follows from the fact that the co mpo s ite of adjunctions in a bicatego ry is an adjunction. The other case is analog ous.  This implies the following generalization of the ca lculus of mates. Prop osition 5.5. L et ( M , N ) and ( P, Q ) b e dual p airs in a fr ame d bic ate gory. Then ther e is a natura l bije ction b etwe en 2-c el ls of t he fol lowing forms: A M | / / f   ⇓ B g   C P | / / D and B N | / / g   ⇓ A f   D Q | / / C . Pr o of. A 2- cell o f the former shap e is equiv alen t to a glo bular 2-cell M g ! → f ∗ P , and a 2-ce ll of the latter shap e is equiv alent to a globular 2-c ell g ! N → Qf ∗ . By Prop osition 5.4, we hav e dual pair s ( M g ! , g ! N ) and ( f ∗ P, Qf ∗ ), so the ordinar y calculus of mates applies.  Examples 5.6. Dual pair s b ehav e signiﬁcan tly diﬀerently in man y of our ex amples. (i) If R is a not-necessar ily comm utative ring , then a right R -mo dule M : Z p → R is r ight dua liza ble in M o d when it is ﬁnitely genera ted pro jectiv e. (ii) The only dual pa irs in S pan ( C ) a r e the ba se change dual pairs. (This is easy in Set , and we ca n then apply the Y oneda lemma for arbitra ry C .) (iii) If M : A p → B is a right dualizable distributor in D ist ( V ), a nd B satisﬁes a mild c o completeness c o ndition dep ending on V (called ‘Cauch y com- pleteness’), then M is neces sarily also of the form f B for some V -functor f : A → B . When V = Set , Cauch y completeness just means that every idempo tent splits. When V = Ab , it means that idemp o ten ts split and ﬁ- nite copro ducts exist. See [Ke l8 2, § 5.5] for mor e a bo ut Cauch y completion of enr ich ed ca tegories. (iv) Dualizable ob jects in E x are s tudied extensively in [MS06, Ch. 18]. 22 MICHAEL SHULMAN Remark 5 .7. There is also a general notion of tr ac e for endomorphisms of a dualizable ob ject in a symmetric monoidal category : if ( X , Y ) is a dual pa ir and f : X → X , then the trace o f f is the comp osite I η − → X ⊗ Y f ⊗ 1 − → X ⊗ Y ∼ = − → Y ⊗ X ε − → I . T ra ces were extended to dual pairs in a bicatego r y in [Pon07], b y eq uipping the bicategory with a suitable structure, called a shadow , to take the place of the symmetry isomorphism. In [P S07] we will c onsider shadows in framed bicateg ories. Dualit y in symmetric monoidal catego ries is mos t interesting when the mo noidal category is clo sed. There is also a classical notion of close d bic ate gory , which means that the comp osition of 1-c e lls has adjoints on b oth sides : B ( M ⊙ N , P ) ∼ = B ( M , N  P ) ∼ = B ( N , P  M ) . Recall that B ( M ⊙ N , P ) denotes the set of g lobular 2 -cells from M ⊙ N to P . In 2-catego rical language , this s ays that rig h t Kan extensio ns and rig h t Kan liftings exist in the bica tegory B . It is prov en in [MS0 6, § 16.4], extending classical results for symmetric monoidal categories , that when M : A p → B is right dualizable, its right dual is alwa ys (iso- morphic to) the ‘canonical dual’ D r M = M  U B ; and conv ersely , whenever the canonical map M ⊙ D r M → M  M is an iso mo rphism, then M is rig h t dualizable. This ca n also be stated as the generaliza tion to bicatego ries of the fact (see [ML98, X.7]) that a functor G ha s an adjoin t when the Kan extension of the identit y along G exists and is preserved by G , and in that case the Kan extension gives the adjoint. Deﬁnition 5.8. A framed bica tegory D is closed just when its underlying hori- zontal bicateg o ry D is clo sed. Examples 5.9 . Many of our examples of framed bicatego ries are clos ed. • M o d is closed; its hom-ob jects are given by P  M = Hom C ( M , P ) N  P = Hom A ( N , P ) . • As long a s V is closed and complete, th en D ist ( V ) is clo sed; its ho m- ob jects are g iven by the cotensor pro duct of distr ibutors (the end co n- struction). • S pan ( C ) is closed precisely when C is lo cally cartesian close d. • E x is a lso clo s ed. This is prov en in [MS06, § 1 7.1]; we will des c r ibe the general metho d of pro of in § 14. Remark 5. 10. A monoidal catego r y is clo sed (on b oth sides) just when its cor- resp onding vertically trivial framed bicateg ory is clos ed. O n the other hand, if a monoidal categor y is symmetric, then the left and right internal-homs ar e iso mor- phic. In § 10 we will prove an a na logue o f this fact for framed bicategories equipp ed with an ‘inv olution’, which includes a ll o f our e x amples. It is not surprising that there is so me r elationship betw een clos e dnes s and ba se change. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 23 Prop osition 5. 11. L et D b e a close d fr ame d bic ate gory. Then for any f : A → C , g : B → D , and M : C p → D , we have f ∗ M g ∗ ∼ = ( g D  M )  C f ∼ = g D  ( M  C f ) and in p articular f C ∼ = C  C f D g ∼ = g D  D Pr o of. Straightforw ard adjunction a r guments.  Note that this implies, by uniqueness of adjoints, that the restriction functor f ∗ can also b e describ ed a s f ∗ N = N  C f . O f cour se there are cor resp onding versions for comp osing on the other side. Moreov er, if co extensions exist, then uniqueness of adjoints also implies that we hav e (5.12) f ∗ M ∼ = M  f C. Conv ersely , if D is closed, then (5.12) deﬁnes a right adjoint to f ∗ ; thus c o extensions exist in any c lo sed framed bicatego ry , and a lso hav e a na tural description in terms of the base change o b jects. 6. The 2-ca tegor y of framed bica tegories W e now introduce the morphisms b etw een framed bicategories. T o b egin with, it is easy to deﬁne mo rphisms of double categories by a nalogy with monoidal cate- gories. Deﬁnition 6.1. Let D a nd E be double categories. A lax double functor F : D → E cons ists of the following. • F unctors F 0 : D 0 → E 0 and F 1 : D 1 → E 1 such that L ◦ F 1 = F 0 ◦ L and R ◦ F 1 = F 0 ◦ R . • Natural transforma tio ns F ⊙ : F 1 M ⊙ F 1 N → F 1 ( M ⊙ N ) and F U : U F 0 A → F 1 ( U A ), who se comp onents ar e globular, a nd whic h sa tisfy the usual co- herence axio ms for a lax monoida l functor o r 2-functor (see, for exam- ple, [ML9 8, § XI.2]). Dually , we have the deﬁnition of an oplax doubl e functor , for which F ⊙ and F U go in the o ppo s ite direction. A strong double functor is a lax double functor fo r which F ⊙ and F U are (globula r ) iso mo rphisms. If just F U is an isomor phism, we say that F is normal . W e o ccas ionally abuse notation by writing just F for either F 0 or F 1 . Observe that a lax double functor prese rves vertical compo sition and identit ies strictly , but preserves horizo ntal comp osition and identiti es only up to constraints. Like the constraints a , l , r for a do uble ca teg ory , the maps F ⊙ and F U are globular , but must be natur a l with resp ect to all 2-cells, not only g lobular ones. If D and E a re just monoidal catego ries, then a double functor F : D → E is the sa me as a monoidal functor (of whichev er sor t). The terms ‘lax’, ‘oplax’, and ‘strong’ a re chosen to generalize this situation; some authors refer to strong double functors as pseudo double funct ors . Since the mono idal functors whic h arise in practice ar e mo st frequently lax, many author s refer to these s imply as ‘monoidal 24 MICHAEL SHULMAN functors’. It is also tr ue for framed bicategories that the lax morphisms ar e often those of most in terest, but we will a lw ays k eep the adjectives for cla r it y . Example 6.2. Let F : C → D b e a lax monoidal functor, wher e the monoidal categories C and D b oth hav e co equalizers pres erved by ⊗ . Then it is well known that F preser ves mono ids , monoid homomorphisms, bimodules, and equiv ar ia nt maps. Mor eov er, if M : A p → B and N : B p → C are bimo dules in C , so that their tensor pro duct is the co equalizer M ⊗ B ⊗ N / / / / M ⊗ N / / M ⊙ N then we hav e the commutativ e diag r am (6.3) F M ⊗ F B ⊗ F N / / / /   F M ⊗ F N / /   F M ⊙ F N   F ( M ⊗ B ⊗ N ) / / / / F ( M ⊗ N ) / / F ( M ⊙ N ) in whic h the top diagra m is the co equalizer deﬁning the tensor pro duct of bimodules in D , and hence the do tted map is induced. Mor eov er, s ince U A in M o d ( C ) is just A regarded as a n ( A, A )-bimo dule, we have F ( U A ) ∼ = U F A . It is stra ig h tforward to chec k that this isomorphism and the dotted map in (6 .3) are the data for a no rmal lax double functor M o d ( F ) : M o d ( C ) → M o d ( D ). Note that F do es no t need to pr eserve co equalizers, so the bo ttom row of (6.3) need not b e a co equa lizer diag r am. How ev er, if F do es pr eserve coequa lizers, and is moreov er a strong monoidal functor, so that the left and middle vertical ma ps are isomorphisms, then so is the right vertical map; hence M o d ( F ) is a stro ng do uble functor in this case. In pa rticular, if C = Mo d R and D = Mo d S for co mmutativ e rings R and S and f : R → S is a homomorphism of commutativ e r ings, then the extension-o f-s calars functor f ! : Mo d R → Mo d S is s tr ong monoida l a nd preser ves co equalizers, hence induces a str o ng double functor. The restriction-o f-scalars functor f ∗ : Mo d S → Mo d R , on the other hand, is o nly lax monoidal, a nd hence induces a normal lax double functor. Example 6.4. Let F : C → D be any functor betw een tw o ca teg ories with pull- backs. Then we hav e an induced norma l opla x double functor S pan ( F ) : S pan ( C ) → S pan ( D ). If F preserves pullbacks, then S pan ( F ) is strong. Now suppose that D and E ar e framed bicategories . Since the characterization of base change ob jects in Theorem 4.1(iii) only inv o lves horizontal comp osition with units, any normal lax (or oplax) fra med functor will pr eserve base change o b je cts up to isomor phism; that is, F ( f B ) ∼ = F f ( F B ). If it is str ong , then it will a lso preserve restrictions a nd extensions, since we have f ∗ M g ∗ ∼ = f B ⊙ M ⊙ D g and similarly . More generally , any la x or oplax double functor F : D → E betw een framed bicategories automatically induces co mpa rison 2-cells such as ( F f ) ! ( F M ) − → F ( f ! M ) and (6.5) F ( f ∗ N ) − → ( F f ) ∗ ( F N ) , (6.6) b y unique factorization through car tesian and o pc a rtesian arrows. As remar ked in § 3, the ﬁrst o f these go es in the ‘lax direction’ while the second go es in the ‘o plax FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 25 direction’. Thu s, for the who le functor to des erve the adjective ‘lax’, the seco nd of these must b e an isomorphism, s o that it has a n in verse which go es in the lax direction. This happ ens just when F preserv es cartesian 2-cells. How e ver, it turns out that this is automatic: any lax double functor b etw een framed bicategories preserves cartesian 2-cells, so that (6.6) is always an iso mor- phism when F is lax. Dually , any o plax do uble funct or preserves o pca rtesian 2-cells , so that (6.5) is an isomo r phism when F is oplax. T o prov e this, we ﬁrst observe that fo r any lax do uble functor F : D → E a nd any ar row f : A → B in D , we hav e the following diag ram of 2-cells in E : (6.7) U F A F U / / op cart   F ( U A ) F (opc art)   F f ( F B ) f F / / cart   F ( f B )   F (cart)   U F B F U / / F ( U B ) . The dotted arrow, given by unique factor ization through the op car tesian one, is the sp e c ia l ca se o f (6.5) when M = U A ; w e denote it by f F . The upp er square in (6.7) commut es by deﬁnition, and the low er squar e also commutes by uniqueness of the factorization. Similarly , we hav e a 2 -cell ( F B ) F f F f − → F ( B f ). If F is opla x instead, we hav e 2-cells in the other dir e c tion. If F is strong (or even just no rmal), the transformations exist in bo th directions, are inv erse isomorphisms, and each is the mate of the inv erse o f the o ther. Prop osition 6.8 . Any lax double funct or b etwe en fr ame d bic ate gories pr eserves c artesian 2-c el ls, and any oplax double functor b etwe en fr ame d bic ate gories pr eserves op c artesian 2-c el ls. Pr o of. Let M : B p → D in D a nd f : A → B , g : C → D . Then the following c o m- po site is cartesia n in D : (6.9) A | f B / / f   cart B | M / / D | D g / / cart C g   B | U B / / B | M / / D | U D / / D and the following comp osite is cartesian in E : (6.10) F A | F f ( F B ) / / F f   cart F B | F M / / F D | ( F D ) F g / / cart F C F g   F B | U F B / / F B | F M / / F D | U F D / / F D. Applying F to (6.9) and factoring the result through (6.10), we obtain a comparison map (6.11) F ( f B ⊙ M ⊙ D g ) − → F f ( F B ) ⊙ F M ⊙ ( F D ) F g , 26 MICHAEL SHULMAN which we wan t to show to b e an isomorphism. W e hav e an obvious candidate for its inv erse, namely the fo llowing comp osite. (6.12) | F f ( F B ) / / f F ⇓ | F M / / | ( F D ) F g / / ⇓ F f | F ( f B ) / / ⇓ F ⊙ | F M / / | F ( D g ) / / | F ( f B ⊙ M ⊙ D g ) / / Consider ﬁrst the comp osite of (6.12) followed b y (6.11 ): | F f ( F B ) / / f F ⇓ | F M / / | ( F D ) F g / / ⇓ F f | F ( f B ) / / ⇓ F ⊙ | F M / / | F ( D g ) / / F ( f B ⊙ M ⊙ D g ) / / ⇓ | F f ( F B ) / / | F M / / | ( F D ) F g / / If we p ostcomp ose this with (6.10), then by deﬁnition o f (6.11 ), we obtain | F f ( F B ) / / f F ⇓ | F M / / | ( F D ) F g / / ⇓ F f | F ( f B ) / / ⇓ F ⊙ | F M / / | F ( D g ) / / F ( f B ⊙ M ⊙ D g ) / / F f   F (cart) F g   | F M / / . By natura lity of the lax constraint for F , this is equal to | F f ( F B ) / / f F ⇓ | F M / / | ( F D ) F g / / ⇓ F f F ( f B ) / / F f   F (cart) | F M / / F ( D g ) / / F (cart) F g   | F ( U B ) / / ⇓ F ⊙ | F M / / | F ( U D ) / / | F M / / FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 27 Because the low er square in (6.7) commutes, this is equal to | F f ( F B ) / / F f   cart | F M / / | ( F D ) F g / / cart F g   U F B / / ⇓ F U | F M / / U F D / / ⇓ F U | F ( U B ) / / ⇓ F ⊙ | F M / / | F ( U D ) / / | F M / / which is equal to (6.10), by the coherence axioms for F . Thus, b y unique factor- ization throug h (6.10), we co nclude that (6.12) followed by (6.11) is the identit y . Now consider the co mp osite of (6.1 1 ) follow ed by (6.1 2). By the construction of factorizations in Theor em 4.1 , (6.11) can b e computed by c ompo sing horizo n tally with op car tesian 2-cells; thus our desired co mpos ite is | U F A / / op cart | F ( f B ⊙ M ⊙ D g ) / / F f   F (cart) | U F C / / F g   op cart F f ( F B ) / / f F ⇓ F M / / ( F D ) F g / / ⇓ F f | F ( f B ) / / ⇓ F ⊙ | F M / / | F ( D g ) / / | F ( f B ⊙ M ⊙ D g ) / / . By deﬁnition of f F a nd F f , this is equal to | U F A / / ⇓ F U | F ( f B ⊙ M ⊙ D g ) / / | U F C / / ⇓ F U F ( U A ) / / F (opc art) | F ( f B ⊙ M ⊙ D g ) / /   F (cart) F ( U C ) / /   F (opc art) | F ( f B ) / / ⇓ F ⊙ F M / / | F ( D g ) / / | F ( f B ⊙ M ⊙ D g ) / / . 28 MICHAEL SHULMAN By natura lity of F ⊙ , this is equal to (6.13) | U F A / / ⇓ F U | F ( f B ⊙ M ⊙ D g ) / / | U F C / / ⇓ F U | F ( U A ) / / ⇓ F ⊙ | F ( f B ⊙ M ⊙ D g ) / / | F ( U C ) / / F ( f B ⊙ M ⊙ D g ) / / F (stuﬀ ) | F ( f B ⊙ M ⊙ D g ) / / where ‘stuﬀ ’ is the comp osite | U A / / op cart | f B / / f   cart | M / / | D g / / cart | U C / / g   op cart | f B / / | U B / / | M / / | U D / / | D g / / which is equa l (mo dulo co ns traints) to the ident ity o n f B ⊙ M ⊙ D g . Thus, applying the coherence axioms for F again, (6.13 ) reduces to the identit y o f F ( f B ⊙ M ⊙ D g ). Therefore, (6 .12) is a tw o -sided inv erse for (6 .11), so the latter is an iso morphism; hence F preserves cartesia n 2-cells. The oplax case is dual.  Here we see again the adv a ntage o f using ﬁbra tions rather than intro ducing ba se change functors explicitly: since ﬁbrations are ‘non- algebraic’, all their constra in ts and coher e nce co me for free. T his lea ds us to the following deﬁnition. Deﬁnition 6.14. A lax framed functor is a lax double functor b et ween framed bicategories. Similarly , a n opl ax or strong framed functor is a double functor of the appropria te t yp e b etw een framed bicatego ries. W e observed in § 1 that while 2-functors give a g o o d notion of morphism be- t ween b oth sorts of bicatego ries, the right notion of tra nsformation for M o d - like bicategories is r ather murkier. Once we include the vertical ar rows to get a fra med bicategory , how ever, it b ecomes muc h clear er what the transfor ma tions should b e. Deﬁnition 6.15. A double transformation b etw een t wo lax double functors α : F → G : D → E consists of natura l transfor ma tions α 0 : F 0 → G 0 and α 1 : F 1 → G 1 (both usually written as α ), such that L ( α M ) = α LM and R ( α M ) = α RM , and s uch that F A | F M / / ⇓ F ⊙ F B | F N / / F C F A F ( M ⊙ N ) / / α A   ⇓ α M ⊙ N F C α C   GA | G ( M ⊙ N ) / / GC = F A α A   ⇓ α M | F M / / F B α B   ⇓ α N | F N / / F C α C   GA | GM / / ⇓ G ⊙ GB | GN / / GC GA | G ( M ⊙ N ) / / GC FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 29 and F A | U F A / / ⇓ F 0 F A F A F ( U A ) / / α A   ⇓ α U A F A α A   GA | G ( U A ) / / GA = F A | U F A / / α A   ⇓ U α A F A α A   GA U GA / / ⇓ F 0 GA GA | G ( U A ) / / GA . The fra med version of this deﬁnition requires no mo diﬁcation at all. Deﬁnition 6.16. A framed transformation b et ween tw o lax framed functors is simply a double tr ansformation b etw een their underlying lax double functors. W e leav e it to the reader to deﬁne transformations betw een oplax functors. In the case of ordinar y bicatego r ies, there is also a notion of ‘mo diﬁca tio n’, or morphism betw een tra nsformations, but with framed bicategories we usually have no need for these. Thus, the fra med bicategor ies, fr a med functors, and fra med transforma tio ns form a C at - lik e bica teg ory , whic h is in fact a strict 2-categ ory . Prop osition 6.17. Smal l fr ame d bic ate gories, lax fr ame d functors, and fr ame d tr ansformatio ns form a strict 2-c ate gory F r B i ℓ . If we r estrict to str ong fr ame d functors, we obtain a 2-c ate gory F r B i , and if we use oplax fr ame d functors inste ad, we obtain a 2-c ate gory F r B i op ℓ . Of cour se, double categor ies, double functors, and double transfor mations also form lar g er 2-categ ories D bl ℓ , D bl , and D bl op ℓ . Remark 6.18. Recall that w e ca n regar d a monoidal category as a framed bi- category whose vertical categor y is trivial, and tha t the framed functors betw een vertically trivial framed bicategorie s a re precise ly the mono idal functors (whether lax, oplax, or strong ). It is easy to chec k that framed transformations are a lso the same a s monoidal transformations; thus M on C at is eq uiv alent to a full sub-2- category of F r B i . This is to b e contrasted with the situation for or dinary ‘unframed’ bicategories . W e ca n also consider monoidal categor ies to b e bicatego r ies with just one 0 - cell, and 2-functors b etw een such bicategories do also cor resp ond to monoida l functors , but mo s t transfor mations betw een suc h 2- functors do not give r ise to an ything resembling a monoida l transforma tion; see [CG06]. Th us, framed bicategories ar e a b etter genera liza tion o f monoida l categor ies than ordinar y bicategories a r e. Example 6.19. Let C a nd D b e monoida l categor ies with co equalizers pr eserved b y ⊗ , and let α : F ⇒ G : C → D b e a mo noidal natural trans fo r mation betw een lax monoidal functors. W e hav e already seen that F a nd G give rise to lax framed functors. Moreover, the fact that α is a monoidal tra nsformation implies that if A is a monoid in C , α A : F A → GA is a monoid homomorphism in D , and similarly for bimo dules. Therefore, we hav e an induced framed transformation M o d ( α ) : M o d ( F ) → M o d ( G ) . This makes M o d ( − ) into a strict 2-functor. Its doma in is the 2-categ ory of monoidal categories with co equalizers preser ved b y ⊗ , lax monoidal functors, and monoidal 30 MICHAEL SHULMAN transformations, a nd its co doma in is F r B i ℓ . If we restrict the domain to stro ng monoidal functors which preserve co e q ualizers, the image lies in F r B i . Example 6. 