arXiv:2006.10890v4 [math.CT] 8 Mar 2021 rtro o h ofiaiyo hi rjcinfunctors. formati projection Keywords: the their of of confinality presentation the applicable for easily criterion an with along allowin by generally emerging (c most the Grothendieck here of categories. present split adjoint we left of which catego The 2-equivalence categories, the the category. of category through mere arrow a factors the of of that object form than an involves It as interest. seen independent functor, of be to promises and insight distinct pr quite three provide give that We methods Berger. using and ac formula, which Batanin ‘twist formula, by general a the independently of proved case (co-)limits: Scher special and a for Chachólski and demonstr by formula’; formulae established We first three as categories. deriving formula, Fubini for small by the of and part category category in the given into a in functor 2-adjuncti diagrams a fundamental small of all web of a category present we theory, category Abstract 00MSC: 2010 functor. quotient (confinal) strictificat cofibration, colimit, (op)lax construction, Grothendieck niern eerhCuclo aaa(NSERC). Canada of Council Research Engineering et fMteaia n ttsia cecs Universit Sciences, Statistical and Mathematical of Dept. 1 w pedcst hswr pl u ndti l eddtoo needed all detail in out spell work this to appendices Two o work earlier extends 2-adjunctions of web our of ‘base’ The fram the Within twofold. are paper this of contributions The mi addresses: Email h eodato’ eerhi upre yaDsoeyGra Discovery a by supported is research author’s second The eateto ahmtc n ttsis okUniversity York Statistics, and Mathematics of Department igas irtos n h eopsto fColimits of Decomposition the and Fibrations, Diagrams, iga aeoy oii eopsto,titdFbn f Fubini twisted decomposition, colimit category, diagram 83,1A5 18D30. 18A35, 18A30, [email protected] Gog Peschke), (George erePeschke George atrTholen Walter 1 [email protected] fAbra dotn let,Cnd,TG2G1 T6G Canada, Alberta, Edmonton, Alberta, of y o flxcmuaiedarm,fe split free diagrams, lax-commutative of ion 1 s. oot,Otro aaa 3 1T6 M3J Canada, Ontario, Toronto, , ulyiiitdti ok n hc was which and work, this initiated tually t(o 020 fteNtrlSine and Sciences Natural the of 501260) (no. nt r e gnrlclmtdecomposition colimit ‘general new a er; n h iga aeoyo narbitrary an of category diagram the ing -firtosadsrcl (co-)indexed strictly and o-)fibrations no utetctgre n novel a and categories quotient of on -iesoa aito ntebase the in variation 2-dimensional g d eeaiaino h well-known the of generalization ed’ the of formation the surrounding ons aino h rtedekcategory Grothendieck the of mation yo oal ml aeois rather categories, small locally of ry t h tlt fteeadjunctions, these of utility the ate h hemn coladGuitart and school Ehresmann the f wr fGohnic’ fibrational Grothendieck’s of ework osfrti oii decomposition colimit this for oofs eeaie utr 2-adjunction Guitart generalized so bainlctgr theory, category fibrational of ls Wle Tholen) (Walter rua slt fibration, (split) ormula, Contents

1 Introduction 3 1.1 Motivation ...... 3 1.2 The Grothendieck construction and the Guitart adjunction ...... 3 1.3 A network of global 2-adjunctions ...... 5 1.4 Organization of the paper and terminological conventions ...... 5

2 Diagram categories 6 2.1 Two types of diagram categories of a given category ...... 6 2.2 Limits and colimits in diagram categories ...... 9 2.3 Left Kan extensions are cocartesian over Cat, invariance under the formation of colimits ...... 10 2.4 Strictification of morphisms in diagram categories ...... 11 2.5 The generic colimit for the standard construction of colimits...... 12

3 The Guitart adjunction 14 3.1 Diag˝ is right adjoint to the Grothendieck construction ...... 14 3.2 Diag˝ has the structure of a normal pseudomonad on CAT ...... 17

4 The twisted Fubini formula and cofibred colimits 18 4.1 The twisted Fubini formulae for colimits and limits ...... 18 4.2 Cofibredcolimits ...... 20

5 The Colimit Decomposition Formula: three proofs 21 5.1 The basic formula and its short first proof ...... 21 5.2 A generalized colimit decomposition formula and the secondproofoftheCDF ...... 22 5.3 Obtaining the Colimit Decomposition Formula via Fubini: the third proof of the CDF ...... 24

6 Extending the Guitart adjunction, making Grothendieck a left adjoint 25 6.1 The diagram category Diag˝pP q of a functor P : E Ñ B...... 25 6.2 Review of the 2-categories CAT 2, CAT{{Cat and CAT{Cat...... 28 6.3 The extended Guitart adjunction ...... 30

7 The Grothendieck equivalence via the extended Guitart adjunction 32 7.1 Strictification of lax-commutative diagrams ...... 33 7.2 Replacingdiagramsbyfibres ...... 34 7.3 The Grothendieck Equivalence Theorem for split cofibrations ...... 36 7.4 The Grothendieck Equivalence Theorem for split fibrations ...... 38

8 A left adjoint to the Grothendieck construction: free split cofibrations 39 8.1 Freesplitcofibrations ...... 40 8.2 A network of global 2-adjunctions ...... 43

9 Diagram categories as 2-(co)fibred categories over Cat 43 9.1 Hermida-Buckley 2-fibrations ...... 43 9.2 How to consider diagram categories as 2-fibred or 2-cofibred over Cat ...... 45

10 Appendix 1: Grothendieck fibrations and the Grothendieck construction 46 10.1 Cartesianmorphisms...... 46 10.2 Grothendieck fibrations ...... 47 10.3 Grothendieck cofibrations and bifibrations ...... 49 10.4 Limits and colimits in a fibred or cofibred category ...... 50 10.5 The Grothendieck construction for indexed categories ...... 51 10.6 Standardexamples ...... 52 10.7 The classifying split (co)fibration ...... 54

11 Appendix 2: Regular epimorphisms in CAT and a confinality criterion 54 11.1 Characterizing regular epimorphisms in CAT ...... 54 11.2 Confinality vs. conneted fibres for regular epimorphismsandfibrations...... 57

2 1. Introduction

1.1. Motivation An initial goal of this work was the development of techniques that may facilitate the compu- tation of “complicated” objects, say in algebra or topology, from smaller and more easily computed “pieces”. In fact, much of this work emanated from a colimit formula established early on, saying that a colimit of a diagram X : K Ñ X in a cocomplete category X may be computed “piecewise”, whenever the small category K is itself expressed as K “ colimΦ in the category Cat of small categories, for some functor Φ: D Ñ Cat. That is, if Kd : Φd Ñ K pd P Dq denotes the colimit cocone of Φ, one has the following Colimit Decomposition Formula:

K dPD Φd colim pXq– colim pcolim pXKdqq. (CDF)

Kd / X / / / Φd colim8 Φ “ K X XpKdxq colim XKd colim8 X r q8 q rrr qq qqq rr qqq qq rr Ke q q rr qqq qqq Φe r q q XpKdyq colim XKe

(Note that this visualization of the formula must not be misinterpreted as a kind of preservation of a colimit by X: the arrows on the left live in CAT, while those on the right are in X .) The CDF (5.1) and its limit analogue (5.2) were presented by the first author in May 2017 at the Workshop on Categorical Methods in Non-Abelian Algebra in Louvain-la-Neuve. The argument con- sisted of brute-force constructions in the category Diag˝pX q of small diagrams in X , see (2.1), with a strong suspicion that fundamental properties of the evident functor Diag˝pX q Ñ Cat would give a transparent and conceptual explanation. From this suspicion the present collaboration evolved. After a talk presented by the second author at CT2019, it turned out that the CDF had actually appeared slightly earlier than the first author’s 2017 presentation, as Lemma 7.13 in the paper [2] by Michael Batanin and Clemens Berger, who give credit to Steve Lack for the short proof they present. We found that Grothendieck’s fibrational category theory provides a good framework for our purposes, and that it yields insights on categories of diagrams which are of interest in their own right. As hoped for, we obtain transparent and conceptual proofs for the CDF and its variations, see Sections 4 and 5. Here is an outline.

1.2. The Grothendieck construction and the Guitart adjunction Consider a Grothendieck fibration P : E Ñ B. By assigning to every object b of the base op category B its fibre Eb in E one obtains a pseudo-functor B Ñ CAT into the huge 2-category of all categories (see the end of this Introduction for notational conventions with respect to CAT and Cat). Conversely, the Grothendieck construction produces for every pseudo-functor Φ: Bop Ñ CAT the ˝ total category of Φ, here denoted by ş Φ, which is fibred over B. In this way, fibred categories over B are equivalently presented as contravariantly indexed categories, that is: as contravariant CAT- valued pseudo-functors defined on B. Dually, Grothendieck cofibrations (nowadays more frequently called opfibrations2 ) E Ñ B correspond equivalently to covariantly indexed categories Φ: B Ñ CAT,

2In this paper we adhere to Grothendieck’s original terminology, but note that, in order to prevent confusion with the terminology used in topology, in recent years Grothendieck cofibrations have more commonly been referred to

3 via the dual Grothendieck construction, here denoted by ş˝ Φ. Under this equivalence, so-called split cofibrations correspond, by definition, to those pseudo-functors that are actually functors, and one may further restrict the equivalence to small-fibred split cofibrations and Cat-valued (as opposed to CAT-valued) functors. These facts are well known and amply documented in the literature, albeit predominantly in a context that leaves the base B fixed; in historical order, references include [18], [16], [17], [3] [28], [40], [4], [5], [25], [13], [23], [27], [41], [39], [44], [26]. Less known is the fact that the Grothendieck construction was studied very early on by Ehresmann [11] and his school, under the name produit croisé. In particular, Guitart (see [19], [20], [21]) showed that the assignment Φ ÞÝÑ ş˝ Φ leads to a left-adjoint functor

ş˝ : CAT{Cat ÝÑ CAT whose right adjoint, here denoted by Diag˝, deserves independent interest. It assigns to a category X the category Diag˝pX q of all small diagrams in X which, curiously enough, may be thought of as arising via the Grothendieck construction. Indeed, with the functor rl, X s “ X l : Catop Ñ CAT one has ˝ Diag˝pX q“ ş X l, to be considered as a (fibred) category over Cat; explicitly, a morphism pF, ϕq : pI,Xq Ñ pJ , Y q with small categories I, J is given by a lax-commutative diagram

I F / J ❄ ❄❄ ϕ:ùñ ⑦⑦ ❄❄ ⑦ X ❄ ⑦⑦Y ❄ ⑦~ ⑦ X in CAT. Since one actually has an adjunction of 2-functors (3.1) , the first nice, but immediate, conse- quence of the Guitart adjunction is the known characterization of the Grothendieck category ş˝ Φ as a lax colimit of Φ: B Ñ CAT in CAT (3.2). Much less known, or expected, is the second con- sequence that we draw from the Guitart adjunction, as it gives us the connection with the CDF. Considering the functor Φ again as a diagram in Cat (and, thus, renaming the category B as D from X X D this viewpoint), a diagram T : ş˝ Φ Ñ in a category corresponds to a -shaped diagram in Diag˝pX q, given by a family T pd, ´q :Φd Ñ X pd P Dq of diagrams in X . If all these have colimits in X , then one has the Twisted Fubini Colimit Formula (4.2):

pd,xqPş Φ dPD xPΦd colim ˝ T pd, xq – colim pcolim T pd, xqq, (TFCF) with the colimit on either side existing if the one on the other side exists. This formula appears in the Appendix of the Memoir [10] by Wojciech Chachólski and Jérˆome Scherer. In Section 5 we explain how the TFCF implies the CDF, using a general confinality criterion (11.3) for quotient functors (= regular epimorphisms in CAT).

as opfibrations. However, this departure from the standard categorical procedure for dualizing terms causes further deviations from established standards, such as “opcartesian" versus “cocartesian". See also [31] for a brief discussion of this terminological difficulty .

4 1.3. A network of global 2-adjunctions

It seems peculiar that, in the Guitart adjunction, the left adjoint ş˝ does not keep track of the fact that, for the Cat-valued functor Φ, the total category ş˝ Φ actually lives over Cat. However, not forgetting this important fact, and still maintaining an adjunction, means that we should extend the functor Diag˝, so that it operates not just on categories, but also on functors, generally to be 2 considered as objects of the arrow category CAT . Accordingly, a major undertaking in this paper 2 is the replacement of CAT by CAT as the domain of the Guitart adjunction, as depicted on the right of the diagram

˝ !l Diag , 2 - CAT l J CAT l J CAT{Cat , Dom ş˝

If we compose this extended 2-adjunction on the right (6.4) with the rather trivial 2-adjunction on the left, one obtains back the original Guitart adjunction. The significance of the extended Guitart adjunction on the right is that it communicates well with the Grothendieck equivalence

(7.3) between split cofibrations with small fibres and Cat-valued functors. In fact, the 2-functor ş˝ above factors as

ş 2 o incl o ˝ o incl CAT SCoFIBsf » CAT{{Cat CAT{Cat , with the (non-full) subcategory SCoFIBsf of small-fibred split cofibrations and their cleavage- 2 preserving morphisms in CAT , and with the left-adjoint (non-full) inclusion functor (8.1) of the comma category CAT{Cat into the lax comma category CAT{{Cat, all considered as 2-categories. (Objects and morphisms of CAT{{Cat are defined as in Diag˝pCatq; just specialize X to Cat above, except that there is no smallness requirement for I, J .)

Of course, dropping the requirement of small-fibredness, we may factor the 2-functor ş˝ of the extended Guitart adjunction also as

ş 2 o incl o ˝ o incl CAT SCoFIB » CAT{{CAT CAT{Cat . Now the full inclusion on the left has become a right adjoint, with its left adjoint producing the free split cofibration generated by an arbitrary functor (7.1). Either way, the extended Guitart adjunction factors through the classical Grothendieck equivalence between (certain) cofibrations and (certain) indexed categories.

1.4. Organization of the paper and terminological conventions To make the paper self-contained and to introduce notation, we collect background material on fibrational category theory and the Grothendieck construction in Appendix 1 (Section 9). In doing so, we clarify some essential details which don’t seem to be documented explicitly in the literature. In Appendix 2 (Section 10) we present a characterization of regular epimorphisms in CAT (likely to be known in this or some similar form) and a confinality criterion for such functors (likely to be new), of which we make essential use in one of the proofs for the CDF. Section 2 introduces diagram categories, shows how to compute colimits and limits in them, identifies a colimit in Cat that is generic for the computation of colimits in terms of coproducts and coequalizers, and finally presents the Guitart adjunction, with the characterization of the

5 Grothendieck construction as a lax colimit in CAT following from it. The Twisted Fubini Colimit Formula appears in Section 3, followed by three independent proofs for the Colimit Decomposition Formula in Section 4. The first one follows [2], the second is based on the new General Colimit Decomposition Formula 5.4, and the third one uses the TFCF. In Sections 5–7 we present successively the extended Guitart adjunction, the Grothendieck equiv- alence, and finally their global interactions. In doing so, we restrict ourselves to the consideration of split (co)fibrations and functorially indexed categories, but take full account of the 2-categorical nature of the global correspondences. However, in the supplementary Section 8 we briefly show how diagram categories may be considered as 2-(co)fibered categories over Cat. The expectation is that there are richer or higher-dimensional contexts (such as those recently considered in [43], [34], [34], [32], [31], [37]), in which this network of adjunctions, as well as the the decomposition formulae, may be established. Throughout the paper, the term category refers to an ordinary category, also called 1-category; when a category carries a higher-dimensional structure and is considered with it, we will say so explicitly. Categories may be large (so that their objects may form a proper class), but they are always assumed to be locally small, so that their hom-functors take values in Set. Categories whose object class is a set are called small. Cat denotes the category of small categories, and CAT is the huge category of all (1-)categories, which contains Set and Cat as particular objects. The huge category of classes and their maps is denoted by SET. These casual conventions may be made more precise and justified through the provision of Grothendieck universes. Acknowledgements. We thank the organizers of the Workshop on Categorical Methods in Non- Abelian Algebra, Louvain la Neuve, June 1-3 2017 for their hospitality and for providing a fruitful environment from which this work evolved. The second author thanks also Paolo Perrone for very helpful comments on parts of this work.

2. Diagram categories

2.1. Two types of diagram categories of a given category Let X be a category. Since every small category I is, via the formation of the functor category X I “rI, X s“ CATpI, X q, exponentiable in CAT, one has the (internal hom-)functor

X l “rl, X s : Catop Ñ CAT, pF : I Ñ J q ÞÝÑ pF ˚ : X J Ñ X I, Y ÞÑ Y F q.

Applying both, the Grothendieck construction and the dual Grothendieck construction (see Section 10.5) to X l yields the diagram categories 3

˝ X ˝X l X X l Diag p q“ ş and Diag˝p q“ ş˝ . For the reader’s convenience, here is a detailed description:

Definition 2.1. The objects of the category Diag˝pX q are pairs pI,Xq with X : I Ñ X a functor of a small category I; a morphism pF, ϕq : pI,Xq Ñ pJ , Y q is given by a functor F : I Ñ J of small categories and a natural transformation ϕ : X Ñ Y F , as depicted on the left of the diagram

3In the literature one finds the notation Cat{{X for either of these categories. We use it later on in special cases.

6 I F / J I o F J ❄ ❄ ❄❄ ϕ:ùñ ⑦⑦ ❄❄ ϕ:ùñ ⑦⑦ ❄❄ ⑦ ❄❄ ⑦ X ❄ ⑦⑦Y X ❄ ⑦⑦Y ❄ ⑦~ ⑦ ❄ ⑦~ ⑦ X X ˝ The category Diag˝pX q has the same objects as Diag pX q, but a morphism pF, ϕq : pI,Xq Ñ pJ , Y q in Diag˝pX q is now given by a functor F : J Ñ I and a natural transformation ϕ : XF Ñ Y , as depicted on the right of the above diagram. ˝ The composite of pF, ϕq followed by pG, ψq : pJ , Y q Ñ pK,Zq in Diag pX q and Diag˝pX q is respectively given by pGF, ψF ¨ ϕq and pF G, ψ ¨ ϕGq . One has the obvious forgetful functors X ˝ op D : Diag pX q Ñ Cat and DX : Diag˝pX q Ñ Cat which remember just the top rows of the above triangles; in the notation of (10.5), they are precisely X l the functors Π and ΠX l , respectively. Consequently one has the following Proposition, which is also easily established “directly”. Proposition 2.2. DX is a split fibration, with cleavage F ˚ J θpJ ,Y q “ pF, 1Y F q : F pJ , Y q “ pI, Y F q Ñ pJ , Y q for F : I Ñ J and Y P X .

Dually, DX is a split cofibration, with cleavage F I δpI,Xq “ pF, 1XF q : pI,Xq Ñ F!pI,Xq “ pJ , XF q for F : J Ñ I and X P X . Since the two types of Grothendieck constructions are dual to each other (as described in 10.5), so are the two types of diagram categories. This fact one may easily see in a direct manner, by an application of the bijective 2-functor lop : CAT co ÝÑ CAT, rα : S ùñ T : C Ñ Ds ÞÝÑ rαop : Sop ðù T op : Cop Ñ Dops (which maps morphisms covariantly but 2-cells contravariantly) to the inscribed triangle on the right of the diagram of Definition 2.1. In this way one establishes an isomorphism between the dual ˝ op of the (ordinary) category Diag˝pX q and the category Diag pX q, coherently so with respect to the forgetful functors, as shown in the commutative diagram

lop Diag X op / Diag˝ X op p ˝p qq – p q

op op pDX q DpX q  lop  Cat / Cat . – ˝ The objects and morphisms of X may be considered as living in both, Diag pX q and Diag˝pX q. Indeed, there are full embeddings X ˝ E : X Ñ Diag pX q and EX : X Ñ Diag˝pX q which interpret every object X of X as a functor 1 Ñ X of the terminal category 1 and every mor- phism f : X Ñ Y in X as a natural transformation, giving respectively the (hardly distinguishable) morphisms

7 Id1 Id1 1 / 1 1 o 1 ❄ ❄ ❄❄ f:ùñ ⑧⑧ ❄❄ f:ùñ ⑧⑧ ❄❄ ⑧ ❄❄ ⑧ X ❄ ⑧⑧Y X ❄ ⑧⑧Y ❄ ⑧ ⑧ ❄ ⑧ ⑧ X X

˝ X in Diag pX q and Diag˝pX q. E and EX cooperate with the dualization isomorphism for the diagram categories, as shown in the commutative diagram X op ▲ op qq ▲ X op pEX q qq ▲▲ pE q qq ▲▲▲ qqq ▲▲ qx q lop ▲% pDiag pX qqop / Diag˝pX opq ˝ –

With (co)completeness to be understood to mean that every small diagram comes with a choice of (co)limit, one has the following “folklore” proposition, whose routine proof we may skip. Proposition 2.3. (1) The category X is functorially4 cocomplete if, and only if, EX is a reflective embedding (with left adjoint colim : Diag˝pX q Ñ X ). (2) The category X is functorially complete if, and only if, EX is a coreflective embedding (with right adjoint lim : Diag˝pX q Ñ X ). ˝ Remark 2.4. (1) We emphasize that, here, we are considering Diag pX q and Diag˝pX q, just like X , X as 1-categories and, thus, ignore their obvious 2-categorical structures which make D and DX 2- functors. In the case of Diag˝pX q, a 2-cell α : pF, ϕq ùñ pF 1, ϕ1q is simply a natural transformation α : F Ñ F 1 with Y α ¨ ϕ “ ϕ1:

pF,ϕq * I α ó 4 J ❅❅ ❅ 1 1 ⑥⑥ ❅❅ pF ,ϕ q ⑥⑥ X ❅❅ ⑥⑥Y ❅ ⑥~ ⑥ X

As DX should preserve the horizontal and vertical compositions of 2-cells, there is no choice of how to define them in Diag˝pX q. All verifications proceed routinely. (2) Likewise, at this point we ignore the obvious fact that the functor X l “ r´, X s : Catop Ñ CAT, as considered at the beginning of this section, is actually a 2-functor: it maps every natural 1 ˚ ˚ 1˚ ˚ transformation α : F ùñ F (covariantly) to the transformation α : F ùñ F with αY “ Y α : Y F ÝÑ Y F 1, preserving both the vertical and horizontal composition of natural transformations. (3) With 1 denoting the terminal category we trivially have

˝ ˝ op Diag pHq – Diag˝pHq – 1 and Diag p1q– Cat, Diag˝p1q– Cat . Much less obvious is the fact that the (ordinary) category Diag˝pCatq is equivalent to a suit- ably defined category which has the split cofibrations of small categories as its objects, and that op Diag˝pCat q is equivalent to the dual of the category of split fibrations of small categories, as one may conclude from the 2-categorical equivalence formulated in Corollary 7.6.

4This means that colimits in X are assumed to be chosen for all small diagrams in X .