20. Let C , D b e catego ries with pullbacks a nd α : F ⇒ G : C → D a natural transfor ma tion. Then α induces a fra med tra nsformation S pan ( α ) : S pan ( F ) → S pan ( G ) in an o b vious wa y . This makes S pan into a strict 2- functor from the 2-categ ory of categories with pullbacks, all functors , a nd all natural transformations, to F r B i op ℓ . If we restrict the domain to functor s which pr eserve pullbacks, the image lies in F r B i . Remark 6.21. It is ea sy to see that any framed functor induces a 2-functor of the appr opriate type b etw een hor izontal bicatego ries, but the situa tio n for fra med transformations is less c lear. W e will consider this further in a ppendix B. 7. Framed equiv alences All the usual no tions of 2-categ ory theor y apply to the study of framed bicat- egories via the 2 -categorie s F r B i ℓ , F r B i , and F r B i op ℓ , and generally reduce to element ary notio ns when expr essed explicitly . Since, a s remarked a bove, the lax framed functors ar e often those of most interest, we work most frequently in F r B i ℓ , but ana lo gous results ar e alwa ys true fo r the other tw o ca ses. One imp orta n t 2 -categor ica l notion is that o f internal e quivalenc e . This is deﬁned to b e a pair of morphisms F : D → E and G : E → D with 2-cell iso morphisms F G ∼ = Id and GF ∼ = Id. The no tio n o f equiv alence for framed bicatego r ies we obtain in this wa y solves another of the pr oblems rais ed in § 1. Deﬁnition 7.1. A framed equiv alence is an internal equiv a lence in F r B i ℓ . Thu s, a framed equiv alence consists of lax framed functors F : D ⇄ E : G with framed natural iso morphisms η : Id D ∼ = GF and ε : F G ∼ = Id E . It might seem strange not to require F a nd G to be st r ong framed functors in this deﬁnition, but in fact this is a utomatic. Prop osition 7.2 . In a fr ame d e quivalenc e as ab ove, F and G ar e automatic al ly str ong fr ame d functors (henc e give an e quivalenc e in F r B i ). W e will prov e this in the next section as Coro llary 8.5. Since strict 2- functors preserve internal equiv alences, our 2- functorial wa ys o f constructing framed bicatego ries give us a ready supply of framed equiv ale nce s . F o r example, any monoidal equiv alence C ≃ D of monoidal categor ie s with co equalizers preserved by ⊗ induces a fr a med equiv alence M o d ( C ) ≃ M o d ( D ). Similarly , any equiv alence o f categories with pullbac ks induces a framed equiv alence betw een framed bicatego ries of spa ns . As for ordinary catego ries, we ca n characterize the framed functors which are equiv alences as those which are ‘full, faithful, and essentially sur jective’. First we in tro duce the terminolog y , beg inning with double catego ries. Recall that we write FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 31 f D g ( M , N ) for the set of 2-cells o f the fo r m A | M / / f   ⇓ α B g   C | N / / D . Deﬁnition 7 .3. A lax or opla x double functor is full (res p. faithful ) if it is full (resp. faithful) on vertical ca tegories and each map (7.4) F : g D f ( M , N ) − → F g E F f ( F M , F N ) is surjective (resp. injective). In the case of a fr ame d functor, how ever, the notions simplify so mewhat. Prop osition 7.5. A lax or oplax framed fun ctor F : D → E is ful l (r esp. faithful) in the sense of Deﬁnition 7.3 if and only if it is ful l (r esp. faithful) on vertic al c ate gories and e ach functor D ( A, B ) → E ( F A, F B ) is ful l (r esp. faithful). Pr o of. Deﬁnition 7.3 clearly implies the given condition. Conv ersely , suppos e that F : D → E is a lax framed functor. W e have a natural bijection g D f ( M , N ) ∼ = D ( M , f ∗ N g ∗ ) which is preserved by F , since it pre s erves restrictio n. In other words, the diagram g D f ( M , N ) ∼ = / / F   D ( M , f ∗ N g ∗ ) F   F g E F f ( F M , F N ) ∼ = * * U U U U U U U U U U U U U U U U E ( F M , F ( f ∗ N g ∗ )) ∼ =   E ( F M , ( F f ) ∗ ( F N )( F g ) ∗ ) commut es. Th us, if the right-hand map is surjectiv e (resp. injective), so is the left-hand map. An analog o us a rgument works for an o pla x fra med functor , using extension instead of restriction.  This is yet another expression of the fact that in a framed bicateg o ry , the g lobular 2-cells carry the infor ma tion ab out all the 2- cells. A similar thing happ ens for essential surjectivity . Deﬁnition 7.6. A lax o r oplax double functor F : D → E is essen tially surjectiv e if we can simultaneously make the following choices: • F or ea c h ob ject C of E , an ob ject A C of D and a vertical isomo rphism α C : F ( A C ) ∼ = C , and • F or eac h horizo n tal 1-cell N : C p → D in E , a horizontal arrow M N : A C p → A D in E and a 2-cell iso morphism F ( A C ) F ( M N ) / / α C |   α M ∼ = F ( A D ) α D   C N | / / D . 32 MICHAEL SHULMAN Prop osition 7.7. A lax or oplax framed functor is essent ial ly surje ctive, in the sense of Deﬁnition 7.6, if and only if it is essential ly surje ct ive on vertic al c ate gories and e ach funct or D ( A, B ) → E ( F A, F B ) is essential ly surje ct ive. Pr o of. Clearly Deﬁnition 7.6 implies the given co ndition. Conv ersely , supp ose that F satisﬁes the giv en condition. Cho ose isomo rphisms α C : F ( A C ) ∼ = C for each ob ject C o f E , whic h exist b ecause F is essentially s urjectiv e on verti- cal catego ries. Then g iven N : C p → D , w e hav e α ∗ C N α ∗ D : F ( A C ) p → F ( A D ), so since F : D ( A C , A D ) → E ( F ( A C ) , F ( A D )) is essen tially surjective, w e hav e an M N : A C → A D and a g lo bular isomo rphism F ( M N ) ∼ = α ∗ C N α ∗ D . Comp osing this with the cartesia n 2-cell deﬁning α ∗ C N α ∗ D , we obtain the desired α M .  The following theorem and its co r ollary are the main p oints of this section. Of course, we deﬁne a double e quiv alence to b e an internal equiv a lence in D bl ℓ . Theorem 7. 8. A str ong double functor F : D → E is p art of a double e quivalenc e if and only if it is ful l, faithful, and essential ly surje ct ive. Pr o of. W e sketc h a construction o f an inv erse equiv alence G : E → D for F . Make choices as in Deﬁnition 7 .6, and deﬁne GC = A C and GN = M N . Deﬁne G o n vertical arrows and 2-cells by co mpos ing with the chosen isomor phisms; vertical functoriality follows from F b e ing full and faithful. W e pro duce the co nstraint cells for G by comp osing these isomorphisms with the inverses of the co ns traint cells for F and using that F is full a nd faithful; this is why we need F to b e strong . The ch oices from the deﬁnition of essenti ally surjective then give directly a dou- ble natural isomo r phism F G ∼ = Id E , a nd we can pro duce a double natural isomo r- phism GF ∼ = Id D b y reﬂecting identit y maps in E . Thus G and F form a double equiv alence.  Corollary 7.9. A str ong fr ame d functor F : D → E is p art of a fr ame d e quivalenc e pr e cisely when • It induc es an e qu ivalenc e F 0 : D 0 → E 0 on vertic al c ate gories, and • Each functor F : D ( A, B ) → E ( F A, F B ) is an e quivalenc e of c ate gories. Pr o of. Combine P rop osition 7.5 a nd Prop osition 7.7 with Theorem 7.8 to see that F has a n inv erse which is a strong double functor , hence also a strong framed functor by Prop osition 6.8 .  A framed equiv a lence F : D ⇄ E : G clearly includes an equiv a lence F 0 : D 0 ⇄ E 0 : G 0 of vertical catego r ies. It is less clea r that it induces a biequiv a lence D ≃ E of hor izont al bicategories . W e w ill se e in app endix B, howev er , that this is true, though not trivial. This lack of triviality , in the following example, was one of the original motiv ations for this work. Example 7. 1 0. Ther e a re a nu mber of fr a med bicategories related to E x , such a s a ﬁber wise version E x B where the ob jects are already para metr iz e d ov er some space B , and an equiv a riant version G E x in which everything ca r ries a n actio n by some ﬁxed gr oup G . In [MS06, 19 .3.5] it was o bserved (essentially) that G E x G/H , the framed bicategory of G -equiv ar iant par ametrized sp ectra all ov er the cos e t space G/H , and H E x , the fra med bicategor y of H -equiv a r iant para metrized sp ectra, are equiv alent. How e ver, as observed in [MS06], the language of bicategor ies do es not rea lly suﬃce to describ e this fact. On ob jects, the equiv alence go es a s follows: if X is a FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 33 G -space ov er G/H , the ﬁber X e is an H -space; while if Y is an H - space, G × H Y is a G -space ov er G/H . But the comp osites in either direction are only homeomorphic, not equa l, wherea s the bicategor y E x descr ibed in [MS06] do es not include a ny information ab out homeomorphisms o f base spaces. 8. Framed adjunctions Adjunctions are one of the most important tools of categor y theory . Thus, from a categor ical p oint o f view, one o f the most serious problems with M o d -like bicategories is the lack o f a g o o d notion of adjunction b etw een them. F or example, Ross Street wrote the following in a review of [CKW91]: Nearly tw o decades a fter J. W. Gray’s work [Gra 7 4], the most useful general notion of adjoin tness for mor phisms b etw een 2 - categories has still no t emerged. Perhaps the go o d notion should depend on the kind of 2-catego ries in mind; 2-categories whose arrows are functions or functors are of a diﬀerent nature from those whose arr ows ar e relations o r profunctors. In fact, motiv a ted by the desire for a go o d notion of adjunction, [CKW91] and related paper s suc h as [V er92, CKVW98] come very close to our deﬁnition of framed bicategory . In app endix C we will make a formal co mparison; for now we simply develop the theory of framed adjunctions. Deﬁnition 8.1. A framed adjunc tion F ⊣ G is an internal a djunction in the 2-catego ry F r B i ℓ . Explicitly , it consists of lax framed functors F : D → E and G : E → D , together with framed tr ansformations η : Id D → GF and ε : F G → Id E satisfying the us ual tria ngle iden tities. Similarly , an op-framed adjunction is an in ternal adjunction in F r B i op ℓ . Exp erience shows that adjunctions in F r B i ℓ arise more frequently than the other t wo t yp es, hence deserve the unador ned name. Ho wev er, we hav e the following fundamen tal res ult. Prop osition 8.2. In any fr ame d adjunction F ⊣ G , t he left adjoint F is always a strong fr ame d functor. Sketch of Pr o of. This actually follows formally from a general 2-categor ical result known as ‘do ctrinal adjunction’; s e e [Kel74]. F or the non-2-catego rically inclined reader we sketc h a more concr ete version of the proo f. W e ﬁrs t sho w that the following co mpos ite is an inv erse to F ⊙ : F M ⊙ F N → F ( M ⊙ N ): (8.3) F ( M ⊙ N ) F ( η ⊙ η ) − − − − − → F ( GF M ⊙ GF N ) G ⊙ − → F G ( F M ⊙ F N ) ε − → F M ⊙ F N F or example, the following diag ram s hows that the comp o site in o ne direction is the identit y . F ( GF M ⊙ GF N ) G ⊙ / / F G ( F M ⊙ F N ) ε / / F G ( F ⊙ )   F M ⊙ F N F ⊙   F ( M ⊙ N ) F ( η ) / / id 5 5 F ( η ⊙ η ) O O F GF ( M ⊙ N ) ε / / F ( M ⊙ N ) . 34 MICHAEL SHULMAN The r ight-hand square co mm utes by natura lit y of ε , the left-hand squa r e commutes beca use η is a framed transfo r mation, and the lower triangle is one of the triangle iden tities. The other direction is ana lo gous. Similarly , w e show that the following comp osite is an inv erse to F U : U F A → F ( U A ): (8.4) F ( U A ) F ( U η ) / / F U GF A G U / / F GU F A ε / / U F A , so that F is strong .  The similarity b etw een (8.3) and (8.4) is obvious. In fact, these co mpos ites a r e the mates o f the co nstraint cells for G under an adjunction in a suitable 2-c a tegory; the re a der may co nsult [Kel7 4] for details. Of co urse, in a n op- fr amed adjunction, the rig h t adjoint is stro ng . W e can now prove Pr o po sition 7.2. Corollary 8.5. Both functors in a fr ame d e quivalenc e ar e str ong fr ame d functors. Pr o of. It is well-known that any classical eq uiv alence of catego ries can b e improv ed to an ‘adjoint equiv alence’, mea ning an equiv alence in which the isomorphisms F G ∼ = Id and Id ∼ = GF are also the unit and counit of an adjunction F ⊣ G , and hence their inv erses are the unit a nd counit of a n a djunction G ⊣ F . This fact can easily b e ‘int ernalized’ to an y 2 -category , such as F r B i ℓ . Thus, any lax framed functor which is par t of a framed equiv alence is a framed left adjoint , and hence b y Prop osition 8.2 is s trong.  As is the ca se for categ ories, we ca n also characterize frame d adjunctions using univ ersa l arrows. A similar r esult for do uble categ ories was given in [Gar0 6]. Recall that g iven a functor G : E → D , a u niversal arr ow to G is a n arrow η : A → GF A in D , fo r some ob ject F A ∈ E , such that any other ar row A → GY factors through η via a unique map F A → Y in E . Similarly , if G : E → D is a framed functor, we deﬁne a universal 2-c el l to be a 2-cell η : M → GF M in D , not in g eneral g lobular, whos e left a nd right frames are universal arrows in D 0 , and such that any 2-c e ll M → GN factors thr o ugh η via a unique 2-ce ll F M → N in E . Prop osition 8.6. L et G : E → D b e a lax fr ame d functor. Then G has a fr ame d left adjoint if and only if the fol lowing ar e true. (i) F or every obje ct A in D , ther e is a u niversal arr ow A → GF A . (ii) F or every horizontal 1-c el l M : A p → B in D , t her e is a universal 2-c el l M → GF M , as describ e d ab ove. (iii) If M → GF M and N → GF N ar e u n iversal 2-c el ls, then so is the c om- p osite   | M / / univ   | N / / univ   GF M / / ⇓ G ⊙ GF N / / | G ( F M ⊙ F N ) / / FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 35 (iv) If A → GF A is universal, then so is t he c omp osite | U A / / univ   ⇓ U univ univ   U GF A / / ⇓ G 0 | G ( U F A ) / / If G is str ong, then (iii) simpliﬁes to ‘the horizontal c omp osite of universal 2-c el ls is universal’ and (iv) simpliﬁes to ‘the horizontal unit of a universal arr ow is a universal 2-c el l’. Sketch of Pr o of. It is stra ightf orward to s how that if G ha s a left a djoint, then the co nditions a re satisﬁed. Conv ersely , co nditions (i) a nd (ii) clearly guarantee that G 0 and G 1 bo th have left a djoints F 0 and F 1 , and that LF 1 ∼ = F 0 L and RF 1 ∼ = F 0 R . Since E is a framed bica teg ory , we ca n r edeﬁne F 1 b y restricting a long these isomor phisms to ensur e that LF 1 = F 0 L and R F 1 = F 0 R . Conditions (iii) and (iv) then s upply the constraints to make F into a strong framed functor. The universal cells g ive a double tr ansformation η : Id → GF a nd the counit ε : F G → Id is constructed as usual. The las t statement follows becaus e anything isomorphic to a universal arrow is universal.  Since strict 2-functors preserve internal adjunctions, our 2 -functorial wa ys of constructing framed bicategor ies also give us a rea dy supply of framed adjunctions. Example 8.7. Since M o d is a 2-functor, an y monoida l adjunction b et ween monoidal categories with co equa lizer s prese r ved b y ⊗ gives rise to a framed adjunction. Here b y a monoidal adjunction we mean an adjunction in the 2-catego ry M on C at ℓ of monoidal catego r ies and lax monoidal functors. F or example, if f : R → S is a homomorphism of commutativ e rings , we hav e an induced mono ida l adjunction f ! : Mo d R ⇄ M o d S : f ∗ and therefore a fr amed adjunction M o d ( f ! ) : M o d ( R ) ⇄ M o d ( S ) : M o d ( f ∗ ) . Example 8.8. Since S pan is a strict 2 -functor, any adjunction f ∗ : E ⇄ F : f ∗ betw een categories with pullbac ks g ives rise to an op-framed adjunction S pan ( E ) ⇄ S pan ( F ). If f ∗ also preser ves pullbacks, then this adjunction lies in F r B i , hence is also a framed adjunction. 