8 2.2. Limits and colimits in diagram categories We continue to work with a fixed category X and now consider the question of how to form limits or colimits in its diagram categories. Using fibrational methods we first confirm the following assertions: Proposition 2.5. (1) Diag˝pX q has coproducts, and DX preserves them. For any small category D, the category Diag˝pX q has all D-shaped limits with DX preserving them if, and only if, X has D-limits. If Diag˝pX q has coequalizers, then so does X , and EX preserves them. ˝ (2) Diag˝pX q has products, and DX preserves them. For small D, the category Diag pX q has all D- colimits with DX preserving them if, and only if, X has D-colimits. If Diag˝pX q has equalizers, then so does X , and EX preserves them. Proof. (2) follows from (1) by dualization (see Proposition 2.2), and the first and last assertion of (1) are quite obvious. The assertion about the existence D-limits for a small category D when these exist in X follows from Theorem 10.4, as soon as we have demonstrated that the comma insertions X X IJ : rJ , X s ÝÑ J ÓD , X ÞÝÑ pIdI Ñ D pI,Xqq preserve D-limits, for any small category J . But this preservation condition just means that, given a D-shaped limit cone λd : Y Ñ Yd in rJ , X s and any D-shaped cone pF, ϕdq : pI,Xq Ñ pJ , Ydq ˝ in Diag pX q, the cone ϕd : X Ñ YdF in rI, X s factors through the cone λdF : Y F Ñ YdF , with a uniquely determined transfomation ϕ : X Ñ Y F . But this is an immediate consequence of the pointwise formation of limits in functor categories, which makes the cone λdF : Y F Ñ YdF a limit cone in rI, X s. We leave to the reader the proof that the existence of D-limits in Diag˝pX q forces the existence of such limits in X .

˝ The question of how to form coequalizers in Diag pX q (or, equivalently, equalizers in Diag˝pX q) may also be addressed with fibrational methods, as follows. For every functor F : I Ñ J of small ˚ J I categories, the functor F : X Ñ X has, by definition, a left adjoint F! if, and only if, for every I X P X , a (chosen) left Kan extension F!X “ LanF X of X along F exists. The existence of this extension is certainly guaranteed when X is cocomplete; conversely, a colimit of X : I Ñ X can be obtained as the left Kan extension of X along the functor I Ñ 1 to the terminal category. Consequently, with Proposition 2.2 and Theorem 10.1 we obtain: Proposition 2.6. The split fibration DX is a bifibration if, and only if, the category X is (small-) cocomplete. Dually, the split cofibration DX is a bifibration if, and only if, X is (small-)complete. With the help of Theorem 10.3, Corollary 10.5, and Proposition 2.6 we can now state: Theorem 2.7. (1) If X is a cocomplete category, then Diag˝pX q is also cocomplete, and DX preserves all colimits. (2) If X is complete, then Diag˝pX q is also complete, and DX preserves all limits (so that DX transforms limits in Diag˝pX q into colimits in Cat). Proof. (1) Cocompleteness of X is needed to make DX a bifibration. Its fibres, i.e., the functor categories of X , have the types of (co)limits that X has. Since the base category, Cat, is bicomplete, the assertion follows with the statements referred to above. (2) With the dualization principle as stated before Proposition 2.3, item (2) follows from an application of (1) to X op, rather than to X .

9 Remark 2.8. Here is a brief illustration of the construction of standard (co)limits in DiagpX q as it follows from the proofs of Propositions 2.5 and 2.6 and Theorem 2.7: Products: The product pI,Xq of the family pIν ,Xν q, ν P N, is given by the products I “ śνPN Iν in Cat and Xppiν qν q“ śνPN Xν piν q in X , with the obvious projections. Equalizers: The equalizer pE, ηq : pE,Zq Ñ pI,Xq of pF, ϕq, pG, ψq : pI,Xq Ñ pJ , Y q is given by the equalizer E : E ãÑ I of F, G in Cat and the equalizers ηi : Zi Ñ XI of ϕi, ψi : Xi Ñ YFi “ Y Gi in X , for all i P E. Coproducts: The coproduct pI,Xq of the family pIν ,Xν q, ν P N, is given by the coproduct I “ šνPN Iν in Cat and X determined by the functors Xν . Coequalizers: To construct the coequalizer pH,γH ¨ κq : pJ , Y q Ñ pK,Zq of pF, ϕq, pG, ψq : pI,Xq Ñ pJ , Y q, one first forms the coequalizer H : J Ñ K of F, G in Cat and then the left Kan extensions of X along HF “ HG and of Y along H, with associated universal transformations λ and κ; now, with α, β determined by ϕ, ψ as in the commutative diagrams

λ λ X / pLanHF XqHF X / pLanHGXqHG ✤ ✤ ϕ ✤ αHF ψ ✤ βHG ✤ ✤  κF   κG  Y F / pLanH Y qHF Y G / pLanH Y qHG one forms the coequalizer

α . γ LanHF X 0 LanH Y / Z β in rK, X s. This completes the construction of the coequalizer pH,γH ¨ κq. For completeness, we record here without proof the following result that is due to Foltz [12]; it will not be used in this paper.

Theorem 2.9. Let X has products. Then Diag˝pX q is cartesian closed with exponentials preserved by DX if, and only if, X is cartesian closed, and then EX preserves exponentials.

2.3. Left Kan extensions are cocartesian over Cat, invariance under the formation of colimits The argumentation given for Proposition 2.6 suggests the following characterization of left Kan extensions with target category X , when viewed as morphisms in Diag˝pX q: Proposition 2.10. A morphism pF, ϕq : pI,Xq Ñ pJ , Y q in Diag˝pX q is DX -cocartesian if, and only if, ϕ : X Ñ Y F exhibits Y as a left Kan extension of X along F . Proof. By definition, DX -cocartesianness of pF, ϕq means that, for all functors G : J Ñ K,Z : K Ñ X , one has a natural bijective correspondence

ψ χ X 3+ ZGF ú Y 3+ ZG that is facilitated by the condition χF ¨ ϕ “ ψ. Such correspondence is certainly in place if Y is a left Kan extension of X along F , with universal transformation ϕ. But this sufficient condition is also necessary, since one may simply consider G “ IdJ .

10 Proposition 2.10 lets us conclude that, for X cocomplete, the reflector colim : Diag˝pX q Ñ X of Proposition 2.3 maps DX -cocartesian morphisms to isomorphisms. In fact, one has the following more refined statement that appeared in essence in [36] (see the remark after the proof of Theorem 5.6 in [36]). Here we formulate it without the imposition of smallness or cocompleteness conditions on the participating categories:

Proposition 2.11. If Y – LanF X, then the canonical morphism colimX Ñ colimY is an isomor- phism, with either colimit existing when the other exists. Proof. For every object z in X , we need to establish a natural bijective correspondence

β X α 3+ ∆z ú Y 3+ ∆z . But this correspondence is readily provided by the correspondence of the proof of Proposition 2.10 where one has to consider Z “ ∆z, G “ IdJ and observe the trivial identity ∆z “ p∆zqF . Proposition 2.11 suggests the viewpoint that Kan extensions are partial colimits, a claim that is pursued in greater depth in [36].

2.4. Strictification of morphisms in diagram categories Other than containing X as a full subcategory, Diag˝pX q contains, of course, also the ordinary comma category Cat{X as a non-full subcategory. In fact, for small (!) X one has the following proposition, embedded in the proof of Lemma 7.13 of [2], with credit given to S. Lack. For arbitrary X , see Remark 2.13(2); we can still draw the essential conclusion of preservation of colimits (see Corollary 2.14) that is needed in the first proof of the CDF (see Section 4). Proposition 2.12. For a small category X , the inclusion functor Cat{X Ñ Diag˝pX q has a right adjoint. Proof. One defines a functor Strict : Diag˝pX q Ñ Cat{X , which transforms lax commutative triangles into strictly commutative triangles. It assigns to an object X in Diag˝pX q the comma category X Ó X, equipped with its domain functor which takes an object pu : a Ñ Xi, iq in X Ó X with i P I to the object a P X . On morphisms it is defined by

StrictpF,ϕq I F / J ÞÝÑ X Ó X / X Ó Y ❄ ❄ ⑦ ❋❋ ① ❄❄ ϕ:ùñ ⑦ ❋❋ ① ❄❄ ⑦⑦ ❋❋ ①① X ❄❄ ⑦ Y domX ❋❋ ①①domY ⑦ ⑦ ❋# ①| ① X Strict X where the functor StrictpF, ϕq takes an object pu : a Ñ Xi, iq in X Ó X with i P I to the object pϕi ¨ u : a Ñ Y pFiq, Fiq in X Ó Y . It is now a routine exercise to establish a natural isomorphism

˝ pCat{X qpX, domY q– Diag pX qpX, Y q in Set.

Remark 2.13. (1) With 2-cells α : F ùñ F 1 given by natural transformations α satisfying Y α “ 1X ,

11 F * I α ó 4 J ❅❅ ❅ 1 ⑥⑥ ❅❅ F ⑥⑥ X ❅❅ ⑥⑥Y ❅ ⑥~ ⑥ X Cat{X becomes a 2-category. In fact, it is the 2-subcategory of Diag˝pX q whose 2-cells are described in Remark 2.4(1). One easily confirms that the isomorphism at the end of the proof of Proposition 2.12 actually lives in Cat. Consequently, one has in fact a 2-adjunction Inclusion % Strict : Diag˝pX q Ñ Cat{X . (2) When X is large, i.e., a CAT-object, we still have a right adjoint, Strict, to the inclusion 2-functor CAT{X Ñ DIAG˝pX q, where DIAG˝pX q is defined like Diag˝pX q, except that the domain I of an object X : I Ñ X is not constrained to be small. We temporarily step into this higher universe to prove the following corollary, which will be used in the proof of Theorem 5.1. Corollary 2.14. For any category X , the inclusion functor Cat{X Ñ Diag˝pX q preserves all colimits. Proof. Trivially, the full embedding Cat Ñ CAT preserves all colimits and, hence, so does Cat{X Ñ CAT{X . By (a large version of) Proposition 2.12, a colimit taken in Cat{X is therefore also a colimit in DIAG˝pX q, and it trivially maintains that role in its “home” category Diag˝pX q. Remark 2.15. We note that, in the notation of 10.2, every small-fibred split fibration P : E Ñ B comes with a mate

Ð ˚ u P : B Ñ Diag˝pEq, pu : a Ñ bq ÞÑ ppu ,θ q : Ja Ñ Jbq, op which, when composed with the split fibration DE : Diag˝pEq Ñ CAT , reproduces the functor pΦP qop : B Ñ CAT op of 10.3. Although ΦP maintains sufficient information about P to reproduce P (as in Proposition 10.6), the mate P Ð may well be regarded as doing so more comprehensively.

2.5. The generic colimit for the standard construction of colimits Let X have coproducts. As is well known, the standard construction of the colimit of a small diagram X : I Ñ X proceeds by forming the diagram

f q Xi .0 Xi / C špu:iÑjq 5 ši 7 O g ❧❧ O ♣♣♣ ❧❧❧ ♣♣ su ❧❧ tj ♣♣ ❧❧t ♣♣κj ❧❧❧ i ♣♣ ❧❧❧ ♣♣ Xi ❧ / Xj ♣ Xu where the left coproduct, with injections su, runs through all morphism u of I and the second, with injections ti, through all objects i of I; the morphisms f and g satisfy f ¨su “ ti and g ¨su “ tj ¨Xu, for all u : i Ñ j in I. Then the colimit C of X with cocone κ : X Ñ ∆C exists in X if, and only if, the coequalizer q : ši Xi Ñ C of f,g exists in X , with κ and q determining each other by q ¨ ti “ κi for all i P I.

12 Example 2.16. (Generic colimit) For every small category I, considering the fibres of the codomain functor I2 Ñ I (see 10.6 (3)), one has a coequalizer diagram

F . Q / špu:iÑjq I{i 0 ši I{i I p˚q G in Cat, where Q restricts to the domain functor Qi, for every summand I{i. Indeed, by direct inspection one sees that the cocone given by the domain functors exhibits I as a colimit of the diagram Φ: I ÝÑ Cat, pu : i Ñ jq ÞÝÑ pu! : I{i Ñ I{jq, as follows: Given any cocone pHi : I{i Ñ Cqi, a functor H : I Ñ C with HQi “ Hi for all i P I must necessarily map every object i to Hip1iq and every morphism u : i Ñ j to the morphism Hj puq in C, since u in I may be seen as the Qj -image of the morphism u : u Ñ 1j in I{j, the domain u of which may be written as the object u “ u!p1iq in I{j. Conversely, putting Hu “ Hj u one similarly confirms the functoriality of H. Justifying its denomination, we now show in which sense Example 2.16 is generic:

Theorem 2.17. The standard construction of the colimit of any small diagram X : I Ñ X in a cocomplete category X is the image under the functor colim : Diag˝pX q Ñ X of a coequalizer diagram in Diag˝pX q whose underlying coequalizer diagram in Cat is given by diagram p˚q that depends only on the shape I of the diagram X.

Proof. Given X, the coequalizer diagram p˚q may be lifted trivially to become a coequalizer diagram in Cat{X :

F Q I{i .0 I{i / I špu:iÑjq ši ♦ ❙❙ G ♦♦♦ ❙❙❙ ♦♦ ❙❙❙ XQ ♦♦♦ XQF ❙❙❙ ♦♦ X ❙❙❙  ♦♦ ❙) X ♦w ♦ By Corollary 2.14, we actually have a coequalizer diagram in Diag˝pX q which, by Proposition 2.3, is preserved by colim : Diag˝pX q Ñ X . We claim that the emerging coequalizer diagram in X ,

f q colimXQF .0 colimXQ / colimX, g coincides (up to isomorphism) with the standard construction of the colimit of X in X . Indeed, in the notation of Example 2.16, XQ is the coproduct of the diagrams XQi in DiagpX q, so that we may compute

XQ / XQi / colimpši I{i X q – ši colimpI{i X q – ši Xi , where, for every object i P I, we were able to evaluate the colimit of XQi trivially as Xi, since 1i is terminal in I{i. With the colimit of XQF computed analogously, one now routinely checks that colim maps the functors F,G,Q (seen as morphisms in Diag˝pX q, and obtained by the standard construction of a colimit in Cat) to the morphisms f,g,q in X , as given by the standard construction of the colimit of X in X .

13 Remark 2.18. Other than p˚q of Example 2.16, there are alternative and quite natural presenta- tions of a small category I as a coequalizer in Cat, to which one may then apply the same procedure for the computation of the colimit of a diagram X : I Ñ X as the one used in the proof of Theorem 2.17. Consider, for example the coequalizer diagrams

P 2 2 1 2 P I ˆI I 1- I / I , P2

M P pI2q2 1- I2 / I , Q2 where P is the the codomain functor, and where M assigns to a commutative square in I, seen as an object of pI2q2, its diagonal morphism, seen as an object in I2. Unlike p˚q, both coequalizer diagrams are even contractible and, hence, absolute. However, the sections of P, P1,M making the coequalizers contractible are all easily seen to be confinal and, hence, under the procedure of the proof of Theorem 2.17, produce just coequalizer diagrams consisting of isomorphisms, for any diagram X : I Ñ X that has a colimit in X .

3. The Guitart adjunction It is hardly surprising that Diag˝pX q, constructed as a Grothendieck category over Cat, behaves 2-functorially in the variable X . But it is a nice twist that the assignment X ÞÑ Diag˝pX q (considered as a category over Cat) has a left adjoint, given again by the Grothendieck construction. This fact was stated by Guitart [20] (see also [21]) in 1-categorical terms. In what follows, we give some details in 2-categorical terms. A generalization is formulated, and proved, as Theorem 6.4.

3.1. Diag˝ is right adjoint to the Grothendieck construction Considering Cat as a 1-category and CAT as a (huge) 2-category, containing Cat as one of its objects, we form the 2-category CAT{Cat as in Remark 2.13(1). Then the 2-functor Diag˝ : CAT ÝÑ CAT{Cat assigns to a category X the fibration DX : Diag˝pX q Ñ Cat; it extends a functor T : X Ñ Y from ordinary to “variable” objects of X by post-composition with T , that is: Diag˝ assigns to T the CAT{Cat-morphism DX Ñ DY , given by the functor T p´q : Diag˝pX q ÝÑ Diag˝pYq, rpF, ϕq : pI,Xq Ñ pJ , Y qs ÞÝÑ rpF,Tϕq : pI,TXq Ñ pJ , T Y qs ; and it assigns to a natural transformation τ : T Ñ T 1 the natural transformation τp´q : T p´q Ñ 1 1 ˝ T p´q, given by pIdI,τXq : pI,TXq Ñ pI,T Xq for all objects pI,Xq in Diag pX q. The 2-functor

CAT ÐÝ CAT{Cat : ş˝ B assigns to a CAT{Cat-object Φ: Ñ Cat its dual Grothendieck category ş˝ Φ, and to a CAT{Cat- morphism Σ:Φ Ñ Ψ with Ψ: C Ñ Cat the functor

pΣ´, “q : ş˝Φ ÝÑ ş˝Ψ, rpu,fq : pa, xq Ñ pb,yqs ÞÝÑ rpΣu,fq : pΣa, xq Ñ pΣb,yqs. 1 A natural transformation σ : Σ Ñ Σ with Ψσ “ 1Φ is sent by ş˝ to the natural transformation 1 pΣ´, “q Ñ pΣ ´, “q whose component at an object pa, xq in ş˝ Φ is the morphism pσa, 1pΦaqxq : 1 pΣa, xq Ñ pΣ a, xq in ş˝ Ψ.

14 ˝ Theorem 3.1. The 2-functor ş˝ is left adjoint to the 2-functor Diag . Proof. It suffices to show that, for every category X and every functor Φ: B Ñ Cat, one has bijective functors l : CATpş Φ, X q Õ pCAT{CatqpΦ,DX q : l p ˝ q X X that are natural in and Φ. To this end, for a functor T : ş˝ Φ Ñ , one lets the functor Tˆ : B Ñ Diag˝pX q map an object a P B to the functor

1 1 Ta :Φa Ñ X , pf : x Ñ x q ÞÝÑ rT p1a,fq : pT pa, xq Ñ T pa, x qs.

Tˆ maps a morphism u : a Ñ b in B to the Diag˝pX q-morphism

Φa Φu / Φb ❈ ❈ u ④ ❈❈ ϕ :ùñ ④④ ❈❈ ④④ Ta ❈ ④ Tb ❈! ④} ④ X ,

u u where the natural transformation ϕ is defined by ϕx “ T pu, 1Φupxqq, for all x P Φa; clearly then, DX Tˆ “ Φ. Also, for a natural transformation τ : T Ñ T 1 one has the 2-cell τˆ : Tˆ Ñ Tˆ1, the components of which are the Diag˝pX q-morphisms

Id Φa Φa / Φa ❆❆ ⑥ ❆ τˆa:ùñ ⑥ ❆❆ ⑥⑥ ❆ 1 Ta ❆ ⑥⑥T ❆ ⑥~ ⑥ a X , defined by pτˆaqx “ τpa,xq, for all a P B, x P Φa. B X X ˇ X Conversely, for a functor Σ: Ñ Diagp q with D Σ “ Φ, one defines the functor Σ: ş˝ Φ Ñ , as follows. For u : a Ñ b in B, writing the DiagpX q-morphism Σu in the form Σu “ pΦu, ϕuq (as in the triangle on the left of the following diagram), one lets Σˇ map a morphism pu,fq : pa, xq Ñ pb,yq in ş˝ Φ to the composite morphism of the triangle on the right:

Φu Φa / Φb pΣbqpΦuqx ❇ u 6 P ❇❇ ϕ :ùñ ⑤⑤ ϕu ♥ PP Σb f ❇ ⑤ x ♥♥♥ PPp q ❇❇ ⑤⑤ ♥♥♥ PPP Σa ❇❇ ~ ⑤⑤ Σb ♥♥♥ PPP ⑤ ♥♥ Σˇpu,fq P( X pΣaqx “ Σˇpa, xq / Σˇpb,yq “ pΣbqy .

1 X Not surprisingly now, a natural transformation σ : Σ Ñ Σ with D σ “ 1Φ gives the natural ˇ ˇ 1 transformation σˇ : Σ Ñ Σ , defined by σˇpa,xq “ pσaqx, for all pa, xqP ş˝ Φ. All verifications proceed in a standard manner.

Let us make explicit how, for a functor Φ: B Ñ Cat, Theorem 3.1 provides an effective characterization of the category ş˝ Φ in the 2-category CAT. A lax cocone over Φ with vertex X is given by a family of functors Σa : Φa Ñ X pa P Bq and a family of natural transformations u ϕ :Σa Ñ ΣbpΦuq pu : a Ñ b in Bq, satisfying the conditions

1a v¨u v u ϕ “ 1Σa and ϕ “ ϕ pΦuq ¨ ϕ ,

15 B for all u : a Ñ b, v : b Ñ c in . We recall that the category ş˝ Φ is the vertex of the lax cocone over Φ, given by the functors 1 1 Ja :Φa Ñ ş˝Φ, ph : x Ñ x q ÞÑ p1a,hq : pa, xq Ñ pa, x q, and the natural transformations u u δ : Ja Ñ JbpΦuq, with δx “ pu, 1Φupxqq : pa, xq Ñ pb, pΦuqxq, for all u : a Ñ b in B and x P Φa (see 10.3). This lax cocone is initial amongst all lax cocones over Φ, in the following sense:

Corollary 3.2 (Lax Colimit Characterization of ş˝ Φ). For every lax cocone over Φ, given by X u X pΣa :Φa Ñ qaPB, pϕ :Σa Ñ ΣbpΦuqqu:aÑb, there is a uniquely determined functor T : ş˝ Φ Ñ u u with TJa “ Σa and Tδ “ ϕ , for all u : a Ñ b in B.

Φa Φu / Φb ❆❚❚ ❥ ❆ ❚❚❚ u ❥❥❥ ⑥ ❆ ❚❚❚ Ja δ :ùñ Jb ❥❥❥ ⑥ ❆❆ ❚❚❚❚ ❥❥❥❥ ⑥⑥ ❆❆ ❚❚❚❚ ❥❥❥❥ ⑥⑥ ❆❆ ❚❚❚* t ❥❥❥ ⑥⑥ ❆ ş Φ ❥ ⑥ ❆❆ u ˝ ⑥⑥ ❆❆ ϕ :ùñ ✤ ⑥⑥ Σ ❆❆ ⑥⑥Σ a ❆❆ ✤ ⑥⑥ b ❆❆ ✤ T ⑥⑥ ❆❆ ⑥⑥ ❆❆ ✤ ⑥⑥ ❆❆ ✤ ⑥⑥ ❆  ⑥~ ⑥ X u B ˝ Proof. The lax cocone pJaq, pδ q describes the adjunction unit J : Ñ Diag pş˝ Φq, and the corol- lary just paraphrases its universal property, as indicated by the diagram

B J / Diag˝p Φq Φ ❍ ş˝ ş˝ ❍❍ ✤ ❍❍ ❍ T p´q T ✤ p“Σq Σ ❍❍ q ❍❍$  ✤ Diag˝pX q X .