9. Monoidal frame d bica tegories Most of o ur examples als o hav e a n ‘external’ mo noidal structure. F or example, if M is an ( A, B )-bimodule and N is a ( C , D )-bimo dule, w e can form the ( A ⊗ C, B ⊗ D )-bimo dule M ⊗ N . The deﬁnition of a ‘monoidal bicateg ory’ inv olves many coher ence a xioms (see [GPS95, Gur06]), but for fr ame d bica teg ories we ca n simply inv oke gener al 2-catego ry theory once again. In a n y 2-categ ory with ﬁnite pro ducts, we hav e the notion of a pseudo-monoid : this is an ob ject A equipp ed with m ultiplication A × A → A and unit 1 → A 36 MICHAEL SHULMAN satisfying the usual monoid axioms up to coherent iso morphism. A pseudo-monoid in C at is precisely an ordinar y monoidal catego ry . Thus, it makes sense to deﬁne a monoidal fra med bicategory to b e a pseudo-mo noid in F r B i . What this means is essentially the following. Deﬁnition 9.1. A monoi dal framed bicategory is a fra med bicategory equipp ed with a stro ng framed functor ⊗ : D × D → D , a unit I ∈ D 0 , and framed na tural constraint iso morphisms satisfying the usual axioms. If we unrav el this deﬁnition more explicitly , it says the following. (i) D 0 and D 1 are b oth monoidal ca tegories. (ii) I is the mono idal unit o f D 0 and U I is the monoidal unit of D 1 . (iii) The functors L a nd R are strict mo no idal. (iv) W e hav e an ‘int erchange’ isomorphism x : ( M ⊗ P ) ⊙ ( N ⊗ Q ) ∼ = ( M ⊙ N ) ⊗ ( P ⊙ Q ) and a unit isomorphism u : U A ⊗ B ∼ = ( U A ⊗ U B ) satisfying appr opriate a xioms (these a rise from the c o nstraint data for the strong framed functor ⊗ ). (v) The asso ciativity and unit isomorphisms for ⊗ a re framed transformations. As we saw in § 6, a s trong framed functor such as ⊗ preser ves cartesian and op cartesian arr ows. Thus we automatically ha v e isomo rphisms such as f ∗ M ⊗ g ∗ N ∼ = ( f ⊗ g ) ∗ ( M ⊗ N ). Examples 9.2. Many of o ur examples of fra med bicateg o ries a re in fa c t monoidal. • The framed bicategory M o d , and mo re generally M o d ( C ) for a symmetric monoidal C , is monoidal under the tensor pro duct of ring s a nd bimo dules. Note that the tensor pro duct of bimodules refer red to here is ‘external’: if M is an ( R, S )-bimodule and N is a ( T , V )-bimo dule, then M ⊗ N is an ( R ⊗ T , S ⊗ V )-bimodule. • If C has ﬁnite limits, then S pan ( C ) is a monoida l framed bicategory under the ca r tesian pro duct o f ob jects and spans. • E x is mo noidal under the ca r tesian pro duct of spaces and the ‘exter nal smash pro duct’ ∧ of par a metrized sp e ctra. • n C ob is monoidal under disjoint union of ma nifolds and cob ordisms. • D ist ( V ) is monoidal under the tensor pro duct of V - categories (see [Kel82, § 1.4]). Example 9. 3 . Recalling that monoidal ca tegories can be ident iﬁed with vertically trivial framed bicategories, it is easy to chec k that a vertically trivial monoida l framed bicategory is the sa me as a catego ry with tw o in terchanging monoidal struc- tures. Mo r e generally , if D is any mo noidal framed bicatego ry , then the category D ( I , I ) inherits t wo int erchanging monoida l structures ⊙ a nd ⊗ . By the Eckmann- Hilton ar gument , any t wo such interc hanging mo noidal structures ag ree up to iso- morphism and are braided. W e emphasize that the asso ciativity and unit constra in ts ar e vertic al isomor - phisms. F o r example, in the monoidal framed bica tegory M o d , the asso ciativity constraint o n ob jects is the ring isomorphism ( A ⊗ B ) ⊗ C ∼ = A ⊗ ( B ⊗ C ). This FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 37 is to b e contrasted with the cla s sical notion of ‘monoidal bicategory’ in which the constraints are 1-cells, which would corresp ond to bimo dules in this case. So while a framed bicatego ry obviously has a n underlying horizontal bicategory , it r e quires pro of that a monoida l framed bicateg ory has a n underlying mono ida l bicategory; see app endix B. W e observe, in pa ssing, that an external mo noidal structure automatica lly pre - serves dual pairs. Prop osition 9. 4. If ( M , N ) and ( P , Q ) ar e dual p airs in a monoidal fr ame d bic at- e gory, then so is ( M ⊗ P , N ⊗ Q ) . Pr o of. It is easy to see that any strong framed functor pr eserves dual pairs, a nd ⊗ is a strong framed functor.  Now, just as an ordina ry monoida l catego r y can b e braided or symmetric, so can a pseudo- monoid in a n arbitra ry 2-categ o ry with pro ducts. W e deﬁne a braided or symmetric mono ida l framed bicateg ory to be essentially a braided o r symmetric pseudo-monoid in F r B i . More explicitly , a braided monoidal framed bicatego ry is a monoidal framed bicategory such that D 0 and D 1 are br aided monoida l with bra iding s s , the functors L and R are br a ided monoidal, and the following diagra ms co mm ute: ( M ⊙ N ) ⊗ ( P ⊙ Q ) s / / x   ( P ⊙ Q ) ⊗ ( M ⊙ N ) x   ( M ⊗ P ) ⊙ ( N ⊗ Q ) s / / ( P ⊗ M ) ⊙ ( Q ⊗ N ) ( U A ⊗ U B ) u / / s   U A ⊗ B U s   U B ⊗ U A u / / U B ⊗ A . A symmetric monoidal framed bica teg ory is a braided monoidal framed bicategory such that D 0 and D 1 are s y mmetric. Examples 9.5. All the examples o f mono idal framed bicatego ries given in Exa m- ples 9.2 are in fact symmetric mo noidal. Example 9.6 . If D is a braided o r symmetric monoidal framed bicategor y , then D ( I , I ) inherits tw o interc hanging monoidal structures, one of which is braided, and therefore it is essen tially a s ymmetric monoidal category . Conv ersely , the vertically trivial mono idal framed bicatego r y corresp onding to any symmetric mono idal cat- egory is a sy mmetric monoidal framed bicategory . W e now deﬁne the morphisms b etw ee n monoidal framed bicateg o ries. As usual, these come in three ﬂavors. Deﬁnition 9.7. A lax monoi dal framed functor b etw een monoidal framed bicategories D , E co nsists of the following s tructure and prop erties. • A lax framed functor F : D → E . • The str ucture of a la x monoidal functor on F 0 and F 1 . • Equalities L F 1 = F 0 L and RF 1 = F 0 R o f lax monoidal functors. 38 MICHAEL SHULMAN • The co mpo sition co nstraints for the lax framed functor F a re mono idal natural tra nsformations. It is strong if F is a strong framed functor a nd F 0 and F 1 are strong monoida l functors. If D and E are braided (resp. symmetric), then F is braided (resp. sym- metric ) if F 0 and F 1 are. W e have a dual deﬁnition of opl ax monoidal framed functor . A monoidal framed transformation is a fra med transforma tio n suc h that α 0 and α 1 are mo noidal transformations . These deﬁnitions give v ar ious 2-ca tegories, each of which has its own attendant notion of equiv alence and adjunction. W e will no t sp ell these out explicitly . Examples 9.8 . The 2-functor M o d lifts to a 2 -functor from symmetric monoidal categories with co equalizers preserved by ⊗ to symmetric monoidal framed bicate- gories. Similarly , S pan lifts to a 2- functor la nding in symmetric monoidal framed bicategories. Finally , we consider what it means for a fra med bicategory to be ‘closed monoidal’. Deﬁnition 9. 9. A monoidal framed bicategor y D is externally closed if for any ob jects A, B , C, D , the functor ⊗ : D ( A, C ) × D ( B , D ) − → D ( A ⊗ B , C ⊗ D ) has rig h t adjoints in each v ar iable, which w e write  and  . Explicitly , this means that for horizont al 1 - cells M : A p → C , N : B p → D , and P : A ⊗ B p → C ⊗ D , there a re 1-cells N  P and P  M and bijections D ( M ⊗ N , P ) ∼ = D ( M , N  P ) ∼ = D ( N , P  M ) . Of cours e, if D is symmetric, then  and  agree, mo dulo suitable isomor phisms. Examples 9.10. The monoidal fra med bicategor y M o d is externally closed, as ar e E x and D i st ( V ). If C is lo ca lly cartesia n clo s ed, then S pan ( C ) is a lso externally closed. 10. Involutions In most of our exa mples, the ‘directionality’ of the horizo n tal 1- cells is to some extent arbitrary . F or exa mple, an ( A, B )-bimodule could just as well b e rega rded as a ( B op , A op )-bimo dule. W e now deﬁne a structure which enco des this fact formally . If D is a framed bicatego ry , we wr ite D h · op for its ‘horizontal dual’: D h · op has the sa me vertical ca tegory as D , but a horizontal 1- cell from A to B in D h · op is a horizontal 1-cell from B to A in D , a nd the 2 -cells ar e similarly ﬂipp ed ho r izontally . Deﬁnition 10 .1. An i n v olution on a framed bicateg o ry D co nsists o f the follow- ing. (i) A stro ng framed functor ( − ) op : D h · op → D . (ii) A framed natural isomor phism ξ : (( − ) op ) op ∼ = Id D such that ( ξ A ) op = ξ A op ; thus ξ and ξ − 1 make ( − ) op in to an a djoin t equiv alence. W e say an in volution is vertica lly strict if the vertical arr ow comp onents of ξ are iden tities. If D , ( − ) op , and ξ are a ll monoidal (resp. symmetric monoida l), we say that the in volution is m onoidal (resp. s ymmetric monoidal ). FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 39 The s tr ong functoriality of ( − ) op implies that we hav e ( U A ) op ∼ = U A op ( M ⊙ N ) op ∼ = N op ⊙ M op . ( f ∗ M g ∗ ) op ∼ = ( g op ) ∗ ( M op )( f op ) ∗ ( f ! M g ! ) op ∼ = ( g op ) ! ( M op )( f op ) ! . In particular, w e hav e ( A f ) op ∼ = f op ( A op ) and dually . If the inv olution is mo no idal, we a lso hav e ( A ⊗ B ) op ∼ = A op ⊗ B op I op ∼ = I . Examples 10. 2. Most of o ur exa mples are equipp ed with vertically strict sym- metric monoidal inv olutions. • The in volution on M o d tak es a ring A to the opposite ring A op , and an ( A, B )-bimodule to the s a me a b elian g roup re g arded as a ( B op , A op )- bimo dule. • The inv olution o n D i s t ( V ) takes a V -catego ry to its oppos ite and reverses distributors in an obvious wa y . • The inv olution o n S pan ( C ) takes ea c h ob ject to itself, a nd a spa n A f ← − X g − → B to the span B g ← − X f − → A . • The inv olution on n C ob takes a manifold M to the manifold M op with the opp osite orientation, and reverses the direction of cob ordisms in an obvious wa y . In all these cas es, the 2-cell comp onents of ξ can a lso b e chosen to b e identit ies, but this is no t true for a ll inv olutions, even vertically strict ones. • The inv olution on E x takes ea ch space to itself, but takes a spec trum E parametrized ov er B × A to the pullback s ∗ E over A × B , where s is the symmetry isomor phism A × B ∼ = B × A . Here s ∗ s ∗ E is o nly canonically isomorphic to E , by pseudofunctoriality . In [MS06, 16.2.1 ] an involution on a bic ate gory was deﬁned to be essentially a pseudofunctor ( − ) op : B op → B eq uipped with a pseudonatura l transfor mation ξ : (( − ) op ) op ∼ = Id B whose 1-cell compo nen ts are iden tities (although the unit axiom for ξ was omitted). It is easy to see that any vertically strict in volution o n D gives rise to an inv olution on D . All the ab ov e examples a re vertically strict, but in § 11 and § 15 we will see examples which a r e not. An y symmetric mono idal ca teg ory , considered as a vertically trivia l framed bicat- egory , has a ca nonical inv olution. The functor ( − ) op is the identit y on 1 - cells (the ob jects of the mo noidal categ ory), and its comp osition constr a int is the s y mmetry isomorphism: ( A ⊙ B ) op = A ⊙ B ∼ = − → B ⊙ A = B op ⊙ A op . All the co mpo nen ts of ξ are identit ies. In fact, to give an inv olution on a vertically trivial framed bica tegory which is the iden tit y on 1-cells and for whic h ξ is an iden tity is essentially to g iv e a symmetry for the cor resp onding mono idal ca tegory . Thu s, we may view an inv olution on a fra med bica tegory as a generalizatio n of a symmetry on a mono idal catego ry . 40 MICHAEL SHULMAN One conseq uence of a monoidal categor y’s b eing symmetric is tha t if it is closed, then the left and r ig ht internal-homs are isomorphic. The original motiv ation in [MS06] for introducing inv olutions was to obtain a similar result for closed bi- categories ; see [MS06, 16 .3.5]. Of course, this is also true for framed bica tegories. Prop osition 10. 3. If D is a close d fr ame d bic ate gory e quipp e d with an involution, then we have M  N ∼ = ( N op  M op ) op . Pr o of. Since ( − ) op is a fra med equiv alence, it is lo ca lly full and faithful. Thus, if M : A p → B , N : C p → B , and P : C p → A , we hav e D ( C, A )( P, M  N ) ∼ = D ( C, B )( P ⊙ M , N ) ∼ = D ( B op , C op )(( P ⊙ M ) op , N op ) ∼ = D ( B op , C op )( M op ⊙ P op , N op ) ∼ = D ( A op , C op )( P op , N op  M op ) ∼ = D  ( C op ) op , ( A op ) op  ( P op ) op , ( N op  M op ) op  ∼ = D ( C, A )( P, ( N op  M op ) op ) so the result follows by the Y o neda lemma.  11. Monoids and modules In most o f our examples of monoidal framed bicatego ries, the external monoidal structure and the horizo n tal compo sition are more clo sely related than is captured b y the in terchange iso morphism: na mely , the horizontal co mpositio n M ⊙ N is a subo b ject or quo tien t of the externa l pro duct M ⊗ N . F or example, in M o d the tensor pro duct M ⊗ R N is a quotient of the e x ternal pro duct M ⊗ N , while in S pan the pullback M × B N is a subo b ject of the ex ternal pro duct M × N . An analogo us r elationship holds b etw een the bicatego rical homs  ,  and the externa l homs  ,  . In this section w e will generalize the construction of the fra med bicatego r y M o d ( C ) of monoids and mo dules from Example 2 .3, r eplacing the monoidal cate- gory C with a framed bicategory D . This describ es o ne gener al class of examples in which the hor izontal comp osition of ‘bimo dules’ is deﬁned as a co equalizer . In §§ 12–1 4, we will in vestigate fra med bicatego ries co nstructed in a way ana logous to S pan . W e will then combine these tw o constructions in § 15 to deﬁne framed bicategories of internal and enriched categories . Deﬁnition 11.1. Let D b e a framed bica teg ory . • A monoid in D consists of a n ob ject R , a horizo nt al 1 -cell A : R p → R , and globular 2-cells e : R → A and m : A ⊙ A → A called ‘unit’ and ‘m ultiplication’ such that the standard diag r ams commute. Thus it is just a monoid in the ordinary monoidal categor y D ( R, R ). • A mo noid homom orphism ( R, A ) → ( S, B ) consists of a vertical arrow f : R → S and a 2-cell φ : A f = ⇒ f B suc h that φ ◦ e = e and φ ◦ m = m ◦ ( φ ⊙ φ ). • A bimo dule from a monoid ( R, A ) to a monoid ( S, B ) is a hor izo nt al 1 - cell M : R p → S together with actio n maps a ℓ : A ⊙ M → M and a r : M ⊙ B → M ob eying the obvious compatibility axioms. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 41 • Let ( f , φ ) : ( R , A ) → ( S, B ) and ( g , ψ ) : ( T , C ) → ( U, D ) b e monoid homo- morphisms a nd M : ( R, A ) p → ( T , C ), N : ( S, B ) p → ( U, D ) b e bimo dules. A ( φ, ψ ) -equiv arian t map is a 2 -cell α : M g = ⇒ f N such that a ℓ ( φ ⊙ α ) = αa ℓ and a r ( α ⊙ ψ ) = αa r . • Let M : R p → S b e an ( A, B )-bimodule and N : S p → T b e a ( B , C )-bimodule. Their tensor pro duct is the following c o e q ualizer in D ( A, C ), if it ex ists: M ⊙ B ⊙ N ⇒ M ⊙ N → M ⊙ B N . Of co urse, if D is a monoidal categ ory , these notions reduce to the usual ones. Example 11.2. If C has pullbacks, then a monoid in S pan ( C ) is an in ternal category in C , and a monoid homomorphism is an in ternal functor. A bimo dule in S pan ( C ) is an ‘in ternal distributor’. Example 11.3. A monoid in M o d consists of a ring R tog e ther with an R -alg ebra A , and a mono id homo morphism ( R, A ) → ( S, B ) consists o f a r ing homomorphism f : R → S and an f -equiv ariant algebra ma p A → B . A bimo dule in M o d is just a bimo dule for the alg ebras. In o rder to deﬁne a framed bicategory o f monoids and bimo dules in D , we need to know that c o equalizers exist and are well-behaved. Deﬁnition 11.4. A framed bica tegory D has lo cal co e qualizers if each cate- gory D ( A, B ) has co equa lizers and ⊙ preser ves co equalize r s in e a ch v a riable. W e in tro duce the following notations. • F r B i q ℓ denotes the ful l sub-2 -categor y of F r B i ℓ determined by the fr a med bicategories with lo cal co equalizers. • F r B i q ℓ,n denotes the locally full sub-2-ca teg ory o f F r B i ℓ determined by the framed bicateg ories with lo cal co equa lizers and the n ormal lax framed functors. • F r B i q denotes the lo cally full sub-2 -category o f F r B i determined by the framed bicategories w ith loca l co equalizer s and the strong framed functors which preserve lo c a l co equalizer s. Note that if D is closed, as deﬁned in § 5, then ⊙ pre serves a ll colimits since it is a left adjoint. The following omnibus theorem combines all our results ab out monoids and mo dules in framed bicategor ies. Theorem 11.5. L et D b e a fr ame d bic ate gory with lo c al c o e qualizers. Then ther e is a fr ame d bic ate gory M o d ( D ) of m onoids, monoid homomorphi sms, bimo dules, and e quivariant maps in D . Mor e over: • M o d ( D ) also has lo c al c o e qualizers. • If D is close d and e ach c ate gory D ( A, B ) has e qualizers, then M o d ( D ) is close d. • If D is monoidal and its external pr o duct ⊗ pr eserves lo c al c o e qualiz- ers, then M od ( D ) has b oth of these pr op erties. If D is symmetric, so is M o d ( D ) . If D is external ly close d and e ach c ate gory D ( A, B ) has e qual- izers, t hen M o d ( D ) is external ly close d. • If D is e quipp e d with an involution, so is M o d ( D ) . If the involution of D is monoidal or symmetric monoidal, so is that of M o d ( D ) . 42 MICHAEL SHULMAN • M o d deﬁnes 2-functors F r B i q ℓ → F r B i q ℓ,n and F r B i q → F r B i q , and similarly for the monoidal versions. Even if F is a strong framed functor, M o d ( F ) is only lax unless F pr eserves lo cal co equalizers. If F is o plax, we cannot even deﬁne M o d ( F ). Of cour se, ther e is a dual construction C om o d , but it ar is es m uch less freq uen tly in practice. Example 11.6. If C is a monoida l category with coequa lizers preserved by ⊗ , then M o d ( C ) has lo ca l co e q ualizers, so we ha ve a fr a med bicategory M o d ( M o d ( C )) of algebras and bimo dules in C . Example 11. 7. I f C is a categor y with pullbacks and co equalizers preser ved by pullback, then S pan ( C ) ha s lo ca l co equa lizers, so we hav e a framed bica tegory M o d ( S pan ( C )) of internal categories and distributors in C . Example 11.8. When V is a co complete closed mono idal categ ory , we can also construct the fra med bicategory D i s t ( V ) of enriched ca tegories and distributors in this wa y . W e ﬁrst deﬁne the framed bica teg ory M at ( V ) a s follows: its vertical category is Set , a nd the ca tegory M at ( V )( A, B ) is the ca tegory of A × B matrices ( M ab ) a ∈ A,b ∈ B of o b j ects of V . Composition is by ‘matrix m ultiplication’. It is then eas y to check that M at ( V ) has lo cal co equa lizers and that M o d ( M at ( V )) ∼ = D ist ( V ). The monoidal catego ry M at ( V )( A, A ) is also called the catego ry of V - gr aphs with ob ject set A . Example 11. 9. Unlike these examples, E x do es not hav e lo cal co equalizers. W e will see a replacement for ‘ M o d ( E x )’ in § 15 . The rest of this section is devoted to the pro of of Theorem 11.5, breaking it up in to a series of pro p os itions for clar it y . Although long , the pro o f is routine and follow-y our-nose, so it can ea sily b e skipp ed. Prop osition 11. 