The dualization of Corollary 3.2 for a functor Φ: Bop Ñ Cat reads as follows: ˝ Corollary 3.3 (Oplax Colimit Characterization of ş Φ). For every oplax cocone over Φ, given by u ˝ pΣa :Φa Ñ X qaPB, pϕ :ΣapΦuq Ñ Σbqu:aÑb, there is a uniquely determined functor T : ş Φ Ñ X u u with TJa “ Σa and Tθ “ ϕ , for all u : a Ñ b in B. Remark 3.4. (1) A “direct” proof of Corollary 3.3 makes essential use of the (vertical, ΠΦ- u cartesian)–factorization pu,fq “ pu, 1Φupyqq ¨ p1a,fq“ θy ¨ Jaf of every morphism pu,fq : pa, xq Ñ ˝ pb,yq in ş Φ. Likewise for Corollary 3.2. (2) Of course, Corollaries 3.2 and 3.3 remain valid verbatim if the functor Φ is CAT-valued (rather than Cat-valued). As another important consequence of Theorem 3.1 we note: ˝ Corollary 3.5. Diag : CAT Ñ CAT{Cat preserves all (weighted) limits, and its left adjoint ş˝ preserves all (weighted) colimits.

16 3.2. Diag˝ has the structure of a normal pseudomonad on CAT That Diag˝ belongs to a (pseudo-)monad on CAT was already observed in [19]. But since a detailed exposition of this claim, even in a 1-categorical form, does not seem to be readily accessible, we outline the construction of the monad; a more detailed exposition and proof of the claim as given by this section’s header appears in [36]. Our 2-functor Diag˝ : CAT ÝÑ CAT, X ÞÝÑ Diag˝pX q, arises by post-composing the right-adjoint of Theorem 3.1 with the forgetful 2-functor CAT{Cat Ñ CAT. The full embedding EX : X Ñ Diag˝pX q of 2.2 may then be considered as the X -component of a (strictly) 2-natural transformation

˝ E : IdCAT ÝÑ Diag since, as one routinely confirms, every natural transformation α : F ñ F 1 : X Ñ Y satisfies Diag˝pF qEX “ EY F and Diag˝pαqEX “ EY α. In order to establish Diag˝ as the carrier of a pseudo-monad, we now define for every category X a functor M X : Diag˝pDiag˝pX qq ÝÑ Diag˝pX q. An object in Diag˝pDiag˝pX qq is a functor Σ: B Ñ Diag˝pX q with B small so that, with Φ:“ DX Σ: B Ñ Cat, for every object a in B one has a functor Σa :Φa Ñ X , and for every morphism u : a Ñ b in B a morphism pΦu, σuq : Σa Ñ Σb in Diag˝pX q. Considering Σ as a lax cocone with vertex X , by Corollary 3.2 we may represent Σ equivalently as a functor X ˇ X u M pΣq :“ Σ: ş˝pΦq ÝÑ , rpu,fq : pa, xq Ñ pb,yqs ÞÝÑ rΣbpfq ¨ σx :Σapxq Ñ Σbpyqs. A Diag˝pDiag˝pX qq-morphism pS, τq :Σ Ñ Ξ with codomain Ξ: C Ñ Diag˝pX q is given by a functor S : B Ñ C of small categories and a natural transformation τ :Σ Ñ ΞS whose component at a P B ˝ a is, in turn, given by a Diag pX q-morphism pRa,ρ q :Σa Ñ Ξa, as in

R Φa a / ΨpSaq ❆ ❆ a ❆❆ ρ :ùñ ②② ❆ ②② Σa ❆❆ ② Ξa ❆ | ②② X ② where Ψ:“ DX Ξ. Now one lets M X assign to pS, τq the Diag˝pX q-morphism pS,ˇ τˇq, as shown by

Sˇ S,DX τ S “ş˝p q B / C ÞÝÑ Φ / Ψ ● ş˝ ş˝ ●● ①① ❅ ●● τ:ùñ ①① ❅❅ τˇ:ùñ ⑦⑦ ●● ①① ❅❅ ⑦⑦ Σ ●● ①① Ξ Σˇ ❅❅ ⑦⑦ Ξˇ ●# ①{ ① ❅ ⑦~ ⑦ Diag˝pX q X where the functor Sˇ and the natural transformation τˇ are defined by ˇ a Spu,fq “ pSu,Rbfq and τˇpa,xq “ ρx, for all morphisms pu,fq : pa, xq Ñ pb,yq in ş˝Φ.

17 4. The twisted Fubini formula and cofibred colimits

4.1. The twisted Fubini formulae for colimits and limits We now exploit the adjunction of Theorem 3.1 for the computation of colimits of those diagrams in a category X whose shape is the dual Grothendieck category of a functor Φ: D Ñ Cat. Such a X X ˆ D ˝ X ˝ X diagram T : ş˝ Φ Ñ in corresponds equivalently to a diagram T : Ñ Diag p q in Diag p q with DX Tˆ “ Φ. As we will show in Theorem 4.2, the colimit of Tˆ facilitates the computation of the colimit of T . The essence of its proof lies in the next lemma, for which we use the following notation. Given Φ: D Ñ Cat and X , every functor F : D Ñ X gives us trivially the functor

Φ F˜ : D Ñ Diag˝pX q, pu : d Ñ eq ÞÝÑ p Φd u / Φe q ❇ ❇❇ ∆F u: ⑤⑤ ❇❇ ùñ ⑤⑤ ∆Fd ❇❇ ⑤⑤∆F e ❇ ⑤~ ⑤ X with DX F˜ “ Φ. (As usual, we use ∆ for constant-value functors and transformations.) For a natural 1 1 X transformation α : F Ñ F , one defines a natural transformation α˜ : F˜ Ñ F˜ with D α˜ “ 1Φ whose components are α˜d “ pIdΦd, ∆αdq. This defines the functor

l : CATpD, X q Ñ pCAT{CatqpΦ,DX q. r Lemma 4.1. For every functor Φ: D Ñ Cat and every category X , the functor l makes the diagram r

l CATpş Φ, X q p / pCAT{CatqpΦ,DX q ˝O – O ∆ l r ∆ X / CATpD, X q commute. If, for all d P D, the category X is Φd-cocomplete, then l has a left adjoint. r Proof. Checking the commutativity of the diagram is a routine matter. In order to construct a left adjoint l % l, assuming X to be Φd-cocomplete and using the same notation as in the proof of s r ˝ X Theorem 3.1, for a functor Σ: D Ñ Diag pX q with D Σ “ Φ, we define Σ:¯ D Ñ X on objects by ¯ Σd “ colimpΣd :Φd Ñ X q; this definition extends canonically to morphisms. (Of course, for X cocomplete, Σ¯ is the composite ˝ functor colim ˝ Σ, with colim % EX : X Ñ Diag pX q, as in Proposition 2.3.) For every functor F : D Ñ X one now obtains a natural bijection

CATpD, X qpΣ¯, F q Ñ pCAT{CatqpΦ,DX qpΣ, F˜q, which associates with a natural transformation α : Σ¯ Ñ F its mate α7 :Σ Ñ F˜, as follows: for every 7 d P D, the natural transformation αd : Sd Ñ ∆F d :Φd Ñ X is simply the composite transformation

∆αd Σd / ∆pcolimpΣdqq / ∆F d .

18 This confirms the adjunction. With the notation used in the proofs of Theorem 3.1 and of Lemma 4.1 one now obtains a general Fubini-type colimit formula that seems to have been proved first by Chachólski and Scherer [10, 40.2], as follows:

Theorem 4.2 (Twisted Fubini Colimit Formula). For a functor Φ: D Ñ Cat, let the category X D X X be Φd-cocomplete, for all d P . Then the colimit of any diagram T : ş˝ Φ Ñ exists in if, and only if, the colimit of the diagram D Ñ X , d ÞÑ colimpTˆ dq, exists in X , and in that case the two colimits coincide: pd,xqPş Φ dPD xPΦd colim ˝ T pd, xq – colim pcolim T pd, xqq. Proof. By the commutative diagram and the adjunction established in Lemma 4.1, cocones T ùñ X ˆ¯ D X X ∆X : ş˝ Φ Ñ correspond bijectively to cocones T ùñ ∆X : Ñ , and naturally so in X P . Consequently, the universal representation of either type of cocone exists if the other does, and they then coincide, up to a canonical isomorphism. D D Remark 4.3. (1) Note that, since all Φd pd P q are small, also ş˝ Φ is small when is small. (2) It is not hard to prove Theorem 4.2 “directly”, without recourse to Theorem 3.1 and Lemma 4.1: one may simply prove that the composite cocone

1 1 T pd , ´q ÝÑ colimxPΦd1 T pd , xq ÝÑ colimdPDpcolimxPΦd T pd, xqq

(as given by the right-hand side of the formula) is well defined and serves as a colimit cocone for T , and conversely. (3) As shown in [36], the Twisted Fubini Colimit Formula for a cocomplete category X is equivalently expressed by the fact that, for the pseudo-algebra pX , colim : Diag˝pX q Ñ X q with respect to the pseudo-monad Diag˝ (see 3.2), the diagram

Diag˝pcolimq Diag˝pDiag˝pX qq / Diag˝pX q

M X colim  colim  Diag˝pX q / X commutes up to isomorphism. (4) Although every colimit may be described as a coend (and conversely), we do not see a convenient way of deriving Theorem 4.2 from the interchange formula for coends (“Fubini”) for coends, as recorded in [33], Section IX.8, or [30], Section 1.3. The dualization of Theorem 4.2 reads as follows:

Corollary 4.4 (Twisted Fubini Limit Formula). For a functor Φ: Dop Ñ Cat, let the category X ˝ be Φd-complete, for all d P D. Then the limit of any diagram T : ş Φ Ñ X exists in X if, and only if, the limit of the diagram D Ñ X , d ÞÑ lim T pd, ´q, exists in X , and in that case the two limits coincide: ˝ limpd,xqPş Φ T pd, xq – limdPDplimxPΦd T pd, xqq.

Theorem 4.2 implies the “untwisted” Fubini formula that is recorded in Mac Lane’s book [33]:

19 Corollary 4.5 (Fubini (Co)Limit Formula). For every functor T : DˆE Ñ X into an E-cocomplete category X , the colimit of T exists in X if, and only if, the colimit of the D-indexed diagram d ÞÑ colimE T pd, ´q exists in X , and then the two colimits coincide: colimpd,eqPDˆE T pd, eq – colimdPDpcolimePE T pd, eqq. Likewise for limits. Proof. Let Φ: D Ñ Cat be the functor which has constant value E (formally assumed to be a small D E category). Then ş˝ Φ “ ˆ , and the assertion of the corollary follows from Theorem 4.2. 4.2. Cofibred colimits The shape of the diagram T considered in Theorem 4.2 is the dual Grothendieck category of a Cat-valued functor Φ. But by the (well-known) dual of Theorem 10.6(i), we may replace ş˝ Φ isomorphically by the domain of any (split) cofibred category over the domain of Φ, which we now call B (rather than D as in Theorem 4.2). Hence, instead of Φ we consider a split cofibration u P : E Ñ B with (in the notation of Section 10.3) cocleavages δ : Ja Ñ Jbu!, for all u : a Ñ b in B. The Twisted Fubini Colimit Formula then tells us that we may evaluate the colimit of an E-shaped diagram in any (sufficiently cocomplete) category by first computing the colimits “fibre by fibre” and then the colimit of the emerging B-shaped diagram in X . This means: Theorem 4.6 (Cofibred Colimit Theorem). Let P : E Ñ B be a (split) cofibration and T : E Ñ X be a diagram such that the colimits of all restricted diagrams TJb : Eb Ñ X pb P Bq exist in X . Then xPEb the left Kan extension LanP T : B Ñ X exists, given by pLanP T qb “ colim T x for all b P B, and one has the formula colimxPE T x – colimbPBpcolimxPEb T xq , with the colimit on the left existing in X if, and only if, the one on the right exists in X .

Proof. By the dual of Theorem 10.6 (i), with ΦP : b ÞÑ Eb we have the bijective functor

KP / E ✤ / ş˝ ΦP pb,yq y ✤ / u ΠΦP P rpu,fq: pa, xq Ñ pb,yqs rf ¨ δx : x Ñ ys   B B which “commutes over B”. An application of Theorem 4.2 to ΠΦP and TKP establishes precisely the claimed colimit formula, up to the negligible isomorphism KP . But we must still confirm the additional claim that the functor

L : B Ñ X , b ÞÑ colimTJb, constitutes a left Kan extension of T along P which, then, provides another proof of the. stated colimit formula, by Proposition 2.11. Indeed, in the notation of the proof of Theorem 3.1, we have L “ colimTK ; explicitly, for zP u : a Ñ b in B, the morphism Lu is defined by the commutative diagram

a TJ λ / ∆La a ✤ T δu ✤ ∆Lu b ✤  λ u!  TJbu! / ∆Lb

20 a Px where λ denotes the colimit cocone of TJa. Let us verify that putting λx :“ λx for all objects x P E defines a natural transformation λ : T Ñ LP . Indeed, for any morphism f : x Ñ y in E, u with u “ Tf and the (cocartesian,vertical) factorization of f “ νf ¨ δx , we have the commutative diagram λPx T x x / LP x

u T δx LP f λP y  u!x  T f Tu x / LPy ! ; ✇✇ T νf ✇✇ ✇✇λP y #  ✇✇ y Ty where the triangle commutes by naturality of the cocone λPy. This confirms the naturality of λ. Given any functor M : E Ñ X and a natural transformation ψ : T Ñ MP , then for every object a P B we have the cocone ψJa : TJa Ñ MPJa “ ∆Ma, which determines the morphism a χa : La Ñ Ma with ∆χa ¨ λ “ ψJa. Its naturality in a is easily checked, and it is clearly uniquely determined by ψ under the condition χP ¨ λ “ ψ. Using the standard construction of colimits in a cocomplete category, one concludes: Corollary 4.7. Given a cofibration P : E Ñ B of small categories, one has a presentation of the colimit of any diagram T : E Ñ X in a cocomplete category X in terms of a coegalisator diagram

f . q / š u:a b colimTJa 0 ša colimTJa colimT, p Ñ q g where TJa : Ea Ñ X is the restriction of T to the fibre of P at a P B. Example 4.8. Returning to the diagram of the generic colimit of Example 2.16 , Φ: I Ñ Cat,i Ñ I I I2 I {i, with a small category , then ş˝ Φ is the arrow category of , considered as a cofibred category over I (see 10.6 (3)). Keeping in mind the trivial existence and evaluation of pI{iq-shaped colimits in any category, the Twisted Fubini Colimit Formula implies that the colimit of any diagram T : I2 Ñ X may be computed by restricting T to the “diagonal” of I2, as in

2 pi,xqPI iPI colim T pi, xq– colim Tpi, 1iq, with either colimit existing when the other exists. Alternatively, this isomorphism of colimits follows also from the fact that the “diagonal” embed- 2 ding I Ñ I ,i ÞÑ pi, 1iq, is confinal.

5. The Colimit Decomposition Formula: three proofs 5.1. The basic formula and its short first proof Given a (small) diagram Φ: D Ñ Cat, we let

Kd :Φd ÝÑ K “ colim Φ pd P Dq denote its colimit cocone in the 1-category Cat. We re-state the formula given in the Introduction and first provide its short proof as given by Batanin and Berger in Lemma 7.13 of [2] (modulo a small correction):

21 Theorem 5.1 (Colimit Decomposition Formula). For every diagram X : K Ñ X in a cocomplete category X , the K-shaped colimit of X may be computed as the D-shaped colimit of the diagram given by the Φd-shaped colimits of XKd, for every d P D:

K dPD Φd colim X – colim pcolim XKdq.

Proof 1 of the CDF. Since, the domain functor Cat{X Ñ Cat reflects colimits, the given colimit in Cat gives us the colimit cocone

K Φd d / K ❇❇ ⑦ ❇❇ ⑦⑦ ❇❇ ⑦⑦ XKd ❇ ⑦ X ❇ ⑦~ ⑦ X in Cat{X . By Corollary 2.14, the inclusion functor Cat{X Ñ Diag˝pX q preserves this colimit, and then, by Proposition 2.3, it is again preserved by the left-adjoint functor colim : DiagpX q Ñ X . But this is precisely the claim of the theorem. The dualization of the theorem reads as follows:

Corollary 5.2 (Limit Recomposition Formula). As above, let K be the colimit of Φ in Cat, with colimit injections Kd. Then, for a diagram X : K Ñ X in a complete category X , the limit of X in X may be computed stepwise, according to the formula

K dPD Φd lim X – lim plim XKdq.

Proof. Apply Theorem 5.1 to Xop : Kop – colimdPDpΦdqop ÝÑ X op.

5.2. A generalized colimit decomposition formula and the second proof of the CDF Our second proof of Theorem 5.1 is based on (what turns out to be) a generalization of the decomposition formula. This generalization follows from the lifting of colimits along a bifibration with cocomplete fibres, as given in Corollary 10.4. By Proposition 2.6, for X cocomplete, we may apply this corollary to the bifibration DX : Diag˝pX q Ñ Cat, keeping in mind that cocartesian liftings are given by left Kan extensions in this case. Hence, in order to obtain the colimit of a diagram T : D Ñ Diag˝pX q, we follow the dualization of the construction given in the proof of Theorem 10.3 and, with Φ “ DX T : D Ñ Cat, form the colimit K of Φ in Cat, as in 5.1. Then, for every u : d Ñ e in D, writing the Diag˝pX q- u object T d as pΦd, Xd : Φd Ñ X q and the morphism Tu as pΦu, ϕ q, as in the triangle on the left,

K Φd Φu / Φe Φd d / K ❈ ❈ ❈ u ⑤ ❈ ⑥ ❈❈ ϕ :ùñ ⑤⑤ ❈❈ κd:ùñ ⑥⑥ ❈❈ ⑤⑤ ❈❈ ⑥⑥ Xd ❈ ⑤ Xe Xd ❈ ⑥ Ld ❈! ⑤} ⑤ ❈! ⑥~ ⑥ X , X ,

22 we form the left Kan extensions Ld :“ LanKd Xd. These extensions come with diagram morphisms as in the triangle on the right and form a D-shaped diagram in X K (the fibre of DX at K). Its dPD ˝ colimit X :“ colim Ld has colimit injections λd : Ld Ñ X. Finally then, the composite Diag pX q- morphisms pKd, λdKd ¨ κdq : pΦd, Xdq Ñ pK,Xq present pK,Xq as a colimit of T in Diag˝pX q. Remark 5.3. In the construction above, we may think of X as the joint left Kan extension of the functors Xd along Kd, characterized by the universal property that, for every functor Y : K Ñ X u and any family pµdqdPD of natural transformations µd : Xd Ñ YKd with µepΦuq ¨ ϕ “ µd for all u : d Ñ e in D, there is a unique natural transformation β : X Ñ Y with βKd ¨ λKd ¨ κd “ µd, for all d P D. Since the left adjoint functor colim of Proposition 2.3(1) preserves colimits, we obtain: Theorem 5.4 (General Colimit Decomposition Formula). For a cocomplete category X and any ˝ X diagram T : D Ñ Diag pX q with D T “ Φ, writing T d as pΦd, Xd : Φd Ñ X q for all d P D, one has K dPD Φd colim X – colim pcolim Xdq dPD dPD in X , where K “ colim Φd with colimit injections Kd in Cat and X “ colim pLanKd Xdq is a colimit of left Kan extensions in the functor category X K. Remark 5.5. Although the formula given in Theorem 5.4 may formally look similar to the CDF of Theorem 5.1, there is a crucial difference between the two statements: whereas in Theorem 5.4 X is formed with the help of the given diagrams Xd, in Theorem 5.1 one proceeds the other way around and defines Xd with the help of X as XKd. Here is the dualization of Theorem 5.4, obtainable with the dualization procedure given after Proposition ??. Corollary 5.6 (General Limit Recomposition Formula). For a complete category X and any di- op op agram T : D Ñ Diag˝pX q, d ÞÑ pΦd, Xd : Φd Ñ X q, with DX T “ Φ and Φ: D Ñ Cat, one has K dPD Φd lim X – lim plim Xdq dPD in X , where K “ colim Φd with colimit injections Kd in Cat, and where X “ limdPDpRanKd Xdq is a limit of right Kan extensions in X K. Let us now show how the General Colimit Decomposition Formula may be used to derive the CDF of Theorem 5.1:

Proof 2 of the CDF. We are given the diagrams Φ: D Ñ Cat and X : K Ñ X , where K is the colimit of Φ, with colimit injections Kd : Φd Ñ K. They allow us to form the diagram ˝ ˝ TX : D Ñ Diag pX q, sending u : d Ñ e in D to the Diag pX q-morphism

Φd Φu / Φe ❇❇ ⑤ ❇❇ 1:ùñ ⑤⑤ ❇❇ ⑤⑤ Xd“XKd ❇ ⑤ XKe“Xe ❇ ⑤~ ⑤ X

23 which actually lives in Cat{X ; so, TX factors through Cat{X . The assertion of Theorem 5.1 will follow from an application of Theorem 5.4 to TX , once we have shown the following lemma, formulated in the terminology introduced in Remark 5.3.

Lemma 5.7. X is the joint left Kan extension of the functors Xd along Kd, d P D. Proof. We have to check the relevant universal property, as described in Remark 5.3. To this end we consider a functor Y : K Ñ X and a family of natural transformations pµd : XKd Ñ YKdqdPD with µepΦuq “ µd for all u : d Ñ e in D and must present µd as µd “ βKd pd P Dq, for a unique natural transformation β : X Ñ Y . But this follows immediately from the fact that the functor p´q op X : Cat Ñ CAT transforms the colimit cocone pKd : Φd Ñ Kq in Cat into a limit cone ˚ K Φd pKd : X Ñ X q in CAT. Indeed, for X small, as a consequence of the cartesian closedness of Cat, this fact follows from the self-ajointness of X p´q : Catop Ñ Cat (see, for example, Proposition 27.7 in [1]); for X large, pro forma one has to step temporarily into the colossal category CAT to generate the needed natural bijective correspondence between families of natural transformations µd and natural transformations β, in the same way as in the small case.

5.3. Obtaining the Colimit Decomposition Formula via Fubini: the third proof of the CDF Our third proof of Theorem 5.1 takes advantage of the twisted Fubini formula of Theorem 4.2 and the confinality criterion of Theorem 11.3 for regular epimorphisms in CAT. Proof 3 of the CDF. Once again, we are given the diagrams Φ: D Ñ Cat and X : K Ñ X , where K is the colimit of Φ, with colimit injections Kd :Φd Ñ K. Via Theorem 5.8, which is of independent K interest, we have the confinal functor Q: ş˝Φ ÝÑ , pd, xq ÞÑ Kdx. With Theorem 4.2, one concludes K ş Φ dPD Φd colim X – colim ˝ XQ – colim pcolim XKdq, for every diagram X : K Ñ X in a cocomplete category X . Theorem 5.8. For a functor Φ: D Ñ Cat, with D small, the comparison functor K Q: ş˝Φ ÝÑ “ colim Φ, ppu,fq : pd, xq Ñ pe,yqq ÞÝÑ pKef : Kdx “ KepΦuqx Ñ Keyq. from the lax to the strict colimit of Φ (see Corollary 3.2) is a confinal regular epimorphism in Cat.