10. If D is a fr ame d bic ate gory with lo c al c o e qualizers, then ther e is a fr ame d bic ate gory M o d ( D ) of monoids, m onoid homomorphisms, bimo dules, and e quivariant maps in D , and it also has lo c al c o e qu alizers. Pr o of. The pro o f that M o d ( D ) is a double categor y is similar to th e case of a monoidal catego ry . F or exa mple, we need the fact that ⊙ pres erves co equalizers to s how that M ⊙ N is a bimo dule a nd that the tensor pro du ct is asso ciative. T o deﬁne the hor izontal comp osite of bimo dule maps α : M ψ = ⇒ φ N and β : P χ = ⇒ ψ Q (where φ : A f = ⇒ f D , ψ : B g = ⇒ g E , a nd χ : C h = ⇒ h F are monoid homomor phisms), we s tart with the co mpos ite (11.11) R M / / f   α S g   P / / β T h   U N / / co eq V Q / / W U N ⊙ E Q / / W which we w ould like to factor through the co equalizer deﬁning M ⊙ B P . How ev er, that co equalizer lives in D ( R, T ), wher e as (11.11) is not glo bular. But since D is a FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 43 framed bicatego ry , we can fac to r (11.1 1) through a cartesian arrow to get a map M ⊙ P → f ∗ ( N ⊙ E Q ) g ∗ in D ( R , T ), and then apply the universal pr op erty of the co equalizer to get a map M ⊙ B P → f ∗ ( N ⊙ E Q ) g ∗ , and hence M ⊙ B P g = ⇒ f ( N ⊙ E Q ). This deﬁnes a ( φ, χ )-equiv ariant map which we c a ll the hor izo nt al comp osite α ⊙ ψ β . The axioms for a double ca tegory follow directly . W e now sho w that M o d ( D ) is a framed bicatego ry . By T heo rem 4.1, it suﬃces to show that it has r estrictions. Thus, suppose that A : R p → R , B : S p → S , C : T p → T , and D : U p → U are monoids in D , M : S p → U is a ( B , D )-bimo dule, and φ : A f = ⇒ f B and ψ : C g = ⇒ g D are monoid homo morphisms. W e then hav e the r estriction f ∗ M g ∗ : R p → T in D . By comp o sing the cartesian arrow in D with φ o r ψ and using the actions of B and D on M , then fac to ring through the cartesian a rrow, we o btain a ctions of A and C on f ∗ M g ∗ . F or e x ample, the action of A on f ∗ M g ∗ is determined by the equality R A / / f   φ R f ∗ M g ∗ / / f   cart T g   S B / / act S M / / U S M / / U = R A / / act R f ∗ M g ∗ / / T R f ∗ M g ∗ / / f   cart T g   S M / / U It is s tr aightforw ard to c hec k that with this structure, the cartesian arr ow f ∗ M g ∗ g = ⇒ f M in D deﬁnes a cartesian a rrow φ ∗ M ψ ∗ ψ = ⇒ φ M in M o d ( D ).  Prop osition 11.12. M o d deﬁnes a 2-functor F r B i q ℓ → F r B i q ℓ,n , which r estricts to a 2-fun ct or F r B i q → F r B i q . Pr o of. Let D , E ∈ F r B i q ℓ and let F : D → E b e a la x framed functor. Then F preserves monoids, monoid homomorphisms, bimo dules, and eq uiv ariant maps, for the sa me re a sons that lax mono idal functors do. W e deﬁne the unit constraint for M o d ( F ) to b e the iden tit y on F A , and the comp osition constraint to b e the res ult of factoring the comp osite (11.13) F M ⊙ F N F ⊙ − → F ( M ⊙ N ) − → F ( M ⊙ B N ) through the co equalizer (11.14) F M ⊙ F N → F M ⊙ F B F N It is straig h tforward to chec k that this ma kes M o d ( F ) into a nor mal lax double functor. Similarly , the comp onents of a fra med tra nsformation F → G deﬁne a framed transfo r mation M o d ( F ) → M o d ( G ). Finally , if F is strong a nd preserves lo cal co equa lizers, then (11.13) is a co equal- izer of the same maps that (11.14) is. Hence the induced co mpo sition constraint is an isomorphism, so M o d ( F ) is strong. It is easy to see that M o d ( F ) a lso preserves lo cal co equa lize r s, so that it lies in F r B i q .  44 MICHAEL SHULMAN Prop osition 11. 15. If D is a monoidal fr ame d bic ate gory with lo c al c o e qualizers pr eserve d by ⊗ , then so is M o d ( D ) . If D is symmetric, so is M o d ( D ) . Pr o of. It is easy to check that the 2-functor M o d : F r B i q → F r B i q preserves pro ducts, so it m ust preserve pseudo-mono ids a nd symmetric pseudo- monoids.  Prop osition 11.16. S upp ose t hat D has lo c al c o e qualizers and e ach c ate gory D ( A, B ) has e qualizers. If D is close d, then M o d ( D ) is close d. If D is monoidal and ex- ternal ly close d with lo c al c o e qualizers pr eserve d by ⊗ , then M o d ( D ) is external ly close d. Pr o of. Just as for mo noidal categ o ries.  Prop osition 11 .17. If D has lo c al c o e qualizers and is e quipp e d with an involution, so is M o d ( D ) . If D , M o d ( D ) , and the involution on D ar e monoidal or symmetric monoidal, so is the involution on M o d ( D ) . Pr o of. It is easy to see that M o d ( D h · op ) ≃ M o d ( D ) h · op , so we can s imply apply the 2- functor M o d to ( − ) op and ξ .  Note, how ever, that since the vertical ar row comp onent s o f ξ in M o d ( D ) ar e deﬁned from the 2-c el l co mpo nen ts of ξ in D , the inv olution of M o d ( D ) may not be vertically strict even if the inv olution of D is s o . 12. Monoidal fibra tions The generalized M o d constructio n from § 1 1 deﬁnes a horizon tal co mpos ition from an external pro duct via a co equalizer. In § 14 w e will explain how in a car tesian situation, horizontal compo s itions can b e co nstructed using a pullback or equalizer- t yp e construction instead. The basic input for this construction is a structure called a ‘monoidal ﬁbration’, which includes base change oper ations and an external pro duct, but a priori no hor izo nt al comp osition. Deﬁnition 12.1. A m onoidal ﬁbration is a functor Φ : A → B such that (i) A and B are monoidal ca tegories; (ii) Φ is a ﬁbration and a strict monoidal functor; and (iii) The tensor pro duct ⊗ o f A pr e s erves ca rtesian arrows. If Φ is also an opﬁbratio n a nd ⊗ preser ves op cartesian ar rows, we say that Φ is a monoidal biﬁbration . W e say that Φ is braided (resp. s ymmetric ) if A , B , and the functor Φ ar e braided (resp. symmetric). W e will also sp eak of ‘monoidal ∗ -ﬁbrations’ a nd ‘mono ida l ∗ -biﬁbr a tions’, but without implying any compatibility be tw een the monoidal structure a nd the right adjoints f ∗ . This is b e c ause in most ca ses there is no such compatibility . Example 12.2. Let C b e a catego ry with ﬁnite limits. Recall that if C ↓ denotes the category o f arrows in C , the co domain functor gives a biﬁbration Arr C : C ↓ → C called the ‘self-indexing’ of C . It is eas y to se e that Ar r C is a monoidal biﬁbration when C and C ↓ are eq uipped with their car tesian pro ducts. Example 12. 3. If D is a mo no idal fr a med bicategory , then ( L, R ) : D 1 → D 0 × D 0 is a monoidal biﬁbra tion. If D is bra ided or symmetric, so is ( L, R ). Example 12 .4. The ﬁbration Mod : Mo d → Ring is a mo no idal ∗ -biﬁbration under the tensor pro duct of rings and the ‘external’ tensor pro duct of mo dules. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 45 F or mos t of our applications, such as Theorem 12.8 b elow and the cons truction of framed bicategor ies in § 14, we will require the base ca tegory B to be car tesian or co cartesian monoidal. How ev er, we see from Ex amples 1 2.3 a nd 1 2.4 that this is not alwa ys the ca se, and the genera l notion o f monoidal ﬁbration is in teresting in its own right. Recall from P rop osition 3.8 that the 2-category of ﬁbrations Φ : A → B is equiv alent to the 2- category of pseudo functors B op → C at . W e int end to prov e an analogo us r esult for monoidal ﬁbrations ov er ca r tesian base categor ies, but ﬁrst we m ust deﬁne the 2-catego ry of mono idal ﬁbrations. Deﬁnition 12.5. Let Φ : A → B and Φ ′ : A ′ → B ′ be mono idal ﬁbrations. • An oplax monoidal m orphism of ﬁbrations is a co mmuting square (12.6) A ′ F 1 / / Φ ′   A Φ   B ′ F 0 / / B (that is, an oplax morphism of ﬁbra tions) together with the data of oplax monoidal functors on F 0 and F 1 such that the ident ity Φ F 1 = F 0 Φ ′ is a monoidal na tur a l transfor mation. • An oplax mor phism is s trong if F 0 and F 1 are str ong monoida l functors and F 1 preserves cartesian arr ows. • A lax morphism is a square (12.6) such that F 0 and F 1 are lax monoidal functors, F 1 preserves cartesian arr ows, and the equality Φ F 1 = F 0 Φ ′ is a monoidal tra nsformation. An y so rt of morphism is ov er B if F 0 is an identit y B ′ = B . If Φ and Φ ′ are braided (re s p. symmetric), then any sort of monoidal morphism is braided (resp. symmetric) if the functors F 0 and F 1 and the equality Φ F 1 = F 0 Φ ′ are br aided (resp. symmetric). If Φ and Φ ′ are monoidal biﬁbrations, then a lax monoidal morphism of biﬁbrations is just a lax mono idal morphism of ﬁbrations, while an oplax (resp. strong ) monoidal morphism of biﬁbrations is an oplax (resp. s tr ong) mo no idal morphism of ﬁbrations which a ls o preserves op c artesian arr ows. A m onoidal transformation of ﬁbrations , o r o f biﬁbra tio ns, is a transfor- mation of ﬁbrations whose co mpo nen ts are mono idal natura l tra nsformations. I f the tw o morphisms a re ov er B , then the transformation is ov er B if its downstairs comp onent is an identit y . Notations 12. 7. Let MF op ℓ (resp. MF , MF ℓ ) b e the 2-catego ry of monoidal ﬁ- brations, oplax (res p. stro ng, la x) mono idal mo r phisms o f ﬁbrations, and monoidal transformations o f ﬁbrations. W e write BM F and SM F for the braided and sym- metric versions. Let M F B denote the sub-2-catego ry of M F c o nsisting of ﬁbrations, morphisms, and transforma tions ov er B , a nd so on. Finally , we write M on C at fo r the 2 - category of monoidal ca tegories, strong monoidal functors , and mono idal nat- ural transforma tions, and s imilarly B r M on C at and S ym M on C at . 46 MICHAEL SHULMAN Theorem 12.8. If B is c artesian monoida l, t he e qu ivalenc e of Pr op osition 3.8 lifts to e quivalenc es of 2-c ate gories MF B ≃ [ B op , M on C at ] BMF B ≃ [ B op , B r M on C at ] SMF B ≃ [ B op , S ym M on C at ] . This means that, in particular, in a monoidal ﬁbra tion with c artesia n ba se, each ﬁb er is monoidal and ea ch transition functor f ∗ is s tr ong monoida l. W e call the monoidal structure on A the external monoida l structure , and the mono ida l structures on ﬁb ers the interna l monoidal structures. In many ca ses, the in ternal monoida l structures on the ﬁb ers ar e more familiar and predate the external mono idal structure. F or example, in Arr C , the ﬁber ov er B is the s lice category C /B , and the int ernal monoidal structure is the ﬁber pro duct ov er B . It is crucia l that B be c artesian monoidal for Theor e m 12 .8 to be true. F or example, the ﬁber of Mod ov er a nonco mm utative r ing R is the categor y Mo d R of R -mo dules, which do es not in gener al have an internal tensor pro duct. But if we r estrict to the monoida l ﬁbra tion CMod of mo dules ov er c ommu tative rings , the tensor pro duct in CRing b ecomes the copro duct, so we can apply the dual result, obtaining the familiar tensor pro duct on Mo d R in the commutativ e case. Notation 1 2.9. In a car tesian monoida l catego r y B , we write π B for any ma p which pro jects B out of a pro duct; thus we have π B : B → 1 , but also π B : A × B × C → A × C . W e a lso write ∆ B : B → B × B for the diago nal, and other maps constructed from it s uc h as A × B × C → A × B × B × C . The or em 12.8. Let Φ : A → B be a monoidal ﬁbra tio n with a chosen cleav a ge, and let B ∈ B . W e deﬁne a monoidal structure on the ﬁber A B as follows. The unit ob ject is I B = π ∗ B I , a nd the pr o duct is given b y (12.10) M ⊠ N = ∆ ∗ B ( M ⊗ N ) where M , N ∈ A B and ⊗ is the mo noidal structure o f A . T o obtain the assoc ia tivit y isomorphism, we tensor the car tes ia n arrow M ⊠ N − → M ⊗ N (whic h lives ov er ∆ B ) with Q to get an a rrow ( M ⊠ N ) ⊗ Q − → ( M ⊗ N ) ⊗ Q which is c a rtesian since ⊗ pr eserves cartesian arrows. W e then comp ose with an- other car tesian arrow over ∆ B to o btain a comp osite car tesian arrow ( M ⊠ N ) ⊠ Q − → ( M ⊗ N ) ⊗ Q . W e do the same on the other side to get a cartesia n arrow M ⊠ ( N ⊠ Q ) − → M ⊗ ( N ⊗ Q ) and the unique factorization of a : ( M ⊗ N ) ⊗ Q ∼ = M ⊗ ( N ⊗ Q ) through these cartesian a r rows gives an asso ciativity is omorphism ( M ⊠ N ) ⊠ Q ∼ = M ⊠ ( N ⊠ Q ) . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 47 for A B . The p ent ago n axio m follows from unique factoriza tion through car tesian arrows a nd the p entagon axiom for A . The unit cons tr aint s a nd ax io ms are a nal- ogous, using the fact that π B ∆ B = 1 B , as is the bra iding when Φ is braided or symmetric. Now consider a map f : A → B ; we show that f ∗ is s tr ong monoidal. W e hav e the co mp osite cartesian a rrows f ∗ M ⊠ f ∗ N − → f ∗ M ⊗ f ∗ N − → M ⊗ N and f ∗ ( M ⊠ N ) − → M ⊠ N − → M ⊗ N , bo th lying ov er ∆ B f = ( f × f )∆ A ; hence we obtain a canonica l isomorphism f ∗ M ⊠ f ∗ N ∼ = f ∗ ( M ⊠ N ) . The unit constraint is similar and, as b efor e, the coherence o f these co nstraints follows from the uniqueness of fa c to rization through c a rtesian arrows, as do es the fact that the isomor phisms ( f g ) ∗ ∼ = f ∗ g ∗ and (1 B ) ∗ ∼ = Id a re monoida l. Therefor e, we have constructed a pseudo functor B op → M on C at fro m a monoidal ﬁbration. It is straightforward to extend this construction to give 2-functors MF B − → [ B op , M on C a t ] BMF B − → [ B op , B r M on C at ] SMF B − → [ B op , S ym M on C at ] . Uniqueness of fac to rization again gives the coherence to show that the r e s ulting pseudonatural transfo r mations are p oint wise monoidal. Conv ersely , given a pseudofunctor B op → M on C at , we deﬁne a ﬁbr a tion over B in the usual wa y , and deﬁne a n external pro duct as follows: g iv en M , N ov er A, B resp ectively , let (12.11) M ⊗ N = π ∗ B M ⊠ π ∗ A N . The external unit is I 1 , the internal unit in the ﬁber ov er 1. F or an asso c ia tivit y isomorphism we use ( M ⊗ N ) ⊗ Q = π ∗ C ( π ∗ B M ⊠ π ∗ A N ) ⊠ π ∗ AB Q ∼ = ( π ∗ B C M ⊠ π ∗ AC N ) ⊠ π ∗ AB Q ∼ = π ∗ B C M ⊠ ( π ∗ AC N ⊠ π ∗ AB Q ) ∼ = π ∗ B C M ⊠ π ∗ A ( π ∗ C N ⊠ π ∗ B Q ) = M ⊗ ( N ⊗ Q ) using the mono idal constraint s for the strong monoidal functors π ∗ , the comp osition constraints for the pseudofunctor, and the asso cia tivit y for the int ernal pro ducts. It is s traightforw ard, if tedious, to chec k that this isomorphism s atisﬁes the p ent ago n axiom. Similarly , we hav e a unit cons traint M ⊗ I 1 = π ∗ 1 M ⊠ π ∗ A I 1 ∼ = M ⊠ I A ∼ = M which ca n b e ch eck ed to be co herent; thus A is monoidal, and Φ is strict monoida l b y deﬁnition. It is ob vious ho w to deﬁne a braiding in the bra ided or symmetric case 48 MICHAEL SHULMAN making A and Φ braided or symmetric. Fina lly , using the comp osition constraints and monoidal constraints, w e have: f ∗ M ⊗ g ∗ N = π ∗ f ∗ M ⊠ π ∗ g ∗ N ∼ = ( f × g ) ∗ π ∗ M ⊠ ( f × g ) ∗ π ∗ N ∼ = ( f × g ) ∗  π ∗ M ⊠ π ∗ N  = ( f × g ) ∗ ( M ⊗ N ) , which we can then use to verify that ⊗ preserves cartesia n arrows. Thu s we hav e constructed a monoidal ﬁbra tion of the desir ed type. It is s traightf orward to extend this to a 2-functor and verify that these co ns tructions a re inv erse equiv alences.  Remark 12.12. Under the ab ov e equiv alence, pseudofunctors which land in c arte- sian monoidal ca tegories cor resp ond to ﬁbratio ns where the total category A is cartesian monoida l. W e end this s e c tion by in tro ducing a few new ex amples of monoidal ﬁbra tions. Example 12.13 . Let C b e a category with ﬁnite limits a nd colimits, and ass ume that pull backs in C pr eserve ﬁnite colimits. (F or example, C could be loca lly cartesian clos ed.) Let Retr ( C ) b e the category of r etr actions in C . Tha t is, an ob ject of Retr( C ) is a pair of maps A s − → X r − → A such that rs = 1 A . This is also known as an ob ject X ‘par ametrized’ ov er A , in which case s is ca lled the ‘section’. W e deﬁne R etr C : Retr( C ) → C to take the a bove retraction to A . It is easy to chec k that pullback and pushout make Φ into a biﬁbra tion, which is a ∗ -biﬁbra tion if C is lo cally ca rtesian closed. The ﬁb er over B ∈ C is the catego ry C B of o b jects parametrized over B . It has ﬁnite pro ducts, given by pullback ov er B , but usua lly the relev an t monoidal structure is not the cartesia n pro duct but the ﬁb erwise smash pr o duct , deﬁned as the pushout X ⊔ B Y / /   X × B Y   B / / X ∧ B Y . The unit is B ⊔ B → B with section given by one o f the copr o jections. Under the assumption tha t pullbac ks preser ve ﬁnite colimits, thi s deﬁnes a symmetric monoidal structure on C B , all the functors f ∗ are strong symmetric monoida l, and the coherence isomorphisms a re also monoidal. Thus by Theor em 12.8, Ret r C is a symmetric monoida l ﬁbration, and it is easy to chec k that it is actually a mono ida l biﬁbration. The external monoidal structure on Retr( C ) is called the external smash pr o duct ⊼ . Example 12.14. Supp ose that C has ﬁnite limits and colimits, and not a ll pull- backs pre s erve ﬁnite co limits, but there is some full sub categor y B o f C suc h that pullbacks alo ng morphisms in B do preserve ﬁnite c o limits. Then we can rep eat the cons tr uction of Example 12 .13 using parametrized ob jects whose base o b ject s are r estricted to lie in B . This is w ha t is do ne in [MS06, § 2 .5], with C = K the category o f k -spaces and B = U the categ o ry o f compactly genera ted spac e s . By a slight abuse of notatio n, w e call the resulting mo noidal ∗ -biﬁbration Re tr T op , since we have only been pr even ted from considering all retractions in T op by p o int -set tech nicalities. The ob jects of Retr( T op ) are called ex-sp ac es . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 49 Example 12.15. F or ea ch space B ∈ U , a categ ory S B of o rthogonal s p ectra parametrized ov er B is deﬁned in [MS06, Ch. 11]. A map f : A → B of spa ces gives r ise to a string of adjoints f ! ⊣ f ∗ ⊣ f ∗ which are pseudofunctorial in f . Each category S B is c lo sed s ymmetric mo no idal under an internal smas h pro duct ∧ B , each functor f ∗ is closed sy mmetric monoidal, and so are the comp osition constraints. Thus, by Theo rem 12.8 , we obtain a symmetric monoidal ﬁbra tio n which we denote Sp . The external smash pro duct ⊼ is deﬁned in [MS0 6, 11 .4.10] just as we hav e do ne in (12.1 1). T o show that S p is in fact a mono idal ∗ -biﬁbra tion, o ne can chec k directly that ⊼ preserves o pca rtesian arrows. How ever, this will a ls o follow from Prop osition 13.3 0 below. Example 12.16. Let B = U as in E xample 12.15, but instead of S B we use its homotopy ca tegory Ho S B . It is pr oven in [MS06, 12.6.7 ] that f ! ⊣ f ∗ is a Quillen adjunction, for a suitable choice of mo del str ucture s on S B , hence it descends to an adjunction on homotopy ca tegories which is still pseudofunctorial; th us we obtain another functor Ho( Sp ) : A → B which is a biﬁbration. The externa l smash pro duct ⊼ is prov en to be a Q uillen left adjoint in [MS06, 12.6.6]; thus it descends to ho mo topy catego ries to make A sy mmetric monoida l. Since [MS06, 13 .7.2] sho ws that ⊼ preserves cartesian arr ows, Ho( Sp ) is a symmetric monoidal ﬁbration, a nd the same methods as in Example 12 .15 show that it is a monoidal biﬁbration. The derived functors f ∗ also hav e right adjoints, althoug h these ar e constructed in [MS06, 13.1.18 ] using Brown representabilit y rather than b y deriving the po in t-set level right a djoin ts; thus Ho( Sp ) is a monoidal ∗ -biﬁbration. 