Proof. Let š Φ denote the coproduct of the categories Φd, d P D, in Cat, with injections Id : Φd Ñ š Φ. As in any cocomplete category, the colimit K of Φ may be constructed as the joint coequalizer C : š Φ Ñ K of the family of pairs pId, IeΦuq, indexed by all the morphisms u : d Ñ e in D. In accordance with the notation of Proposition 11.1, the equivalence relation induced by C on the set of objects of š Φ is the least equivalence relation «C with Idpxq «C IeΦupxq, for all u : d Ñ e and objects x P Φd. Hence, one has Idx «C Iey in š Φ precisely when there are morphisms ui : di Ñ ei, vi : di`1 Ñ ei in D and objects xi P Φdi with

u0 v0 u1 v1 un d “ d0 / e0 o d1 / e1 o ... dn / en “ e and x “ x0, Φuipxiq“ Φvipxi`1q pi “ 0, ..., n ´ 1q, Φunpxnq“ y. Since CId “ Kd for all objects d P D, the equivalence relation «Q induced by Q on the set of objects of ş˝Φ is described by

pd, xq«Q pe,yq ðñ Idpxq«C Iepyq. Hence, any two objects pd, xq, pe,yq in the same fibre of Q are linked by a string of morphisms

24 δu0 δv0 δu1 δun x0 x1 x1 xn pd, xq “ pd0, x0q / pe0, Φu0px0qqo pd1, x1q / ... / pen, Φunpxnqq “ pe,yq

Φ Q Qδui K 1 in ş˝ . These morphisms actually live in the same fibre of since xi “ di p Φuipxiqq and vi 1 K Qδxi`1 “ Kdi`1 p Φvipxi`1qq are identity morphisms in . Consequently, the fibres of Q are con- nected. In order for us to conclude with Theorem 11.3 that Q is confinal, we must also confirm that Q is actually a regular epimorphism. But this is a generally valid consequence of the fact that Q is a second factor of the regular epimorphism C, with the first factor E being easily recognized as an epimorphism in Cat:

ş Φ = ˝ ④ ❇❇ E ④④ ❇❇Q ④④ ❇❇ ④④ ❇❇ ④ C š Φ / K . D Explicitly, E is the functor with EId “ Jd :Φd Ñ ş˝Φ, x ÞÑ pd, xq, for all d P and x P Φd. Remark 5.9. We note that, unlike Q, the regular epimorphism C as above fails to be confinal, as soon as there exists a morphism with distinct domain and codomain in D.

6. Extending the Guitart adjunction, making Grothendieck a left adjoint

We return to the Guitart adjunction

Diag˝ - CAT l J CAT{Cat .

ş˝

B of Theorem 3.1. Realizing that ş˝ Φ is cofibred over the domain of any functor Φ: Ñ Cat, so that 2 the 2-functor ş˝ actually takes values in the morphism category CAT of CAT, in this section we 2 indicate how to extend the 2-functor Diag˝ and, in fact, the entire 2-adjunction, from CAT to CAT .

We also make precise that the 2-functor ş˝ may actually be defined on the “lax slice” CAT{{Cat, rather than on its subcategory CAT{Cat.

6.1. The diagram category Diag˝pP q of a functor P : E Ñ B. We start by giving a fibered version of the formation of the category Diag˝pX q, replacing X by a functor P , in such a way that the original construction is described as Diag˝pX Ñ 1q, with 1 the terminal category. Hence, for any functor P : E Ñ B we define the category Diag˝pP q, as follows:

´1 • objects are triples pa, I,Xq, with a an object in B and X : I Ñ Ea “ P paq a functor of a small category I;

• a morphism pu,F,ϕq : pa, I,Xq Ñ pb, J , Y q is given by a morphism u : a Ñ b in B, a functor F : I Ñ J , and a natural transformation ϕ : JaX ÝÑ JbY F with P ϕ “ ∆u (the constant transformation with value u); Ja,Jb are inclusion functors, as in

25 I F / J

X Y   Ea ϕ:ùñ Eb ❉❉ ③ ❉❉ ③③ ❉❉ ③③ Ja ❉ ③ Jb ❉❉" | ③③ E③ E

P  a u / b B

• composition: pv, G, ψq ¨ pu,F,ϕq “ pv ¨ u,GF,ψF ¨ ϕq. The category Diag˝pP q comes equipped with the obvious functors BP : Diag˝pP q Ñ B, pu,F,ϕq ÞÑ u, DP : Diag˝pP q Ñ Cat, pu,F,ϕq ÞÑ F, P ˝ E : E Ñ Diag pP q, x ÞÑ pP x, 1, ∆x:1 Ñ EPxq, pf : x Ñ yq ÞÑ pP f, Id1, ∆fq. EP is a full embedding which makes the diagram

EP E / Diag˝pP q ❂ ❂❂ ✈ ❂❂ ✈✈ ❂ ✈ P P ❂ ✈✈B ❂ { ✈✈ B ✈ commute. For DP and BP one easily proves:

Proposition 6.1. (1) DP is a split fibration. (2) BP is a (split) (co)fibration if, and only if, P has the corresponding property.

P Proof. (1) The D -cartesian lifting of F : I Ñ J at pb, J , Y q may be taken to be p1b,F, 1JbY F q : pb, I, Y F q Ñ pb, J , Y q. (2) For a (split) fibration P , with P -cartesian liftings denoted by θ, we claim that the BP - u cartesian lift of u : a Ñ b at pb, J , Y q may be taken to be pu, IdJ ,θ Y q. Indeed, the required universal property, as depicted by the left side of the diagram below, follows from a pointwise application of the corresponding P -cartesian property, as depicted on the right side:

26 H

K ❴❴❴❴❴❴❴❴❴ / J */ J H IdJ

Z u˚Y Y ψ θu    k ˚ Y Hk u Zk / u pY Hkq / Y Hk Ec ψ:ùñ Ea θ Y :ùñ Eb ❴❴❴ 7 ❉ ■ ❉❉ ✇✇ ■■ ①① ❉ ✇✇ ■■ ①① χ ❉❉ ✇✇ ■■ ①① k Jc ❉❉ ✇✇ Ja Ja ■■ ①① Jb ❉! ✇{ ✇ ■$ ①{ ① E χ:ùñ 9 E IdE c v / a u / b v u 8 c / a 4/ b w w Conversely, if BP is a (split) fibration, a P -cartesian lifting of u : a Ñ b at y may be realized as a P B -cartesian lifting of u : a Ñ b at pb, 1, ∆y : 1 Ñ EBq. When P or BP is a (split) opfibration, the proof proceeds analogously. We should clarify further the interdependency of the diagram constructions for categories and for functors. Trivially, for a category X , one has Diag˝pX q – Diag˝pX Ñ 1q. Less trivially, when P : E Ñ B is a split cofibration, with the help of the Grothendieck construction we may build ˝ ˝ Diag pP q from the categories Diag pEaq pa P Bq, as follows. Consider the functor

ΘP : B Ñ CAT, pu : a Ñ bq ÞÑ ru!p´q : Diag˝pEaq Ñ Diag˝pEbqs, 1 ˝ ˝ where the functor u!p´q maps pF, ψq : pI,Xq Ñ pJ ,X q in Diag pEaq to pF,u!ψq in Diag pEbq:

Id Id I F / J ) I F /) J ❄❄ ❄❄ ❄ ψ:ùñ ⑦ ❄ u!ψ:ùñ ⑦ ❄❄ ⑦⑦ ❄❄ ⑦⑦ X ❄ ⑦ 1 u X ❄ ⑦ 1 ❄❄ ⑦⑦ X ! ❄❄ ⑦⑦ u!X ⑦~ u! ⑦ Ea / Eb Proposition 6.2. If P : E Ñ B is a split cofibration, then, as a cofibred category over B, the ˝ category Diag pP q is isomorphic to the dual Grothendieck category ş˝ ΘP , by an isomorphism that maps objects identically. Proof. For every morphism u : a Ñ b in B, where a “ P x and b “ Py with x, y P E, one has the natural bijection u Ebpu!x, yq Ñ Eupx, yq, pf : u!x Ñ yq ÞÑ f ¨ δx , ´1 where Eupx, yq“ Epx, yqX P puq (see 10.2). Given functors X : I Ñ Ea, Y : J Ñ Eb, F : I Ñ J , exploiting the above bijection for x “ Xi, y “ YFi pi P Iq, one obtains the natural bijection u tψ | ψ : u!X Ñ Y F nat.tr.uÑtϕ | ϕ : JaX Ñ JbY F nat.tr., P ϕ “ ∆uu, ψ ÞÑ Jbψ ¨ δ X. Equivalently, writing pa,Xq instead of pa, pI,Xqq, we have the natural bijection ˝ u tψ | pu,F,ψq P pş˝ΘP qppa,Xq, pb, Y qqu Ñ tϕ | pu,F,ϕqP Diag pP qppa,Xq, pb, Y qqu, ψ ÞÑ Jbψ ¨ δ X. ˝ With objects kept fixed, this defines a bijective functor ş˝ ΘP Ñ Diag pP q which obviously com- mutes with the B-valued split cofibrations:

27 – / ˝ ş˝ ΘP Diag pP q ❉❉ ✇✇ ❉❉ ✇✇ Π ❉ ✇ P ΘP ❉❉ ✇ B ❉! { ✇✇ B ✇

˝ In the same way as one arrives at the definition of morphisms of Diag˝pX q once those of Diag pX q have been defined, one may also define the morphisms of the category Diag˝pP q; that is: keeping the same objects, but inverting the direction of the functor F while keeping the direction of the natural transformation ϕ in the definition of a morphism pu,F,ϕq in Diag˝pP q, one defines the morphisms of the category Diag˝pP q. The dualization of Proposition 6.2 then says that, when P is ˝ P a split fibration, Diag˝pP q is isomorphic to ş Θ as a fibred category over B, with P op ˚ Θ : B Ñ CAT, pu : a Ñ b in Bq ÞÑ ru p´q : Diag˝pEbq Ñ Diag˝pEaqs . 2 6.2. Review of the 2-categories CAT , CAT{{Cat and CAT{Cat. In order to extend the transitions E B P ˝ B B pP : Ñ q ÞÑ pD : Diag pP q ÞÑ Catq, pΦ: Ñ Catq ÞÑ pΠΦ : ş˝Φ Ñ q, 2 2-functorially, we form the 2-categories CAT , CAT{{Cat and CAT{Cat in a standard manner: 2 • The objects of CAT are functors P : E Ñ B of 1-categories (= CAT-objects); a morphism pS,T q : P Ñ Q is given by functors that make the square on the left of the diagram

T T * E / F E β ó 4 F T 1 P Q P Q S  S    B / C B α ó 4* C S1

commutative; and a 2-cell pα, βq : pS,T q ùñ pS1,T 1q is a pair of natural transformations α : S Ñ S1, β : T Ñ T 1 with Qβ “ αP ; their horizontal and vertical compositions are inherited from the 2-category CAT in each of the two components. • The objects of CAT{{Cat are functors Φ: B Ñ Cat of 1-categories; for Ψ: C Ñ Cat, a morphism pΣ, τq : Φ Ñ Ψ is given by a functor Σ: B Ñ C and a natural transformation τ : Φ Ñ ΨΣ; a 2-cell pσ, µq : pΣ, τq ùñ pΣ1, τ 1q is a natural transformation σ : Σ Ñ Σ1 together with a modification5 µ :Ψσ ¨ τ Ñ τ 1; this means that, for every object a P B, we 1 have a natural transformation µa : pΨσaqτa Ñ τa, such that, for every morphism u : a Ñ b in B, the following two natural transformations coincide: 1 1 1 1 1 pΨΣ uqµa : pΨΣ uqpΨσaqτa Ñ pΨΣ uqτa and µbpΦuq : pΨσbqτbpΦuq Ñ τbpΦuq. (These two transformations have the same domain and codomain, by the naturality of σ, τ.)

5For this term to make sense here, we consider the ordinary category B as a discrete 2-category (i.e., as having 1 identical 2-cells, so that Φ, ΨΣp q become 2-functors and τ, τ 1 2-natural transformations, for free.)