13. Closed monoidal fibra tions W e now consider tw o diﬀere nt notions of when a monoidal ﬁbra tion is ‘clo sed’. T o ﬁx terminolog y and no ta tion, we say a n ordinar y monoidal catego ry C with pro duct ⊠ is close d if the functors ( M ⊠ − ) and ( − ⊠ N ) hav e right adjoints ( −  M ) and ( N  − ), resp ectively , for all M , N . Of course, if C is symmetric, then P  M ∼ = M  P . If C a nd D are closed monoidal categor ies a nd f ∗ : C → D is a strong monoidal functor, then there are canonica l natural transfor mations f ∗ ( N  P ) − → f ∗ N  f ∗ P (13.1) f ∗ ( P  N ) − → f ∗ P  f ∗ N . (13.2) When these transfor mations ar e iso morphisms, we say that f ∗ is close d monoidal . Of cours e , in the symmetric case, (13.1) is an isomorphism if and only if (13.2) is. Deﬁnition 13.3. Let Φ : A → B b e a monoidal ﬁbratio n wher e B is car tesian monoidal (so t hat eac h ﬁbe r is a monoidal c a tegory). W e sa y Φ is interna lly closed if each ﬁb er A B is closed monoidal and each functor f ∗ is closed monoidal. How e ver, in any monoidal ﬁbration, w e can also ask whether the external pro duct ⊗ : A A × A B → A A ⊗ B has adjoints  ,  , with deﬁning iso morphisms A A ⊗ B ( M ⊗ N , P ) ∼ = A A ( M , N  P ) ∼ = A B ( N , P  M ) . 50 MICHAEL SHULMAN If so , then for any f : C → A and g : D → B there a re canonical transforma tions f ∗ ( N  P ) − → N  ( f ⊗ 1) ∗ P (13.4) g ∗ ( P  M ) − → (1 ⊗ g ) ∗ P  M (13.5) deﬁned ana logously to (13.1) a nd (13.2). F o r example, (13.4) is the adjunct of the comp osite f ∗ ( N  P ) ⊗ N ∼ = − → ( f ⊗ 1) ∗  ( N  P ) ⊗ N  − → ( f ⊗ 1) ∗ P. Deﬁnition 13.6. Let Φ : A → B b e a monoidal ﬁbration. W e s ay that Φ is externally closed if the a djoin ts  ,  exist and the maps (13 .4) a nd (13.5 ) are isomorphisms for all f , g . Examples 13.7. If C is lo ca lly cartesia n closed, then Arr C is in ternally and exter- nally closed. If C also has ﬁnite colimits, then Retr C is internally and externally closed. Example 13.8. The ﬁbration Sp of parametr ized orthogonal sp ectra ov er spaces is in ternally and externally closed; its int ernal homs are deﬁned in [MS06, 1 1.2.5] and the base change functors ar e shown to be clo sed in [MS06, 11.4.1 ]. W e po stpo ne consideration of Ho( Sp ) until later. Example 13. 9. The ﬁbration Mod : Mo d → Ring is externally closed. If N is a B -mo dule and P is an A ⊗ B -mo dule, the externa l-hom N  P is Hom B ( N , P ), which r e ta ins the A -mo dule structure from P . In this case, internal closure makes no sense b ecause the ﬁb ers are no t even monoidal. Example 13.10. The monoidal ∗ -biﬁbration CMod o f mo dules over c ommutative rings is also externally clo sed. In this cas e the ﬁb ers Mo d R are clo sed mono idal, but neither f ! nor f ∗ is a close d mono idal functor. Example 13.11. If D is a monoidal fr a med bica tegory , then the monoidal biﬁ- bration ( L, R ) is externally closed just when D is ex ternally close d in the s ense o f § 9. The fact that (13.4) a nd (13 .5 ) are isomorphisms in this case will follow from Prop osition 13 .30, below. Remark 13. 12. Contrary to what one might expe c t ( see, for example, [MS06, § 2.4]), exter na l closedness do es not imply that the monoidal categ ory A is closed in its own r ight. F or one thing, N  P is only deﬁned when N ∈ A B and P ∈ A A × B . But ev en when deﬁned, N  P is not an in ternal-hom for A : if M ∈ A C , then the morphisms M → N  P in A are bijective not to all mo rphisms M ⊗ N → P , but only those lying over f × 1 for some f : C → A . In the rest o f this s ection, we will prov e that under mild hypotheses, int ernal and externa l closedness are equiv alent, a nd g iv e useful dual versions o f the maps (13.1), (13.2), (13.4), and (13.5). W e b egin by compar ing the internal and external homs. Prop osition 13.1 3. Le t Φ b e either (i) a monoidal ∗ -ﬁbr ation in which B is c artesian monoidal , or (ii) a monoidal biﬁbr ation in which B is c o c artesian monoidal. Then the right adjoints  ,  exist if and only if Φ has close d ﬁb ers (i.e. the right adjoi nts  ,  ex ist ). FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 51 Pr o of. Suppos e ﬁr s t that B is cartesian and each ﬁb er is closed. Then for N ∈ A B and Q ∈ A A × B we deﬁne (13.14) N  Q = ( π B ) ∗  π ∗ A N  Q  . and similarly for  . Conv ersely , if  ,  exist, then for N , Q ∈ A A we deﬁne (13.15) N  Q = N  ((∆ A ) ∗ Q ) and s imilarly for  . It is easy to chec k, using the relationships b etw een ⊗ and ⊠ established in Theorem 12 .8, that these deﬁnitions suﬃce. In the co car tesian case, these relationships b ecome M ⊠ N ∼ = ∇ ! ( M ⊗ N ) M ⊗ N ∼ = η ! M ⊠ η ! N , where ∇ A : A ⊔ A → A denotes the ‘fold’ or co dia g onal, and η : ∅ → A is the unique map from the initial ob ject. Therefor e, the ana logous deﬁnitions: M  N = M  ( ∇ ∗ N ) M  N = η ∗  η ! M  N  . allow us to pass back and forth b et ween in ternal a nd external clo sedness.  This equiv alence is v a luable b ecause so metimes one of the tw o types of rig h t adjoints is muc h eas ier to constr uct than the other . Example 13.16. The homotopy-lev el ﬁbration Ho( Sp ) has the a djoin ts  ,  , since the adjunction b etw een ⊼ and  in Sp is Q uillen (see [MS06, 12.6.6 ]). This then implies, by P rop osition 13.13, that the ﬁbers of Ho( Sp ) are all closed monoidal. This would b e diﬃcult to prov e directly , since we hav e no ho mo to pical control over the internal monoidal structure s in Sp . In order to prov e a full eq uiv alence of lo cal and external closedness, we need to assume a n extra condition on the commutativit y of right and left adjoints. Suppo se that Φ : A → B is a ﬁbration a nd that the square (13.17) A h / / k   B g   C f / / D commut es in B . Thus we obtain a square (13.18) A A ∼ = A B h ∗ o o A C k ∗ O O A D f ∗ o o g ∗ O O which commutes up to canonical isomor phism. If Φ is a biﬁbra tion, there is a canonical natural transfor ma tion (13.19) k ! h ∗ − → f ∗ g ! , namely the ‘mate’ of the isomor phism (13.18 ). Explicitly , it is the co mpos ite k ! h ∗ η − → k ! h ∗ g ∗ g ! ∼ = k ! k ∗ f ∗ g ! ε − → f ∗ g ! . 52 MICHAEL SHULMAN Similarly , if Φ is a ∗ -ﬁbration, there is a canonical transfor ma tion (13.20) g ∗ f ∗ − → h ∗ k ∗ . Deﬁnition 13.21. If Φ is a biﬁbration (resp. a ∗ -ﬁbra tion), we say that the square (1 3.17) sa tisﬁes the Beck-Chev alle y condition if the natural transfor- mation (13.19) (resp. (13.2 0)) is an isomor phism. W e say that Φ is strongly BC if this condition is satisﬁed b y every pullback square, and w eakly BC if it is sa tis- ﬁed by every pullback square in which one of the legs ( f or g , ab ov e) is a pro duct pro jection. If instead all pushout squares satisfy the Beck-Chev alley condition, we say that Φ is s trongly co-BC . If Φ is a ∗ -biﬁbratio n, then (13.19) a nd (13 .20) are mates under the comp osite adjunctions f ∗ g ! ⊣ g ∗ f ∗ and k ! h ∗ ⊣ h ∗ k ∗ , so that one is an isomo rphism if and only if the other is. Thus, a ∗ - biﬁbra tion is strongly or weakly BC as a biﬁbra tion if and only if it is so as a ∗ -ﬁbr a tion. Examples 13.22 . The monoidal biﬁbrations Arr C and Ret r C are alwa ys strongly BC, as is the mo no idal ∗ -biﬁbration S p (see [MS06, 11 .4.8]). Example 13.23. The monoidal ∗ -biﬁbration CMod , whose bas e is co car tesian monoidal, is strong ly co-BC. Example 13. 24. The homotopy-lev el monoidal ∗ -biﬁbration Ho( Sp ) is only w eakly BC; it is proven in [MS06, 13.7.7 ] that the Beck-Chev alley condition is satisﬁed for pullback squares one of whose legs is a ﬁbration in the to po lo g ical sense (which in- cludes pro duct pro jections, of co urse). It do es not sa tisfy the B e c k-Chev alley con- dition for arbitra ry pullback squa res; a co ncrete coun terexample is given in [MS06, 0.0.1]. One intuitiv e r eason for this is that since Sp also incorp orates ‘homo topical’ information ab out the base spac e s, w e should only ex p ect the der ived o per ations to b e well-behav ed on homotopy pullback s quares. T his is o ur main mo tiv ation for in tro ducing the notion of ‘weakly BC’. Of cours e, the idea of co mmuting adjoints is older than the term ‘Beck-Chev alley condition’. In the theory of ﬁb ered categor ies, what we ca ll a ‘stro ngly BC biﬁbra- tion’ is referred to a s a ‘ﬁbration with indexed copro ducts’. W e will even tually use Beck-Chev alley conditions in our constr uctio n o f a framed bicategory fro m a monoidal ﬁbration (Theorem 14.4), but we mention them in this section for the purp o ses of the following res ult. Prop osition 13.25. L et Φ : A → B b e a monoida l ∗ -ﬁbr ation in which B is c artesian monoidal. Then (i) if Φ is int ernal ly close d and we akly BC, then it is external ly close d, and (ii) if Φ is ex ternal ly close d and st ro ngly BC, then it is int ernal ly close d. In p articular, a str ongly BC monoida l ∗ -ﬁbr ation is internal ly close d if and only if it is ex ternal ly close d. Sketch of Pr o of. Under the equiv alences (13.1 4) and (13.15), each o f the maps (13.1) and (13.4 ) is equa l to the comp osite of the o ther with a Be ck-Chev alley transforma- tion, and similarly for (13.2 ) and (13.5). It turns out that bo th of these transfor - mations co me from pullback squares; thus since π appea rs in (13.1 4) but ∆ a ppea rs in (13.15 ), the weak condition is go o d enough in o ne case but not the other.  FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 53 Now, if f ∗ is strong monoidal and has a left a djoin t f ! , there is a cano nica l map (13.26) f ! ( M ⊠ f ∗ N ) − → f ! M ⊠ N . When the monoidal categ ories in question a re closed, this is the mate of (13.1), so one is an isomo r phism if a nd only if the other is. In par ticular, if Φ is a monoidal biﬁbration with cartesian monoidal base and clo sed ﬁb ers, then Φ is internally closed if and o nly if the maps (13 .26), together with the analog ous maps (13.27) f ! ( f ∗ N ⊠ M ) − → N ⊠ f ! M , are all isomor phisms. This dual condition is so metimes easier to chec k. Example 13.28. T op olo gical a rguments inv olving excellent presp ectra are used in [MS06, 13.7.6] to show that the derived ma ps (13 .26) are isomorphisms, a nd therefore Ho( Sp ) is internally closed. Since it is weakly BC, we can then conclude, b y P rop osition 13.2 5(i), that it is externally closed as wel l. In a similar wa y , if Φ is a n y mo noidal biﬁbra tion with rig h t a djoin ts  ,  , then (13 .4 ) has a mate (13.29) ( f × 1 ) ! ( M ⊗ N ) − → f ! M ⊗ N which is an iso morphism if a nd only if (13.4 ) is. But (13.29) is an iso morphism just when − ⊗ N preser ves the o p car tesian ar row M → f ! M , so we hav e the following. Prop osition 13 . 30. L et Φ b e a monoidal ﬁ br ation which is also an opﬁbr ation and such t hat the right adjoints  ,  exist. Then ⊗ pr eserves op c artesian arr ows (that is, Φ is a monoidal biﬁbr ation) if and only if Φ is external ly close d. Example 13. 31. As remarked earlier , this implies that a monoidal framed bicate- gory D is externally clos e d in the sense of § 9 if a nd only if the monoidal biﬁbration ( L, R ) is externally clo sed in the sense of this s e c tio n. Example 13. 32. In the conv erse direction, Prop osition 13.30 can b e use d to show that Ho( Sp ) and S p are monoidal biﬁbrations, s ince we know that they are exter- nally close d. This could also b e shown directly . Corollary 13.33. L et Φ b e a str ongly BC monoidal ∗ -biﬁbr ation over a c artesian b ase and having close d ﬁb ers. Then Φ is int ernal ly and ext ernal ly close d. Pr o of. Since Φ is a ∗ -ﬁbration, b y Pr o po sition 13.13 it also has right adjoints  ,  . Then, since it is a monoida l biﬁbra tio n, it is exter nally closed by Prop osi- tion 13.30. But since it is strongly BC, Pro po s ition 13.25 (ii) then implies that it is also internally closed.  14. From f ibra tions to framed bica tegories W e now prove that any well-behaved monoidal biﬁbration gives rise to a framed bicategory . The reader may no t b e to o surprised that there is some r elationship, since many of our examples of monoida l biﬁbra tions lo ok v ery similar to our exam- ples of fra med bicategories . In this section we s ta te our r esults; the pro ofs will b e given in §§ 16– 17 after we co nsider an imp o rtant class of e x amples in § 15. T o motiv a te the pr ecise co nstruction, consider the relationship b etw een the framed bicategory S pan ( C ) and the monoida l biﬁbratio n A rr C : C ↓ → C . A ho r- izontal 1-cell M : A p → B in S pan ( C ) is a s pan A ← M → B , which can als o b e considered as an arrow M → A × B , and hence an o b ject of C ↓ ov er A × B . The 54 MICHAEL SHULMAN horizontal comp osition of M : A p → B and N : B p → C is given by pulling back along the maps to B , then remembering only the maps to A and C : M × B N y y r r r r r r % % L L L L L L M ~ ~ | | | | & & L L L L L L L N x x r r r r r r r A A A A A B C But this can also b e phrased in ter ms of the maps M → A × B and N → B × C b y tak ing the pro duct map M × N → A × B × B × C , pulling back along the diagona l ∆ B : M × B N / /   M × N   A × B × C / / A × B × B × C and then comp osing with the pr o jection π B : A × B × C → A × C . In terms of the monoidal biﬁbration A rr C , this can b e wr itten as (14.1) M × B N = ( π B ) ! ∆ ∗ B ( M × N ) . Similarly , the unit ob j ect U A in S pan ( C ) is the span A ← A → A , alter natively viewed as the diagona l ma p A → A × A . This can b e obtained (in a so mewha t per verse wa y) by pulling back the terminal ob ject 1 a long the map π A : A → 1 , then comp osing with the diag onal ∆ A : A → A × A . In the language o f Arr C , we hav e (14.2) U A = (∆ A ) ! π ∗ A 1 . W e now o bserve that the express ions (14.1 ) and (14.2) can easily be genera lized to any monoidal biﬁbration in which the base is cartesian monoidal, so that we have diagonals and pro jections. This may help to motiv ate the following result. Deﬁnition 14.3. W e say that a mono idal biﬁbration Φ : A → B is frameable if B is ca rtesian monoidal and Φ is either (i) strongly B C or (ii) w eakly BC and internally closed. Theorem 14. 4. L et Φ : A → B b e a fr ame able monoidal biﬁbr ation. Then t her e is a fr ame d bic ate gory F r (Φ) with a vertic al ly st rict involution, deﬁne d as fol lows. (i) F r (Φ) 0 = B . (ii) F r (Φ) 1 , L , and R ar e deﬁne d by the fol lo wing pul lb ack squar e. F r (Φ) 1 / / ( L,R )   A Φ   B × B × / / B . Thus the horizontal 1-c el ls A p → B ar e t he obje ct s of A over A × B , and the 2-c el ls M g = ⇒ f N ar e t he arr ows of A over f × g . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 55 (iii) The horizontal c omp osition of M : A p → B and N : B p → C is M ⊙ N = ( π B ) ! ∆ ∗ B ( M ⊗ N ) , and similarly for 2-c el ls. (iv) The horizontal unit of A is U A = (∆ A ) ! π ∗ A I . (v) The involution is the identity on obje cts and we have M op = s ∗ M , wher e s is the symmetry isomorphism. If Φ is extern al ly close d and a ∗ -biﬁbr ation, then F r (Φ) is close d in t he sense of § 5. If Φ is symmetric, then F r (Φ) is symmetric monoidal in the sense of § 9 and its involution is also symmetric monoidal. Examples 14.5. As alluded to ab ov e, if C has ﬁnite limits, the symmetric monoidal biﬁbration Arr C gives rise to the s y mmetric mono idal framed bicatego ry S pan ( C ), which is clos ed if C is lo cally car tesian closed. Example 14.6. If C ha s ﬁnite limits and co limits preserved by pullback, then the monoidal biﬁbration Re tr C gives rise to a symmetric mo noidal framed bicatego ry of p ar ametrize d obje cts , which we deno te E x ( C ). It is a lso clo sed if C is lo cally cartesian close d. Applied to the monoidal ∗ -biﬁbration Retr T op of e x -spaces from Example 1 2.14, we obtain a fra med bicategory E x ( T op ) o f par ametrized spaces which is bo th sym- metric monoidal and clo sed. Example 14.7. The monoidal ∗ -biﬁbra tion Sp of para metrized or tho g onal sp ectra gives r ise to a po in t-set level framed bicategor y of pa rametrized sp ectra, which w e may denote S p . It is sy mmetric monoidal and clos e d. Example 14.8. The ho motopy-category monoidal ∗ - biﬁbration Ho( Sp ), which is weakly BC and in ternally closed, g iv es r ise to a framed bicategory Ho( S p ). This is the sa me a s the framed bica tegory we hav e b een calling E x ever since § 2. Similarly , Ho( Retr T op ) giv es ris e to a homotopy-lev el framed bicategory Ho ( E x ( T op )) of parametrized spac e s . Both o f these framed bicateg ories are symmetric monoidal and closed. These ar e the only o nes of o ur examples which are weakly rather than strongly BC. The dua l version of Theo rem 14.4 says the following. Theorem 14.9. If Φ : A → B is a str ongly c o-BC monoidal biﬁbr ation wher e B is c o c artesian monoidal, then ther e is a fr ame d bic ate gory F r (Φ) with a vertic al ly strict involution, deﬁne d as in The or em 14.4, exc ept that c omp osition is given by M ⊙ N = η ∗ ∇ ! ( M ⊗ N ) and units ar e given by U A = ∇ ∗ η ! I . If Φ is external ly close d and is a ∗ -biﬁbr ation, then F r (Φ) is close d. If Φ is sym- metric, then F r (Φ) and its involution ar e symmetric monoidal. Example 14 .10. The mono idal ∗ -biﬁbration CM od gives rise to the fra med bicat- egory CM o d , which is symmetric monoidal and clo sed. How e ver, M o d cannot be co nstructed in this wa y , b ecause the category of noncommut ative rings is not co cartesian mono ida l. 56 MICHAEL SHULMAN Like o ur o ther wa ys of constructing framed bicategories, these results are 2- functorial. T o state this precis e ly , we need to deﬁne the rig ht 2- categories. In § 12 we deﬁned la x, stro ng , a nd o plax monoidal morphisms of biﬁbrations to be those that preserve b oth the monoidal structure and the ba se c hange in appropria te w ays. These morphisms, together with the monoida l transforma tions deﬁned there, give us 2-ca tegories M bi F , M bi F , and M bi F ℓ . Let MF fr denote the full sub-2-ca tegory of M bi F spanned by the frameable mo no idal biﬁbrations, and similarly for M F fr op ℓ and MF fr ℓ . Theorem 14.11 . The c onstruction of The or em 14.4 exten ds to a 2-functor F r : MF fr − → F r B i and similarly for oplax and lax morphisms. Example 14.12. The 2-functor S pan : C art → F r B i clearly factors through F r via a 2-functor Arr : C art → MF fr . Example 14.13. Let bi C art psc denote the 2-category of categories with ﬁnite limits and ﬁnite colimits preser ved by pullback, functors which preserve ﬁnite limits and colimits, and natural transformations. Then we ha ve a 2-functor Retr : bi C art psc → MF fr . Co mp osing this with F r deﬁnes a 2- functor E x : bi C art psc → F r B i . As with our o ther 2 -categor ies, we automa tically o btain notions of equiv alence and adjunction betw e e n monoidal biﬁbratio ns, and these are preser ved by 2-functors such a s F r . As usual, we c a n a lso characterize these mo re explicitly; we omit the pro of of the following. Prop osition 14.1 4. An adjunction F ⊣ G in M F fr ℓ b etwe en Φ : A → B and Φ ′ : A ′ → B ′ c onsists of the fol lowing pr op erties and stru ctur e. (i) F is a str ong monoidal morphism of biﬁbr ations and G is a lax monoidal morphism of biﬁbr ations; (ii) We have monoidal adjunctions F 0 : B ⇄ B ′ : G 0 and F 1 : A ⇄ A ′ : G 1 ; (iii) We have e qualities Φ ′ F 1 = F 0 Φ and Φ G 1 = G 0 Φ ′ which ar e monoidal tr ansformatio ns; and (iv) The adjunction F 1 ⊣ G 1 ‘lies over’ F 0 ⊣ G 0 in the sense t hat the fol lo wing squar e c ommutes: A ( M , G 1 N ) ∼ = / / Φ   A ′ ( F 1 M , N ) Φ ′   B ( A, G 0 B ) ∼ = / / B ′ ( F 0 A, B ) . Remark 14.15. In fact, these conditions are s omewhat redundant. F or example, left adjoints automa tically preser ve opcar tesian arrows and rig ht adjoints a utomat- ically preserve ca rtesian ones, and the r ight a djoin t of a strong monoidal functor is alwa ys lax monoidal. These ar e consequences of ‘do ctrina l adjunction’ (see Pr op o - sition 8.2 and [Ke l74]) and a prop erty ca lled ‘lax-idemp otence’ (see [KL9 7]). In man y cases, F 0 and G 0 are the identi ty , and the entire adjunction is ‘ov er B ’ in the sense int ro duced in Deﬁnition 12.5. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 57 Example 14.16. Let C hav e ﬁnite limits and ﬁnite colimits prese r ved by pull- back. Then there is a for getful lax mono idal mo r phism Retr C → Arr C lying ov er C . Cartes ia n arr ows are given by pullback in b oth cases, and hence are pr e s erved strongly , but op cartesian ar rows are given by pushout in Retr C and mere comp osi- tion in Arr C , hence ar e preserved only lax ly . The lax monoidal constraint is given b y the quotient map M × N → M ⊼ N . This for getful morphism ha s a left adjoint (14.17) ( − ) + : A rr C − → Retr C . which takes an ob ject X → A ov er A to the retraction A − → X + = X ⊔ A − → A. W e say that the functor ( − ) + adjoi ns a disjoint se ction . It is straig htforward to chec k tha t ( − ) + is a strong monoidal mor phism of biﬁbra tio ns and the pair satisﬁes Prop osition 14 .14. As C v aries, the forg etful morphisms deﬁne a 2-natural transforma tion fro m the 2- functor Retr to the 2- functor Arr , while the morphisms ( − ) + form an opla x natural tra nsformation Ar r → R etr . This r emains tr ue up on comp osing with the 2-functor F r , so we obtain framed adjunctions ( − ) + : S pan ( C ) ⇄ E x ( C ) : U where the r ight a djoin t is 2-na tural in C and the left adjoint is oplax na tural in C . Example 14.18. It is essentially shown in [MS06, Ch. 11 ] that we have an adjunc- tion (14.19) Σ ∞ : R etr T op ⇄ S p : Ω ∞ . of mo noidal biﬁbratio ns lying ov er U . Applying F r , w e obtain a fra med adjunction (14.20) Σ ∞ : E x ( T op ) ⇄ S p : Ω ∞ . The ﬁb er adjunctions are shown to be Quillen in [MS06, 12.6.2], so pass ing to homotopy ca tegories of horizontal 1-cells, we obtain a framed adjunction (14.21) Σ ∞ : Ho( E x ( T o p )) ⇄ E x : Ω ∞ . 15. Monoids in monoid al fibra tions and examples W e have seen that w e ll- behaved monoidal biﬁbra tions g ive rise to fra med bi- categories , and that framed bicategories with lo cal co equalizers admit the M o d construction, so it is natur a l to ask what conditions o n a monoidal ﬁbration ensure that the resulting fra med bicateg ory has lo cal co equa lizers. Deﬁnition 15.1. Let Φ : A → B b e a ﬁbratio n. • W e say that Φ has ﬁb erwise co e qualizers if each ﬁb er A B has co equal- izers a nd the functors f ∗ preserve coe q ualizers. Note that this latter con- dition is auto ma tic if Φ is a ∗ -ﬁbration, since then f ∗ is a left adjoint. • Similarly , we say that Φ has ﬁb erwise equalizers if each ﬁb er A B has equalizers and f ∗ preserves equalizer s, the seco nd condition b eing auto- matic if Φ is a biﬁbration. 58 MICHAEL SHULMAN • If Φ is a monoidal ﬁbr ation with ﬁb erwise co e q ualizers, we s ay these co- equalizers are preserved by ⊗ if the functors ⊗ : A A × A B − → A A ⊗ B all prese r ve c o equalizers in each v a riable. This is a utomatic if the r ight adjoints  ,  exist. Prop osition 15.2. L et Φ b e a fr ame able monoidal biﬁbr ation with ﬁb erwise c o- e qualizers pr eserve d by ⊗ . Then: (i) The fr ame d bic ate gory F r (Φ) has lo c al c o e qualizers, so t her e is a fr ame d bic ate gory M o d ( F r (Φ)) . (ii) If Φ is symmetric, then F r (Φ) is a monoidal fr ame d bic ate gory with lo c al c o e qualizers pr eserve d by ⊗ ; henc e M o d ( F r (Φ)) is also monoida l. (iii) If Φ is external ly close d, a ∗ -ﬁbr ation, and has ﬁb erwise e qu alizers, t hen F r (Φ) is close d and its hom-c ate gories have e qualizers; henc e M o d ( F r ( Φ)) is also close d. Pr o of. Since the hom-c a tegory F r (Φ)( A, B ) is just the ﬁber o f Φ over A × B , it has co eq ualizers. And since M ⊙ N = π ! ∆ ∗ ( M ⊗ N ), where ⊗ and ∆ ∗ preserve co equalizers by assumption a nd π ! preserves all colimits as it is a left adjoin t, these co equalizers are preserved by ⊙ ; thus F r (Φ) ha s lo cal co equalizer s. Item (i) then follows fro m Theorem 11.5. F or (ii) , we need to know what the exter nal mono idal structure of F r (Φ) is. When we prov e in Prop osition 17.1 that F r (Φ) is monoidal, we will deﬁne this external pr o duct to be es s en tially that of Φ, but with a slight twist. Namely , if Φ( M ) = A × B and Φ( N ) = C × D , then we hav e Φ( M ⊗ N ) = ( A × B ) × ( C × D ), whereas their pro duct in F r (Φ) should lie ov er ( A × C ) × ( B × D ). Thus we deﬁne the external pro duct M ⊗ ′ N o f F r (Φ) to b e the base change of M ⊗ N a lo ng the constraint iso morphism ( A × C ) × ( B × D ) ∼ = ( A × B ) × ( C × D ) . In particular, we hav e M ⊗ ′ N ∼ = M ⊗ N ; thus ⊗ ′ preserves co equalizers b ecause ⊗ do es. It then follows from Theor em 11.5 tha t M o d ( F r (Φ)) is monoidal. Finally , (iii) follows directly from Theo rems 14 .4 and 11.5.  Example 15. 3. If C has ﬁnite limits a nd co equalizers pr e s erved by pullback, then its self-indexing Arr C satisﬁes the conditions o f Prop osition 1 5.2, and we obtain the framed bica tegory M o d ( S pan ( C )) o f internal categories and distributors which we mentioned in Example 1 1.7. How e ver, we can also obtain enriche d categor ies and distributors, by sta r ting with a diﬀerent monoidal biﬁbration. Example 15.4. Given an y ordinary categor y V , let F am ( V ) b e the categ o ry of families of ob jects of V . That is, an ob ject of F am ( V ) is a set X tog ether with an X -indexed fa mily { A x } x ∈ X of ob jects in V . Then there is a ﬁbr ation Fam V : F am ( V ) → Set whic h is sometimes called the naive indexing of V ; its ﬁb er ov er a set X is the catego ry V X . The reader may ch eck the following. • If V is a monoida l categ ory , then Fam V is a monoidal ﬁbration; the external pro duct of { A x } x ∈ X and { B y } y ∈ Y is { A x ⊗ B y } ( x,y ) ∈ X × Y . The ﬁb erwise monoidal structure is the obvious one. If V is bra ided or symmetric, then so is Fa m V . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 59 • If V has small copro ducts (r e s p. pr o ducts), then F am V is a str ongly B C biﬁbration (resp. ∗ -ﬁbration). If V is als o monoidal and ⊗ preserves co- pro ducts, then Fam V is a monoida l biﬁbration. • If V has co equa lizers preserved by ⊗ , then Fam V has ﬁb erwise co eq ua lizers preserved b y ⊗ . If V has eq ualizers, then Fam V has ﬁb erwise equalizers. • If V is closed, then Fam V is in ternally closed. Thus, b y Corolla ry 13 .33, if V also has s mall pro ducts, then F am V is externally closed. In particular, when V is monoidal a nd has co limits preserved by ⊗ , Fam V is frame- able and F r ( Fam V ) has lo cal co equalizers, so we can deﬁne M o d ( F r ( Fam V )). It is easy to see that F r ( Fam V ) is equiv alent to the framed bicateg ory M at ( V ) deﬁned in Example 11.8, where we observed that M o d ( M at ( V )) ≃ D ist ( V ). This sug gests that we can view a fra meable closed s y mmetric monoidal ∗ -biﬁbra tion Φ with ﬁb erwis e equalizers and co equalizers as a ‘par ametrized’ version of a com- plete and c o complete closed symmetric monoidal category V , a nd th at w e ca n view the a sso ciated framed bicatego r y M o d ( F r (Φ)) as a parametrized version of D ist ( V ). The other e x ample of A rr C seems to b e a r this out. In fact, we can view the monoidal biﬁbrations Fam V and A rr C as living at op- po site ends of a co nt inuum. In Fa m V , the base category Set is fairly unin teresting, while all the interesting things happ en in the ﬁber s. On the other hand, in Arr C , the base catego ry C can b e interesting, but the ﬁb ers carry ess e ntially no new information, b eing determined by the ba se. Other monoida l biﬁbratio ns will fall somewhere in b e tw een the tw o, and monoids in the resulting framed bicategories can b e thought of as ‘catego r ies whic h a re b o th internal and enriched’. Example 15. 5. If C has ﬁnite limits and ﬁnite colimits preserved by pullback, then Prop osition 15 .2 applies to the monoidal biﬁbratio n Retr C , so that M o d ( E x ( C )) is a symmetric mono idal and clo sed framed bicatego r y . A mo no id in E x ( C ) may be thought of as a ‘p ointed internal catego ry’ in C . F o r ex ample, a monoid in E x ( Set ) is a small categor y enriched over the categor y Set ∗ of p oint ed s ets with smash pro duct, mea ning that each hom-set has a chosen basep oint a nd comp osi- tion pre serves basep oints. Similar ly , a monoid in E x ( T op ) is a ‘based top olog ic a l category’. If its space of ob jects is discrete, then it is just a small category enriched ov er based topo lo gical s paces, but in general it will b e ‘b oth internal and e nr ich ed’. Applying M o d to the dis jo in t-sections functor ( − ) + from E xample 14 .1 6, w e obtain a fra med functor M o d ( S pan ( C )) → M o d ( E x ( C )). Thus, a n y internal cate- gory can b e made into a p oint ed in ternal ca tegory by ‘adjoining disjo int basep oints to hom-ob jects’. Example 15.6. Prop osition 15.2 applies to the p oint-set ﬁbr ation Sp of paramet- rized sp ectra, so M o d ( S p ) is a symmetric mono idal and closed framed bicategory . A mono id in S p can b e viewed as a categor y ‘internal to s paces and enriched o ver sp e c tr a’; if its space of ob jects is discrete, then it is just a small categor y enriched ov er o rthogonal sp ectra . T o obtain other examples, we can apply M o d (Σ ∞ ) to a ny ba s ed top ological category a s in Example 15.5 , and thereby to any internal category in T o p with a disjoint section adjoined. Certain monoids in S p ar ising in this way from the topo logized fundamental group oid Π M or path-gr o upo id P M of a s pace M play an impo r tant r o le in [Pon07]. 60 MICHAEL SHULMAN A go o d case can b e made (see [MS0 6]) that a mo noid in S p is the right paramet- rized analogue of a classical ring sp ectrum, since when its space of ob ject s is a point, it reduces to an orthogona l ring s pectr um. The more na ive notion o f a monoid in Sp B with resp ect to the internal smash pro duct ∧ B is p o orly b ehav ed be c a use, un- like the situatio n for the exter na l smash pro duct ⊼ , we hav e no homo topical control ov er ∧ B . The ab ov e tw o e x amples give framed bicategor ies with inv olutions which a r e not vertically s tr ict, since the 2 -cell comp onents of ξ in F r (Φ) are not identities. Example 15 .7. Pro p os ition 15.2 do es not apply to the homotopy-level monoidal ﬁbration Ho( Sp ), since the stable homotopy catego ries o f parametrized sp e ctra do not in general admit co equalizer s . Rather than M o d ( E x ) = M o d (Ho( S p )), the correct thing to co nsider is ‘Ho( M o d ( S p ))’. Here the o b j ects are honest monoids in S p , whose multi plication is asso c iative and unital on the point-set level, just like in M od ( S p ), but we pass to homotopy categor ies of horizont al 1 - cells. W e then need to use a ‘homotopy tensor pro duct’ to deﬁne the derived horizontal compo sition, as was done in [Pon07]. W e hop e to investigate the homotopy theory of framed bicategories more fully in a later pa per . Example 15.8. W e can construct v a rious monoidal ﬁbra tio ns over any ‘ Set -like’ category E by mimicking the constructions of class ic a l Set -bas ed monoidal cate- gories. F or example, w e ha ve a ﬁbration Ab E ov er E whose ﬁbe r over B is the cate- gory Ab ( E /B ) of ab elian g roup ob jects in E /B . If E is lo ca lly cartesian clos ed, has ﬁnite colimits, and the for getful functors Ab ( E /B ) → E /B have left adjoints, then this is a strong ly BC ∗ -biﬁbration (see [Joh02 b, D5.3.2]). If E is co complete, the tensor pro duct of abelia n group ob jects can b e deﬁned int ernally and makes Ab E a monoidal biﬁbra tion; mono ids in the corr espo nding framed bicategory A b ( E ) give a notion of ‘ Ab -category in E ’. F or exa mple, if E is a catego r y of topo logical spaces, then a n y vector bundle ov er a space B gives an o b j ect of Ab ( E /B ). One might argue, analogously to Example 15.6 , that monoids in A b ( E ) give a g o o d notion o f a ‘bundle o f rings’. The theor y o f such r elative enriche d c ate gories app ears to b e fairly unexplored; the only references we know are [GG76] and [Prz0 7]. W e will e xplo re this theory more extensively in a later pap er; in man y ways, it is v ery similar to cla ssical enriched category theory . W e end with o ne further example o f this phenomenon. If V is an o rdinary monoidal categor y with copr o ducts pres erved by ⊗ , then any small unenriched ca tegory C gives rise to a ‘free’ V -categor y V [ C ] whose hom- ob jects are given b y co powers o f the unit o b ject: (15.9) V [ C ]( x, y ) = a C ( x ,y ) I . F or a monoida l ﬁbration Φ : A → B , the analogue of an unenriched catego ry is an in ternal categ ory in B . The following is an analogue of this cons truction in our general context. Prop osition 15.10. L et B have ﬁn ite limits and let Φ : A → B b e a str ongly BC monoidal biﬁbr ation. Then t her e is a c anonic al str ong monoidal morphism of biﬁbr ations (15.11) Arr B − → Φ FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 61 which t akes an obje ct X f − → B of B /B to the obje ct f ! π ∗ X I of A B . Conse quently, ther e is a c anonic al fr ame d functor (15.12) S pan ( B ) − → F r (Φ) and thus, if Φ h as ﬁb erwise c o e qualizers pr eserve d by ⊗ and B has c o e qualizers pr eserve d by pul lb ack, a fr ame d functor (15.13) M o d ( S pan ( B )) − → M o d ( F r (Φ)) . Example 15.14 . When Φ = Fa m V for a n ordinary monoidal category V , the morphism Ar r B → F am V takes a set A to ` A I . Thus the induced framed functor D ist ( Set ) → D is t ( V ) is exa ctly the ‘free V - c a tegory’ op eration (15.9) describ ed ab ov e. Examples 15.15. More int erestingly , when Φ is Re tr C , the mor phism Arr C → Retr C is the disjoint-section op eration describ ed in E x ample 14.16. And w hen Φ is the mono idal ﬁbration Sp of orthogo na l sp ectra, the morphism Arr U → Sp ﬁrst adjoins a disjoint section, then applies the pa rametrized susp ension- spec tr um functor Σ ∞ from Example 1 4.18. Therefor e, if C is an internal ca tegory in top o- logical spaces , the ‘top olo g ically int ernal and sp ectrally enr ich ed ca teg ory’ Σ ∞ C + considered in Example 15.6 is in fact ‘freely gener ated by C ’ in this canonical wa y . 16. Tw o technical lemmas In preparation for our pro of o f Theor em 14.4 in § 17, in this section we refor mulate the Beck-Chev alley co ndition and internal closedness in terms of car tesian arrows. Lemma 16.1. L et Φ : A → B b e a biﬁbr ation. Th en a c ommuting squar e (16.2) A h / / k   B g   C f / / D in B satisﬁes the Be ck-Chevalley c ondition if and only if for every M ∈ A B , the squar e (16.2) lifts to some c ommutative squar e (16.3) M ′ φ / / χ   M ξ   M ′′′ ψ / / M ′′ in A in which φ and ψ ar e c artesian and χ and ξ ar e op c artesian. Note that given φ, χ, ξ lifting h, k , g with χ op cartesian, there is exactly o ne ψ lifting f which makes (16.3) commute. Thus the condition can also b e stated as “Given any ca rtesian φ and op cartesian χ, ξ , the unique mor phism ψ ov e r f making (16.3) commute is cartesia n”. Pr o of. Cho ose a cleav age. Then by the universal pro per ties of cartesian and op- cartesian arrows, there is a unique do tted arr ow living ov er 1 C which makes the 62 MICHAEL SHULMAN following p entagon commute: h ∗ M cart / / op cart y y s s s s s s s s s s M op cart   k ! h ∗ M % % g ! M f ∗ g ! M . cart : : v v v v v v v v v W e cla im that in fact this dotted a rrow is the comp onent of the Beck-Chev alley natural transfor mation (13.19) at M . T o see this, we ﬁll out the dia gram as follows. h ∗ M cart / / op cart   h ∗ η & & M M M M M M op cart   η z z u u u u u h ∗ g ∗ g ! M cart / / op cart   ∼ = ' ' O O O O O O g ∗ g ! M cart   . . . . . . . . . . . . . . . . . . . . . . . k ∗ f ∗ g ! M cart   op cart   k ! h ∗ M k ! h ∗ η / / _ _ _ . . k ! h ∗ g ∗ g ! M ∼ = / / _ _ _ k ! k ∗ f ∗ g ! M εf ∗ g ! & & M M M M M f ∗ g ! M cart / / g ! M . Here the dashed arr ows ar e unique factorizations through (op)ca r tesian arr ows. This ex hibits the dotted arr ow as the comp osite of a unit, canonical isomorphism, and counit, which is the deﬁnition of the tr ansformation (13.19). This proves our claim. Therefore, if (13.1 9) is an isomorphism, the co mpos ite k ! h ∗ M ∼ = f ∗ g ! M → g ! M is cartesian, and hence we hav e a c o mm uting square of cartesia n and o pca rtesian arrows as desired. Conv ersely , if we hav e s uc h a commuting sq ua re, then clearly for some choice of cleav age, the dotted arr ow is the ident ity; hence it is a n isomor phism for all cleav ages.  As alwa ys, it simpliﬁes our life gr eatly to work with ca rtesian arrows rather than chosen cleav age s . F or example, we can now easily show the following. Corollary 16.4. If D is a monoidal fr ame d bic ate gory, t hen the squar e (16.5) A ⊗ C f ⊗ 1 / / 1 ⊗ g   B ⊗ C 1 ⊗ g   A ⊗ D f ⊗ 1 / / B ⊗ D in D 0 satisﬁes the Be ck-Cheval ley c onditio n for the biﬁbr ations L and R . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 63 Pr o of. Let M : B ⊗ C p → E b e a horizontal 1-cell and consider the following diagra m in R − 1 ( E ). ( f B ⊗ C ) ⊙ M op cart   cart / / ( B ⊗ C ) ⊙ M op cart   ∼ = / / M ( f B ⊗ D g ) ⊙ M cart / / ( B ⊗ D g ) ⊙ M The arrows lab eled ca r tesian or o pc a rtesian are obtained from the car tesian f B → B and op c a rtesian C → D g via ⊗ and ⊙ . The square co mm utes by functoria lit y of ⊗ , so the result follows from L e mma 16.1.  Corollary 16.6. If D is a monoidal fr ame d bic ate gory in which D 0 is c artesian monoidal, then t he monoidal biﬁbr ation ( L, R ) is we akly BC. Pr o of. T aking D = 1 in the s q uare (16 .5) s hows immediately that L and R are weakly BC; an analog ous square in D 0 × D 0 applies to ( L, R ).  T o deal with the ‘weakly BC and internally closed’ ca se of Theorem 1 4.4, we als o need a statement ab out ca rtesian arr ows that makes use of the closed s tr ucture. Recall that if f ∗ is strong mono idal and ha s a left adjoint f ! , then f ∗ is closed monoidal if and o nly if the dual maps f ! ( M ⊠ f ∗ N ) − → f ! M ⊠ N (16.7) f ! ( f ∗ N ⊠ M ) − → N ⊠ f ! M (16 .8) are isomorphisms. This la tter co ndition is amenable to restatement in ﬁbra tional terms, using the characteriza tio n of ⊠ in ter ms of ⊗ a nd ∆ ∗ . Lemma 16.9. L et Φ : E → B b e an internal ly close d monoidal biﬁbr ation, wher e B is c artesian monoidal . Then for any f : A → B in B , and any M , N in E with Φ( M ) = A, Φ( N ) = B , the s qu ar e (16.10) A f / / ∆ A   B ∆ B   A × A f × f / / B × B in B lifts to a squar e (16.11) ∆ ∗ A ( M ⊗ f ∗ N ) op cart / / cart   ∆ ∗ B ( f ! M ⊗ N ) cart   M ⊗ f ∗ N op cart ⊗ cart / / f ! M ⊗ N in A , and dual ly. Pr o of. A cleav a ge gives us an op cartesia n M → f ! M , a cartesia n f ∗ N → N , and cartesian arrows on the left a nd righ t, inducing a unique a r row on the top which lifts f . W e then obser ve that ∆ ∗ A ( M ⊗ f ∗ N ) ∼ = M ⊠ f ∗ N and ∆ ∗ B ( f ! M ⊗ N ) ∼ = f ! M ⊠ N , so factor ing the top arrow thr o ugh an o pca rtesian a r row gives us precisely (16.7 ). Since Φ is internally close d, this is an isomorphism, s o the top ar r ow must b e op cartesian.  64 MICHAEL SHULMAN 17. Pr oofs of Theorems 14.4 and 14.11 This section is devoted to the pro o fs of Theorems 14 .4 and 14.1 1 (and the dual version Theorem 14 .9). T o make the pro ofs more mana geable, we split them up in to se veral prop os itio ns . Prop osition 17 .1. L et Φ : A → B b e a str ongly BC monoidal biﬁbr ation, wher e B is c artesian monoidal. Then ther e is a fr ame d bic ate gory F r (Φ) deﬁne d as fol lows. (i) F r (Φ) 0 = B . (ii) F r (Φ) 1 , L , and R ar e deﬁne d by the fol lo wing pul lb ack squar e. F r (Φ) 1 / / ( L,R )   A Φ   B × B × / / B . Thus the horizontal 1-c el ls A p → B ar e t he obje ct s of A over A × B , and the 2-c el ls M g = ⇒ f N ar e t he arr ows of A over f × g . (iii) The horizontal c omp osition of M : A p → B and N : B p → C is M ⊙ N = ( π B ) ! ∆ ∗ B ( M ⊗ N ) , and similarly for 2-c el ls. (iv) The horizontal unit of A is U A = (∆ A ) ! π ∗ A I . If Φ is symmetric, then F r (Φ) is symmetric monoidal. Pr o of. Throughout the pro of, we will write P = F r (Φ) for brevity . Since we intend to construct an algebr a ic s tructure (a framed bicatego ry), we cho ose once and for all a cleav ag e (and op cleav age) on Φ , and reserve the notations f ∗ , f ! and so on for the functors given b y this cleav ag e. Howev er, we will still use (op)ca r tesian arrows which a r e not in this cleav age in order to c o nstruct the constraints and coherence. W e have the structure and op erations , a t leas t, of a double categor y essentially already deﬁned, except for the functoriality of ⊙ and U . It is e a sy to see that ⊙ is a functor, since ⊗ , ∆ ∗ , and π ! are functors. The functoriality of U is similar, but per ha ps not as obvious since it is a functor P 0 → P 1 . Its action on a n arrow f : A → B is g iven by the unique facto rization U f as follows. I π ∗ A I cart o o op cart / /      U A U f      π ∗ B cart ` ` @ @ @ @ @ @ @ @ op cart / / U B Thu s, to show that P is a double category , it suﬃces to construct coherent as so cia- tivit y and unit constraints. The following ar gumen ts should re mind the reader of the pro o f of Theor em 12.8, a lthough they are more complicated. Note ﬁrst that for horizontal 1-cells M : A p → B and N : B p → C in P , we hav e Φ( M ) = A × B and Φ( N ) = B × C , and the chosen cleav age gives us cano nical morphisms (17.2) M ⊗ N cart ← − ∆ ∗ B ( M ⊗ N ) op cart − → ( π B ) ! ∆ ∗ B ( M ⊗ N ) = M ⊙ N . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 65 W e b egin with the asso ciativity isomor phism. So supp ose in addition to M , N w e hav e Q : C p → D . Then since ⊗ pr eserves (op)car tesian ar rows, w e can c o nstruct the following diagram. ( M ⊙ N ) ⊗ Q 8 8 opcart p p p p p p p p p p g g cart O O O O O O O O O O ( M ⊗ N ) ⊗ Q ∆ ∗ B ( M ⊗ N ) ⊗ Q cart o o ∆ ∗ C ` ( M ⊙ N ) ⊗ Q ´ opcart / / ( M ⊙ N ) ⊙ Q. ∆ ∗ BC ` ( M ⊗ N ) ⊗ Q ´ cart f f N N N N N cart f f opcart 7 7 Here the so lid a rrows are part of the chosen cleav age. The da shed a rrow is a unique factorization, which is ca r tesian by Pro po sition 3.4 (ii). The do tted arrow, also a unique factoriza tion, is op cartesia n by the Beck-Chev alley condition (Lemma 16.1), beca use the square in question lifts the pullback square A × B × C × D ∆ C / / π B   A × B × C × C × D π B   A × C × D ∆ C / / A × C × C × D . Comp osing the t wo op cartesian ar rows on the right, w e obtain a spa n (17.3) ( M ⊗ N ) ⊗ Q ∆ ∗ B C  ( M ⊗ N ) ⊗ Q  cart o o op cart / / ( M ⊙ N ) ⊙ Q. W e per form a n analogous construction for M ⊙ ( N ⊙ Q ), then factor the ass o ciativity isomorphism for ⊗ through these cartesian and op cartesia n arr ows to o bta in an asso ciativity is omorphism for ⊙ : (17.4) ( M ⊗ N ) ⊗ Q ∼ =   ∆ ∗ B C  ( M ⊗ N ) ⊗ Q  cart o o op cart / / ∼ =   ( M ⊙ N ) ⊙ Q ∼ =   M ⊗ ( N ⊗ Q ) ∆ ∗ B C  M ⊗ ( N ⊗ Q )  cart o o op cart / / M ⊙ ( N ⊙ Q ) . This isomorphism is na tural because it is deﬁned b y unique factorization. The pro of that it satisﬁes the p entagon axiom is s imila r to its construction: we tensor (17 .3) with R : D p → E , then use the Beck-Chev alley c o ndition a g ain for the squar e AB C D E / /   AB C D D E   ADE / / ADD E (where we omit the symbol × ) to obtain a spa n (17.5) (( M ⊗ N ) ⊗ Q ) ⊗ R ∆ ∗ BC D ` (( M ⊗ N ) ⊗ Q ) ⊗ R ´ cart o o opcart / / (( M ⊙ N ) ⊙ Q ) ⊙ R. By uniqueness of factorizations, the isomor phism (( M ⊙ N ) ⊙ Q ) ⊙ R ∼ = ( M ⊙ ( N ⊙ Q )) ⊙ R, 66 MICHAEL SHULMAN obtained by applying the functor − ⊙ R to (17.4 ), is the s a me as the isomor phism obtained by factoring the iso morphism (( M ⊗ N ) ⊗ Q ) ⊗ R ∼ = ( M ⊗ ( N ⊗ Q )) ⊗ R through the spa n (1 7 .5). Therefore, by insp ecting the following diag ram and us- ing unique factor ization again, we s ee that the pentagon axiom for ⊗ implies the pentagon axiom for ⊙ . (( M ⊙ N ) ⊙ Q ) ⊙ R ( M ⊙ N ) ⊙ ( Q ⊙ R ) ∼ =   O O   ∼ = Z Z Z Z Z Z Z Z Z Z Z - - Z Z Z Z Z Z Z Z Z Z Z ∼ = d d d d d d d d d d d q q d d d d d d d d d d d ( M ⊙ ( N ⊙ Q )) ⊙ R ∼ =   $ $ I I I I I I I I I I I _ _ ? ? ? ? ∼ =   (( M ⊗ N ) ⊗ Q ) ⊗ R ∼ = u u u u u z z u u u u ∼ = I I I I I $ $ I I I I z z u u u u u u u u u u u ? ?     ∼ =   ( M ⊗ N ) ⊗ ( Q ⊗ R ) ∼ =   ( M ⊗ ( N ⊗ Q )) ⊗ R ∼ =   M ⊗ ( N ⊗ ( Q ⊗ R )) M ⊗ (( N ⊗ Q ) ⊗ R ) ∼ = o o 5 5 k k k k k k k k       i i S S S S S S S S   ? ? ? ? ∼ = o o M ⊙ ( N ⊙ ( Q ⊙ R )) M ⊙ (( N ⊙ Q ) ⊙ R ) ∼ = o o ∼ = c c c c c c q q c c c c c c ∼ = [ [ [ [ [ [ - - [ [ [ [ [ [ Now we consider the left unit transformation. Let M : A p → B and recall that U A = (∆ A ) ! π ∗ A I , so that we hav e I π ∗ A I cart o o op cart / / U A . T ensor ing this with M a nd a dding the arrows fr o m (17.2) for U A ⊙ M , we have (17.6) U A ⊗ M I ⊗ M π ∗ A I ⊗ M cart o o opcart 8 8 r r r r r r r r r r ∆ ∗ A ( U A ⊗ M ) cart g g O O O O O O O O O O O opcart / / U A ⊙ M . M cart f f M M M M M M l ∼ = c c opcart 7 7 The so lid arrows marked car tesian or op cartesian are part of the chosen cleav age. The other so lid a rrow is the le ft unit co nstraint for ⊗ , which is an isomorphism, hence also cartesian. The da s hed arrow is cartesian by Pr op osition 3.4 (ii), and the dotted arrow is op car tesian b y the Beck-Chev alley condition for the pullbac k square (17.7) A × B ∆ / / ∆   A × A × B ∆ × 1   A × A × B 1 × ∆ / / A × A × A × B . Since the comp osite o f the tw o op cartesia n a r rows on the r ig h t is op cartesia n a nd lies ov er 1 A × B , it is an isomorphism M ∼ = U A ⊙ M FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 67 which we take a s the left unit iso morphism for ⊙ . Its natura lity follows, as b efore, from unique factoriza tion. The rig h t unit isomorphism is analo g ous. W e now show the unit axiom. W e tenso r the diagra m I ⊗ M M ∼ = o o ∼ = / / U A ⊙ M with N and co mpo se with the deﬁning ca rtesian and o pc a rtesian arrows for ⊙ to obtain the following diagram. (17.8) N ⊗ ( U A ⊙ M ) N ⊗ ( I ⊗ M ) N ⊗ M ∼ = o o ∼ = ; ; x x x x x x x x ∆ ∗ A ( N ⊗ ( U A ⊙ M )) cart g g P P P P P P P P P P P opcart / / N ⊙ ( U A ⊙ M ) . ∆ ∗ A ( N ⊗ ( I ⊗ M )) cart c c F F F F ∼ = opcart 7 7 W e do the same for ( N ⊙ U A ) ⊙ M . By universal factorization, the tw o unit isomorphisms N ⊙ ( U A ⊙ M ) ∼ = N ⊙ M ( N ⊙ U A ) ⊙ M ∼ = N ⊙ M are given b y factor ization through these (op)cartesian ar r ows, as is the asso c ia tivit y isomorphism ( N ⊙ U A ) ⊙ M ∼ = N ⊙ ( U A ⊙ M ) . Thu s, a s for the p entagon, the unit ax iom for ⊗ implies the unit axio m for ⊙ . This shows that P is a do uble categ ory . Since the pullbac k of a biﬁbr a tion is a biﬁbration, ( L, R ) is a biﬁbratio n. Thus, by Theorem 4.1, P is a framed bicatego ry . W e now assume that Φ is s ymmetric and show that P is a symmetric monoida l framed bicatego ry . Since P 0 = B , it is alr e ady (cartesian) s ymmetric monoidal. The monoida l structure of P 1 is almo s t the same as that of A , but with a slight t wist. If M : A p → B and N : C p → D , s o that Φ( M ) = A × B and Φ( N ) = C × D , then we hav e Φ( M ⊗ N ) = ( A × B ) × ( C × D ) whereas the pr o duct of M a nd N in P should b e an ob ject of A lying over ( A × C ) × ( B × D ). But the chosen cleav age gives us a cartesian a r row ending a t M ⊗ N lying over the unique constr aint ( A × C ) × ( B × D ) ∼ = ( A × B ) × ( C × D ) , and we call its doma in M ⊗ ′ N . Since cartesian arr ows ov er isomorphisms are isomorphisms, we hav e M ⊗ ′ N ∼ = M ⊗ N . Similarly , the unit for A should be U 1 = (∆ 1 ) ! ( π 1 ) ∗ I , and since π 1 = 1 1 and ∆ 1 is the unique isomorphism 1 ∼ = 1 × 1 we hav e U 1 ∼ = I ; we deﬁne I ′ = U 1 . The co ns traints and coherence axioms for ⊗ and I pa ss acr oss these isomo r phisms to make P 1 a symmetric monoidal categ ory under ⊗ ′ , with ( L, R ) a strict symmetric monoidal functor. Thu s, to make P a symmetric mo noidal framed bicateg ory , it rema ins to con- struct co herent in terchange and unit isomo rphisms and show that the monoidal asso ciativity and unit constraints are framed tra nsformations. Our by-now familiar 68 MICHAEL SHULMAN pro cedure gives the fo llowing diagr am for the interc hange isomorphism. ( M ⊗ ′ P ) ⊗ ( N ⊗ ′ Q ) ∼ =   ∆ ∗  ( M ⊗ ′ P ) ⊗ ( N ⊗ ′ Q )  cart o o op cart / / ∼ =   ( M ⊗ ′ P ) ⊙ ( N ⊗ ′ Q ) ∼ =   ( M ⊗ N ) ⊗ ′ ( P ⊗ Q ) ∆ ∗ ( M ⊗ N ) ⊗ ′ ∆ ∗ ( P ⊗ Q ) cart o o op cart / / ( M ⊙ N ) ⊗ ( P ⊙ Q ) . F or the the unit isomor phism w e hav e I ∼ =   π ∗ A × B I cart o o op cart / / ∼ =   U A × B ∼ =   I ⊗ I π ∗ A I ⊗ π ∗ B I cart o o op cart / / U A ⊗ ′ U B . As b efore , b y facto ring known co mm uting diag rams thro ugh cartesia n and op carte- sian ar rows, we can show that these cons tr aint s are framed transforma tions and satisfy the monoidal coherence axio ms.  Corollary 17. 9. If Φ : A → B is a str ongly c o-BC monoidal biﬁbr ation wh er e B is c o c artesian monoidal, t hen ther e is a fr ame d bic ate gory F r (Φ) deﬁne d as in Pr op osition 17.1, exc ept t hat c omp osition is given by M ⊙ N = η ∗ ∇ ! ( M ⊗ N ) , units ar e given by U A = ∇ ∗ η ! I , and similarly for t he other data. If Φ is symmetric, then F r (Φ) is symmetric monoidal. Pr o of. Simply apply Pr op osition 17.1 to the strongly BC mo no idal biﬁbration Φ op : A op → B op , since B op is c a rtesian monoidal.  W e no w consider the case when Φ is only weakly BC. Most of the pullback squares for which we use d the Beck-Chev alley condition in P rop osition 17.1 had one of their legs a pro duct pro jection, so those parts of the pro o f car ry ov er with no problem. How ever, there was one which in volv ed only diagonal maps, and this is the problem that Lemma 16.9 w as designed to so lve. This is essent ially the same method as that used in [MS06, Ch. 1 7] for the c a se of Ho( Sp ). Prop osition 17.10. L et Φ : A → B b e a we akly BC and internal ly clo se d monoidal biﬁbr ation, wher e B is c artesian monoidal. Then t he same deﬁnitions as in Pr op o- sition 17.1 give a fr ame d bic ate gory, which is symmetric monoidal if Φ is. Pr o of. There is only o ne place in the pro of of Theorem 1 4.4 wher e we used a Bec k- Chev alley pr op e r t y for a ‘bad’ square: in proving that the unit transformation is an isomorphism, using the squar e (17 .7). In this case, the dotted ar r ow in (17.6) which we wan t to b e op cartesian is deﬁned by unique factorization from a sq uare of the form (17.11) M cart   / / ∆ ∗ A ( U A ⊗ M ) cart   π ∗ A I ⊗ M op cart ⊗ 1 / / U A ⊗ M . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 69 This is almost of the form (16 .11), but not quite, s ince it lies ov er the square AB ∆ A B / / ∆ A B   AAB A ∆ A B   AAB ∆ A AB / / AAAB which is not of the fo r m (16.10 ). But we can decomp ose it into another pa ir of squares: AB ∆ A B / / ∆ AB   AAB ∆ AAB   AB AB π B   ∆ A B ∆ A B / / AAB AAB π A π B   AAB ∆ A AB / / AAAB Here the top square is of the form (16.10), where f = ∆ A B . If we then con- struct (17.1 1 ) b y lifting in stag e s , w e obta in (17.12) M cart   / / ∆ ∗ A ( U A ⊗ M ) cart   π ∗ AB I ⊗ M / / _ _ _ cart   π ∗ B U A ⊗ π ∗ A M cart   π ∗ A I ⊗ M op cart ⊗ 1 / / U A ⊗ M . where the outer re c ta ngle is the same a s (17.11 ). W e ca n then obtain the b o ttom square as the pro duct of a s quare π ∗ AB I / / _ _ _ cart   π ∗ U A cart   π ∗ A I op cart / / U A , where the dashed arrow is opcar tesian by the Beck-Chev alley condition, and a square M / / _ _ _ π ∗ A M cart   M M . where the dashed arrow is car tes ia n by Prop osition 3.4(ii). Thus the da s hed arrow in (17 .1 2) is of the fo r m op car t ⊗ cart, so by Lemma 16.9, the dotted arrow is op cartesian as desired. Since the unit transforma tion that we hav e just shown to b e an is omorphism is the sa me as the transfo r mation deﬁned in the pr o of of Theorem 14.4 , the same pro o f of the coherence axioms a pplies.  70 MICHAEL SHULMAN Note that Corollary 1 6.6 shows that any monoidal fra med bicatego ry with carte- sian base is weakly BC, so b eing weakly BC is a n e c essary condition for the co n- struction of Theorem 14.4 to give a framed bicateg ory . W e do not know whether being weakly BC is s uﬃcien t for fra meabilit y without closedness, but we susp ect not. Prop osition 17.13. If Φ is a fr ame able monoidal biﬁbr ation, t hen F r (Φ) has a vertic al ly strict involution given by t he identity on obje cts and M op = s ∗ M on 1-c el ls. If Φ is symmetric, this involution is symmetric monoidal. Pr o of. Left to the r eader.  Prop osition 17.14 . L et Φ : A → B b e a fr ame able monoidal ∗ -biﬁbr ation which is extern al ly close d. Th en the r esulting fr ame d bic ate gory F r (Φ) is close d. Pr o of. Deﬁne N  P = N  ∆ ∗ π ∗ P . W riting D for the hor izont al bicategor y of F r (Φ), we hav e D ( M ⊙ N , P ) = D ( π ! ∆ ∗ ( M ⊗ N ) , P ) ∼ = D (∆ ∗ ( M ⊗ N ) , π ∗ P ) ∼ = D ( M ⊗ N , ∆ ∗ π ∗ P ) ∼ = D ( M , N  ∆ ∗ π ∗ P ) = D ( M , N  P ) . The construction of  is similar.  Prop osition 17.15. L et Φ b e an external ly close d and str ongly BC monoidal ∗ - biﬁbr ation in which B is c o c artesian monoidal. Then the r esulting fr ame d bic ate gory F r (Φ) is close d. Pr o of. Deﬁne N  P = N  ∇ ∗ η ∗ P . Again writing D f or the horizontal bicategory of F r (Φ), we have D ( M ⊙ N , P ) = D ( η ∗ ∇ ! ( M ⊗ N ) , P ) ∼ = D ( ∇ ! ( M ⊗ N ) , η ∗ P ) ∼ = D ( M ⊗ N , ∇ ∗ η ∗ P ) ∼ = D ( M , N  ∇ ∗ η ∗ P ) = D ( M , N  P ) . The construction of  is similar.  Finally , we sketc h the pro o f of Theor em 14.1 1 . Prop osition 17.1 6. The c onstruction of The or em 14.4 extends t o a 2-functor F r : MF fr − → F r B i and similarly for oplax and lax morphisms. Sketch of Pr o of. Let F : Φ → Ψ b e a morphism in the appropriate domain category . W e deﬁne F r ( F ) to b e F 0 on vertical ca tegories. If M : A p → B , so that Φ( M ) = A × B and th us Ψ( F 1 ( M )) = F 0 ( A × B ), we let F r ( F )( M ) = ( F × ) ! F 1 ( M ), where F × : F 0 ( A × B ) → F 0 A × F 0 B is the unique oplax constraint downstairs (which is an isomo rphism if F is stro ng or la x). FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 71 The horizontal comp osition and units are built out of the monoidal structure and the functors f ∗ and f ! , so the lax or opla x co nstraints for these induce lax or oplax co nstraints for a strong double functor. F or exa mple, s upp ose F : Φ → Ψ is a lax mo noidal morphism of ﬁbrations and tha t M : A p → B and N : B p → C are horizontal 1-cells in F r (Φ). Then M ⊙ N comes with a diagra m (17.17) M ⊗ N ∆ ∗ ( M ⊗ N ) cart o o op cart / / M ⊙ N lying over (17.18) A × B × B × B A × B × C o o / / A × C. Applying F to (17.18), we obtain the following diagr am (omitting the symbol × ). (17.19) F ( AB B C ) F ( AB C ) o o / / F ( AC ) ( F A )( F B ) ( F B )( F C ) ∼ = O O ( F A )( F B ) ( F C ) o o / / ∼ = O O ( F A )( F C ) . ∼ = O O Applying F to (17.17), a nd adding the deﬁning arr ows for F M ⊙ F N , we obtain F ( M ⊗ N ) F (∆ ∗ ( M ⊗ N )) cart o o / / F ( M ⊙ N ) F M ⊗ F N O O ∆ ∗ ( F M ⊗ F N ) cart o o op cart / / O O    F M ⊙ F N O O The da shed and dotted arrows follo w by factoring the lax constraint of F through the given ca rtesian and o pca rtesian arrows. Since F do es not pr eserve op cartesia n arrows, the top-right solid a rrow is not necessar ily op car tesian, but this do es not matter. The unit co nstraint is similar. Finally , the opla x case is dual to this; the o nly diﬀerence is that all the vertical arrows g o the o ther wa y , and in (17.1 9) they ar e no longer isomorphisms.  Appendix A. Connection p airs As men tioned in § 1 , the questions whic h led us to framed bicatego ries have been address ed b y others in s everal ways. In this section we explain how framed bicategories are r elated to connection pairs on a double category; in the other app endices we consider their relationship to v ar io us parts of bicategory theor y . F or further detail on co nnection pairs, we refer the rea der to [BS76, BM9 9] and also to [Fio 06], which proved that connection pa irs ar e eq uiv alent to ‘foldings’. Our presentation of the theo ry diﬀers from the usual one b e cause we fo cus o n the pseudo case, which turns out to simplify the deﬁnition greatly . The following terminolo gy is from [GP04, PPD06]. Deﬁnition A. 1. Let D b e a do uble categ ory a nd f : A → B a vertical arrow. A companion for f is a horizontal 1-cell f B : A p → B together with 2-cells | f B / / f   ⇓ | U B / / and | U A / / ⇓ f   | f B / / 72 MICHAEL SHULMAN such that the following equations hold. | U A / / ⇓ f   f B / / f   ⇓ | U B / / = | U A / / f   ⇓ U f f   | U B / / | f B / / ∼ = | U A / / ⇓ | f B / / f   ⇓ | f B / / | U B / / | f B / / ∼ = = | f B / / | f B / / A conjoint for f is a horizo n tal 1-cell B f : B p → A toge ther with 2-cells | B f / / ⇓ f   | U B / / and | U A / / f   ⇓ | B f / / such that the following equations hold. | U A / / f   ⇓ B f / / ⇓ f   | U B / / = | U A / / f   ⇓ U f f   | U B / / | B f / / ∼ = | f B / / ⇓ | U A / / f   ⇓ | U B / / | B f / / | B f / / ∼ = = | B f / / | B f / / Comparing this deﬁnition with Theorem 4.1 (iii), the following b ecomes evident. Theorem A.2. A double c ate gory is a fr ame d bic ate gory exactly when every vertic al arr ow has b oth a c omp anion and a c onjoint. One can prov e in general that c o mpanions a nd co njoin ts ar e unique up to canon- ical isomorphism, that ( f B , B f ) is a dual pair if b oth are deﬁned, and that the op erations f 7→ f B and f 7→ B f are pse udo functorial insofar as they ar e deﬁned. The following deﬁnition is then easily seen to be e q uiv alent to those given in [BS76, BM99, Fio06]. Because it was or iginally motiv ated by double categor ies like the ‘quintets’ of a 2-ca tegory , it includes only c o mpanions and not conjoints. Deﬁnition A.3. Let D b e a strict double category . A connection pair o n D is a choice of a companion f B each vertical arrow f suc h that the pseudofunctor f 7→ f B is a strict 2- functor . Thu s, an arbitra ry choice of co mpanions on a non-strict double c ategory may b e called a ‘pseudo connection pair’, and a choice of conjoin ts may b e ca lled a ‘pseudo op-connection pa ir’. Theor em A.2 then states that a double catego ry is a framed bicategory pr ecisely when it admits b oth a pseudo co nnection pair and a pseudo op-connection pair. FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 73 Appendix B. Biequiv alences, biadjunctions, and monoidal bica tegories W e now consider the question of how muc h of the structure of a framed bica te- gory D is r eﬂected in its underlying bicateg ory D . Note that any bicategory may be considered as a framed bicateg o ry with only identit y vertical arr ows; we call such framed bicategories vert ically discrete . If F r B i 0 denotes the underlying 1-catego ry of F r B i and Bicat denotes the 1-categ o ry of bicateg ories and pseudo 2-functors, we hav e an a djunction (B.1) Bicat ⇄ F r B i 0 . in which the left adjoint considers a bicatego r y as a vertically discrete framed bica t- egory , while the rig h t adjoint takes a framed bicategor y to its underlying horizo ntal bicategory . The left adjoint Bicat → F r B i 0 do es not e xtend to a 2-functor or 3-functor, but in the other dir e ction, a n y framed transfor ma tion α : F → G : D → E can b e ‘lifted’ to a n o plax transfor mation b e tw een the underlying pseudofunctors as follows. F o r an ob ject A ∈ D , we deﬁne e α A = ( α A ) ∗ ( U GA ) : F A p → GA. F or a ho rizontal 1 -cell M : A p → B , we deﬁne e α M : F M ⊙ e α B ∼ = ( F M )( α B ) ! − → ( α A ) ∗ ( GM ) ∼ = e α A ⊙ GM to b e the globula r 2 -cell corres po nding to α M : F M α B = ⇒ α A GM . It is eas y to ch eck that this deﬁnes a n oplax natura l transfor ma tion betw een the pseudo 2-functors induced by F and G . If D and E a r e bicategories, we wr ite Bicat op ℓ ( D , E ) for the bicategor y of pseudo 2-functors, oplax natura l tra nsformations, and mo diﬁcations from D to E . By the pseudofunctoriality of base change, Pro po sition 4.1 6, the ab ov e construction deﬁnes a pseudofunctor (B.2) F r B i ( D , E ) − → Bicat op ℓ ( D , E ) . W e ca n also allow lax or oplax functors o n b oth sides. Note, how ever, that fra med transformations alwa ys give rise to oplax natura l transformations. W e would like to say that this construction extends to a functor from F r B i to the tricategory of bica tegories, but unfortunately there is no tricategory of bicatego ries which includes oplax natural tra nsformations, since the comp osition op eration Bicat op ℓ ( F , E ) × Bicat op ℓ ( D , E ) → Bicat op ℓ ( D , F ) would b e only an opla x 2-functor. W e could allow the co do main to b e a so r t of ‘oplax tricategor y ’, such as the ‘bicatego r y op-enr iched categ ories’ o f [V er92, 1.3 ], but this would take us to o far a ﬁeld. Instead, we merely o bs e r ve that if α happens to b e a framed natur a l isomor phism, then e α is a pseudo natura l equiv a lence. This suﬃces to prov e the following. Prop osition B.3. An e quivalenc e of fr ame d bic ate gories induc es a bie quivalenc e of horizontal bic ate gories. Pr o of. If F , G are in verse equiv alences in F r B i , then they give r is e to pseudo- functors, and by the ab ov e observ ation, the framed natural iso mo rphisms F G ∼ = Id and Id ∼ = GF g iv e rise to pseudo natura l equiv a lences.  74 MICHAEL SHULMAN F or example, this implies that in Example 7.10, the hor izont al bica teg ories G E x G/H and H E x actually ar e biequiv alent . Ho wev er, we b elieve the e quiv alence is more naturally stated, and easier to work with, in F r B i . In a simila r way , we ca n lift a mono ida l structure on a framed bicateg ory to a monoidal structure o n its horizontal bicategory . Many examples o f monoidal bicat- egories a ctually arise from monoidal fra med bica tegories. This is useful, b e c a use monoidal bicategories a re complicated ‘tr ica tegorical’ ob jects, wherea s monoidal framed bicatego ries a r e muc h easier to get a ha ndle on. See [GPS95, Gur06, CG0 7] for a deﬁnition of monoidal bicategor y . Theorem B. 4. If D is a monoidal fr ame d bic ate gory, then any cle avage f or D makes D into a monoidal bic ate gory in a c anonic al way. Sketch of Pr o of. D alr eady has a pro duct and a unit o b ject induced fr om D , s o it suﬃces to construct the co nstraints and coherence. W e consider the ass o ciativit y constraints, lea ving the unit constra in ts to the reader. Since D is a monoidal double category , it has a vertic al asso ciativity constraint a : ( A ⊗ B ) ⊗ C ∼ = A ⊗ ( B ⊗ C ) . But since D is a fra med bicategor y , this v ertical isomo rphism ca n b e ‘lifted’ to an equiv alence in D : e a = a ∗  ( A ⊗ B ) ⊗ C  : ( A ⊗ B ) ⊗ C p → A ⊗ ( B ⊗ C ) with adjoint in verse  ( A ⊗ B ) ⊗ C  a ∗ ; this will b e the asso cia tivit y equiv alence for the mono idal bicategory D . W e need further a ‘p entagonator’ 2-iso mo rphism e a ◦ e a ◦ e a ∼ = e a ◦ e a . But the coherence p entagon for the vertical iso morphism a tells us that a ◦ a ◦ a = a ◦ a and since ba se change ob jects are pseudofunctorial by Propo sition 4.1 6, this e quality in D 0 beco mes a cano nical isomorphi sm in D , which w e take as the p entagonator. It rema ins to chec k that this pentagonator satisﬁes the ‘co cycle equatio n’ for relations b etw een quintupl e pro ducts. Ho wev er, s ince all t he p entagonators are deﬁned by universal prop erties (being cano nical isomor phisms b etw een tw o carte- sian arrows), b oth sides of the co cycle equatio n a re also characterized by the same univ ersa l prop erty , and therefor e must be equal.  In co n trast to these well-behav ed cases, a fra med adjunction F : D ⇄ E : G does not generally give r ise to a biadjunction D ⇄ E . It do es, how ever, give rise to a lo c al adjunction in the s ense o f [BP 8 8]; this co nsists of an oplax 2-functor F : D → E , a lax 2- functor G : E → D , and a n a djunction (B.5) D ( A, GB ) ⇄ E ( F A, B ) . In a biadjunction, F and G would b e pseudo 2- functors and (B.5) would b e an equiv alence. When F and G a rise from a framed or op-fra med adjunction, a lo cal adjunc- tion (B.5) is g iven by D ( A, GB ) ( F − ) ε ! / / E ( F A, B ) η ∗ ( G − ) o o . FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 75 Of course, in a framed adjunction F is strong, while in an op-fra med adjunction G is strong . A bit more 2 -categor y theory than we hav e discussed here (see [Kel74]) gives us a no tion of ‘lax/o plax’ framed adjunction, in w hich the left adjoin t is oplax and the rig h t adjoint is lax; these a lso give rise to lo cal adjunctions b et ween horizontal bicategories. In this w ay , pra ctically any fra med-bicategorica l notion giv es rise to a counterpart on the purely bica tegorical level. F or example, by a process similar to that in Theorem B.4, any in volution on D gives r ise to a ‘bicategorica l in volution’ o n D . Of course, we can also deﬁne monoids and bimo dules in a n y bicategory ; in this context monoids ar e often called monads , since in C at they reduce to the usual notion of monad. The fact that b oth internal and enriched categories are mo noids in appropria te bicategories is well-known, and bicategory theorists hav e studied categories enriched in a bicateg o ry as a gener alization of categories enriched in a monoidal catego r y; see [W al81, Str8 1, Str83 a, Str83b, CJ SV94, KLSS02]. How e ver, pure bicateg ory theo ry usually starts to break down whenever we need to us e vertical a r rows. F or exa mple, it is ha r der to get a handle on internal or en- riched functors purely bicatego rically . In the next app endix we int ro duce a str uc- ture called an e quipment w hich is sometimes used for this purp ose, for exa mple in [LS02]. Appendix C. Equipments F or the theor y of equipments we refer the r eader to [W o o8 2, W o o8 5, CJSV94, V er92]. F r om o ur p oint of view, it is natura l to introduce them by a s king how the vertical arrows of D are reﬂected in D . W e know that there is a ps e udo functor D 0 → D sending f : A → B to the ba se change ob ject f B ; this pseudo functor is bijectiv e on ob jects and eac h f B has a right adjoint in D . Thus we almost hav e an instance of the following structure. Deﬁnition C.1. A proarro w equipm en t is a pseudo 2-functor ( − ) : K → M betw een bicateg ories such that (i) K and M have the same o b ject s and ( − ) is the identit y o n ob jects; (ii) F or every arrow f in K , f has a right adjoint e f in M ; a nd (iii) ( − ) is lo ca lly full a nd faithful. The o nly diﬀerence is that in an equipment, K is a bicatego ry rather than the 1- category D 0 , but condition (iii) means that the 2-cells in K are determined by those in M anyw ay . Th us, g iven a framed bicateg ory D , we c an factor the base-change ob ject pseudofunctor D 0 → D a s D 0 i − → K ( − ) − → D where i is bijective on ob jects and 1- cells a nd ( − ) is lo cally full and faithful. The ob jects and morphisms of K are those of D 0 , and its 2 -cells from f → g a r e the 2-cells f B → g B in D . W e hav e prov en the following. Prop osition C. 2. If D is a fr ame d bic ate gory, t hen the ab ove pseudofunctor ( − ) is a pr o arr ow e quipment. Note that in the proarr ow equipmen t arising from a framed bicategory , the bi- category K is actually a strict 2-ca tegory . Ho wev er, this is ess en tially the only restriction on the equipments which arise in this wa y . 76 MICHAEL SHULMAN Prop osition C.3. L et ( − ) : K → M b e a pr o arr ow e quipment s u ch that K is a strict 2-c ate gory. Deﬁne a double c ate gory D whose • Obje cts ar e those of K (and M ); • V ertic al arr ows ar e the arr ows of K ; • Horizontal 1-c el ls ar e the arr ows of M ; and • 2-c el ls A M / / f   ⇓ α B g   C N / / D ar e the 2-c el ls α : M ⊙ g − → f ⊙ N in M . Then D is a fr ame d bic ate gory. Sketch of Pr o of. First we s how that D is a double category . The v ertical comp osite M / / f   ⇓ α g   N / / h   ⇓ β k   P / / is deﬁned to b e the comp osite M ⊙ k g ∼ = M ⊙ ( g ⊙ k ) ∼ = ( M ⊙ g ) ⊙ k α − → ( f ⊙ N ) ⊙ k ∼ = f ⊙ ( N ⊙ k ) β − → f ⊙ ( h ⊙ P ) ∼ = ( f ⊙ h ) ⊙ P ∼ = hf ⊙ P. The co herence theore ms for bicategorie s a nd pseudo functor s imply that this is ver- tically a sso ciative and unital. Horizontal comp osition of 1-cells is deﬁned as in M , a nd horiz o nt al co mpos ition of 2-cells is deﬁned analogo usly to their vertical comp osition. The constra in ts come fro m those o f M . Finally , fo r a n a rrow f : A → B in K , and e f the right adjoint of f , it is easy to chec k that the 2-cells f / / f   ⇓ 1 f U B / / , e f / / ⇓ ε f   U B / / , U A / / f   ⇓ η e f / / , and U A / / ⇓ 1 f f   f / / , deﬁned by iden tities and by the unit and counit of the adjunction f ⊣ e f , sa tisfy the equa tions of Theo rem 4.1(iii). Thus D is a framed bicategor y .  A t the level of ob jects, it is easy to show that the tw o constructions a re in verses up to isomorphism. In o rder to state this as an equiv a lence of 2-categor ies, ho wev er, FRAMED BICA TEGORIES AND MONOIDAL FIBRA TIONS 77 we w ould need mo rphisms and esp ecially tra nsformations b etw een equipments, and it is not immediately obvious how to deﬁne these. The appr o ach to c o nstructing a 2-category of equipments taken in [V er92] is essentially to ﬁr st make equipments into double categ o ries, as we hav e done, a nd deﬁne morphisms and tra nsformations of equipmen ts to b e morphisms b etw e e n the corr esp onding double ca tegories. This makes our des ired equiv alence true by deﬁnition. Actually , [V er92] uses ‘doubly w eak’ double categories t o deal with equipmen ts wher e K is not a s tr ict 2-categ o ry , and th us obta ins a tricateg ory rather than a 2-ca tegory , but the idea is the s ame. Thu s, fra med bica tegories ca n b e reg arded as a characterization o f the double categories whic h arise from equipmen ts. How ever, since the correct notions of morphism and transformatio n are appa r ent only from the side of do uble ca tegories, we b elieve it is mo re natura l to work directly with framed bicatego ries. Remark C.4. The a uthor s of [CKW91, CK VW98] consider a related notion of ‘equipmen t’ where K is repla ced b y a 1-ca teg ory but the hor izontal comp ositio n is forgotten. If D is a framed bica tegory , then the span (C.5) D 0 L ← − D 1 R − → D 0 has the prop erty that L is a ﬁbra tion, R is an opﬁbration, and the t wo t ypes of base change commute, making it into a ‘t wo-sided ﬁbration’ from D 0 to D 0 in the sense of [Str80]; these a re essentially what [CKVW98] studies under the na me ‘equipmen t’. The fact that L is also a n opﬁbratio n, a nd R a ﬁbration, in a co m- m uting wa y , make (C.5) in to what they c a ll a starr e d p ointe d e quipment . This structure incorp orates les s of the structure of a framed bicatego r y , but it was suf- ﬁcien t in [CKW91, CKVW98] to o btain a 2 -category o r tricategory o f equipments and a notion of equipment adjunction. It is ea sy to chec k that any framed adjunc- tion gives rise to an equipment adjunction in their sens e . Appendix D. Epilogue: framed bica tegories versus bica tegories W e end with some more philo s ophical remarks ab out the rela tionship of framed bicategories to pure bicatego ry theory . F or any bicategor y B , there is a canonical proarr ow equipment ( − ) : K → B , where K is the bicategory of a djunctions f ⊣ e f in B . When B is a strict 2-categ o ry , so is K , and the resulting fr amed bicategor y is what we called A dj ( B ) in Example 2 .8 . In gener al, we obtain a ‘doubly weak’ framed bicatego ry which w e also call A dj ( B ). Thu s, we can rega rd the theory of framed bicategor ies , or of equipmen ts, as a generalizatio n o f the theory of bicategor ies in which we sp ecify which adjunctions are the base change ob jects, rather than using a ll of them. A certain amount of pure bicategory theory can b e reg arded as implicitly working with the fra med bicategor y A dj ( B ); freq uen tly 1-c e lls with right a djoin ts ar e ca lled maps and ta ke on a sp ecial role. See, for exa mple, [Str81] and [CKW87]. This purely bicategorica l appr oach works well in bicategories like D ist ( V ), be- cause, as we mentioned in Example 5.6(iii), the mild condition of ‘Cauc hy complete- ness’ on the V -catego ries inv olved is s uﬃcien t to ensur e that any distributor with a right adjoint is isomo rphic to a bas e c hange o b ject. How ever, in other fra med bicategories, suc h as M o d and E x , t here will not b e a go o d supply o f ‘Ca uch y complete’ ob jects, so framed bicategor ies o r equipmen ts ar e neces sary . Moreover, even when working with D ist ( V ), framed bicategories are implicit in some of the 78 MICHAEL SHULMAN bicategorica l literature, such as the ‘calculus o f mo dules’ for enriched catego ries; see, for example, [SW78 , W o o8 2, CKW87, Str83 a]. 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Diﬀ´ er entiel le , 22(3):283–286, 1981. Third Col l oquium on Categories, Part IV (Amiens, 1980). [W oo82] R. J. W oo d. Abstract proarro ws. I. Cahiers T op olo gie G´ eom. Diﬀ´ er enti e l le , 23(3):279– 290, 1982. [W oo85] R. J. W o o d. Proarrows. II. Cahiers T op olo gie G ´ eom. Diﬀ´ er entiel le Cat´ eg. , 26(2):135– 168, 1985. Dep ar tment of Ma thema tics, University of Chicago, 5734 S. University A v e, Chicago IL 60615 E-mail add r ess : shulman@ma th.uchicago .edu

Framed bicategories and monoidal fibrations

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