28 pΣ,τq Σ * B / C B pσ,µq ó 4 C ❈❈ ⑤ ❊❊ ③ ❈❈ τ:ùñ ⑤⑤ ❊❊ pΣ1,τ 1q ③③ ❈❈ ⑤ ❊ ③③ Φ ❈ ⑤⑤Ψ Φ ❊❊ ③③Ψ ❈! ⑤} ⑤ ❊" ③| ③ Cat Cat

The horizontal and vertical compositions are such that the CAT-valued assignment pΣ, τq ÞÑ Σ becomes a 2-functor. Important Note: CAT{{Cat has a richer 2-categorical structure than DIAG˝pX q (as defined in Remark 2.13(2)), which is due to the fact that, when considering 2-cells in CAT{{Cat, we are envoking the 2-categorical structure of Cat, in order to form modifications. These are all identities when Cat is replaced by a 1-category X , considered as a 2-category with identity 2-cells.

• CAT{Cat, as already defined in Remark 2.13(1), is the sub-2-category of CAT{{Cat whose 1 morphisms pΣ, τq :Φ Ñ Ψ satisfy τ “ 1Φ, so that Φ “ ΨΣ; consequently, a 2-cell σ :Σ ùñ Σ in CAT{Cat is just a natural transformation satisfying Ψσ “ 1Φ.

P Proposition 6.3. The transitions P ÞÑ D and Φ ÞÑ ΠΦ are the object assignments of 2-functors

˝ 2 2 Diag : CAT Ñ CAT{Cat and ş˝ : CAT{{Cat Ñ CAT . Proof. We just describe the assignments for morphisms and 2-cells and leave all routine verifications to the reader. Diag˝ assigns to a morphism pS,T q : P Ñ Q the functor

Σ “ Diag˝pS,T q : Diag˝pP q Ñ Diag˝pQq which, in turn, is given by the morphism assignment

ppu,F,ϕq : pa, I,Xq Ñ pb, J , Y qq ÞÑ ppSu,F,Tϕq : pSa, I,TaXq Ñ pSb, J ,TbY qq; here Ta is the restriction of T that makes the square of the diagram below commute.

Ta ˝ Σ ˝ Ea / FSa Diag pP q / Diag pQq ❏❏ t ❏❏ tt Ja JSa ❏❏ tt DP ❏❏ tt DQ  T  ❏$ tz t E / F Cat Trivially, the triangle on the right commutes as well, so that Σ is indeed a morphism in CAT{Cat. For a 2-cell pα, βq : pS,T q ùñ pS1,T 1q, one defines the natural transformation

σ “ Diag˝pα, βq :Σ “ Diag˝pS,T q ÝÑ Σ1 “ Diag˝pS1,T 1q by

1 1 1 σpa,I,Xq “ pαa, IdI ,βJaXq :Σpa, I,Xq “ pSa, I,TaXq ÝÑ Σ pa, I,Xq “ pSa , I,TaXq, ˝ for all objects pa, I,Xq in Diag pP q. Note that σpa,I,Xq is well defined since QpβJaXq“ αP JaX “ ∆αa.

29 I IdI / I

X X

 Ja Ja  Ea / E o Ea

1 1 Ta T β:ùñ T Ta   Ò  FSa / Fo FS1a F JSa JS1a Q α  Sa a / S1a C

2 ş˝ assigns to the (CAT{{Cat)- morphism pΣ, τq :Φ Ñ Ψ the CAT -morphism given by the square T / ş˝ Φ ş˝ Ψ

ΠΦ ΠΨ   B Σ / C , where the functor T maps a morphism pu,fq : pa, xq Ñ pb,yq to

T pu,fq “ pΣu, τbpfqq : T pa, xq “ pΣa, τapxqq ÝÑ T pb,yq “ pΣb, τbpyqq; note that the naturality of τ makes τbpfq have the correct domain, namely τbpΦupxqq “ ΨpΣuqpτapxqq. Given a 2-cell pσ, µq : pΣ, τq ùñ pΣ1, τ 1q in CAT{{Cat, we need to define a natural transformation 1 1 1 1 1 β : T Ñ T , where pΣ,T q“ ş˝pΣ, τq, pΣ ,T q“ ş˝pΣ , τ q, that satisfies ΠΨβ “ σΠΦ. To this end, 1 for pa, xq P ş˝ Φ, we put βpa,xq “ pσa, pµaqxq, which is a well-defined morphism T pa, xq Ñ T pa, xq 1 1 in ş˝ Ψ since pµaqx is a morphism Ψσapτapxqq Ñ τapxq in ΨpΣ aq.

6.3. The extended Guitart adjunction 2 We are now ready to prove that the restriction of the 2-functor ş˝ : CAT{{Cat Ñ CAT to CAT{Cat is left adjoint to Diag˝ of Proposition 6.3:

˝ 2 Theorem 6.4. ş˝ % Diag : CAT Ñ CAT{Cat is an adjunction of 2-functors. Proof. In generalization of the adjunction established in the proof of Theorem 3.1, for all functors Φ: B Ñ Cat,Q : F Ñ C we must, naturally in Φ and Q, establish functors

l 2 p / Q CAT pΠΦ,Qq n pCAT{CatqpΦ,D q l q that are inverse to each other. In doing so, we follow the notation used in the proof of Proposi- tion 6.3, with slight adjustments. In particular, we write pc,Zq instead of pc, K,Zq for objects of Diag˝pQq. “Ñ”: First, given the commutative square on the left, we must define the functor Σ “ pS,T q of { the commutative triangle on the right:

30 T Σ ş Φ / F B / Diag˝pQq ˝ ❆ ❆ ✉ ❆❆ ✉✉ ΠΦ Q ❆❆ ✉✉ Φ ❆ ✉✉ DQ   ❆ ✉z ✉ B S / C Cat ˝ Σ sends an object a P B to the Diag pQq-object pSa,Taq, with the functor

1 1 Ta :Φa Ñ FSa , pf : x Ñ x q ÞÑ pT p1a,fq : T pa, xq Ñ T pa, x qq, and a morphism u : a Ñ b in B is sent to the Diag˝pQq-morphism

u Σu “ pSu, Φu,Tδ q :Σa “ pSa,Taq ÝÑ Σb “ pSb,Tbq,

u where δx “ pu, 1Φupxqq : pa, xq Ñ pb, Φupxqq is the ΠΦ-cocartesian lift of u at x P Φa. The u u commutativity of the square above guarantees QpTδ q“ SΠΦδ “ ∆Su, as required.

Φa Φu / Φb ❉❉ ③ ❉❉ δu:ùñ ③③ Ta ❉❉ ③③ Tb Ja ❉ ③ Jb  ❉" ③} ③  F F Sa ş˝Φ Sb ❉❉ ③ ❉❉ ③③ ❉❉ T ③③ JSa ❉ ③ JSb ❉❉"  } ③③ F ③ We note that the emerging functor Σ satisfies DQΣ “ Φ, as required. To establish the functoriality of l, for a 2-cell pα, βq : pS,T q ùñ pS1,T 1q one defines the natural transformation p σ “ pα, βq :Σ “ pS,T q ÝÑ Σ1 “ pS1,T 1q { { { 1 1 ˝ 1 by σa “ pαa, IdΦa,βpa,´qq : pSa,Taq ÝÑ pS a,Taq in Diag pQq, with βpa,xq : Tax Ñ Tax px P Φaq. Q Note that one has D σ “ 1Φ, as required. “Д: Conversely now, given the functor Σ of the commutative triangle on the right of the above diagram, we must define the pair of functors pS,T q“ Σ, making the square on the left commute. q With BQ : Diag˝pQq Ñ C as in 6.1, we put S “ BQΣ. For objects a P B and x P Φa, having the functor Σa :Φa Ñ FSa, we put T pa, xq“ Σapxq. For a morphism u : a Ñ b in B, we may write the Diag˝pQq-morphism Σu :Σa Ñ Σb as

u u u Σu “ pSu, Φu, ϕ q :Σa Ñ Σb, with ϕ : JSa Σa Ñ JSb Σb Φu and Qϕ “ ∆Su.

For pu,fq : pa, xq Ñ pb,yq in ş˝ Φ, we can now define T pu,fq : T pa, xq Ñ T pb,yq as the composite arrow

ϕu Σbpfq Σapxq x / ΣbpΦupxqq / Σbpyq .

Its second morphism is Q-vertical since the functor Σb takes its values in FSb. Consequently,

u QT pu,fq“ Qpϕxq“ Su “ SΠΦpu,fq,

31 1 so that we indeed have QT “ SΠΦ. Clearly, l becomes a functor since, for a 2-cell σ : Σ ùñ Σ , 1q 1 1 Q we may define pα, βq “ σ : pS,T q“ Σ ùñ pS ,T q“ Σ , by putting α “ B σ, then writing σa as a q 1 q q a pαa, IdΦa, β :JSaΣa Ñ JS1aΣ aq and finally setting βpa,xq “ βx, for all objects pa, xqP Gr˝pΦq. Finally, we must confirm that the functors l, l are inverse to each other. First, given pS,T q : 2 ˜ ˜p q ˜ ΠΦ Ñ Q in CAT , let Σ “ pS,T q and pS, T q “ Σ. Then, trivially, S “ BQΣ “ S, and for ˜ { q ˜ all pa, xq P ş˝ Φ one has T pa, xq “ Σapxq “ Tapxq “ T pa, xq; likewise, T p1a,hq “ T p1a,hq for every morphism h in Φa. For an arbitrary morphism pu,fq : pa, xq Ñ pb,yq in ş˝pΦq, with its u u (cocartesian,vertical)-factorization pu,fq “ p1b,fq ¨ pu, 1Φupxqq“ Jbpfq ¨ δx , and with ϕ as in “Д, one obtains ˜ u u u T pu,fq“ Σbpfq ¨ ϕx “ Tbpfq ¨ Tδx “ T pJbpfq ¨ δx q“ T pu,fq. This shows T˜ “ T . Conversely, given Σ:Φ Ñ DQ in CAT{Cat, one argues very similarly that the transitions Σ ÞÑ Σ “ pS,T q ÞÑ Σ˜ “ pS,T q actually return Σ. Indeed, having u : a Ñ b and writing q { Σu “ pS, Φu, ϕuq as above, we deduce

u u u Tδx “ T pu, 1Φupxqq“ Σbp1Φupxqq ¨ ϕx “ ϕx, for all x P Φa and, hence, Σ˜u “ pSu, Φu,Tδuq “ pSu, Φu, ϕuq“ Σu. This then shows that l, l are inverse to each other on the objects of their (co)domains. Showing 2 that the same happens forp theq morphisms (i.e., the 2-cells in CAT and CAT{Cat) involves only easy routine checks.

Remark 6.5. The Guitart adjunction of Theorem 3.1 follows from the extended Guitart adjunction of Theorem 6.4, with the help of (the quite trivial) adjunction

2 Dom % !l : CAT Ñ CAT ,

2 where the right adjoint to the domain functor Dom (which exhibits CAT as fibered over CATq assigns to a category X the functor !X : X Ñ 1 (which happens to be a bifibration), considered 2 as an object of CAT . Post-composing this adjunction of 2-functors with the extended Guitart adjunction produces the Guitart adjunction, as the composite adjunction

˝ !l Diag , 2 - CAT l J CAT l J CAT{Cat . Dom ş˝

7. The Grothendieck equivalence via the extended Guitart adjunction

In this section we show how the 2-equivalence of split cofibrations P : E Ñ B and CAT-valued functors Φ: B Ñ CAT, with functorial and natural changes of the base category B permitted, may be obtained from the fundamental adjunction of Theorem 6.4. Initially we will restrict ourselves to the consideration of split cofibrations with small fibres. We also formulate the dualized statement for split fibrations.

32 7.1. Strictification of lax-commutative diagrams As the Grothendieck equivalence for strict cofibrations with small fibres involves the 2-category CAT{{Cat, rather than its subcategory CAT{Cat, our first goal is to map the latter 2-category into the former, with a right adjoint to the inclusion functor. To explain the importance of this step we 2 start with the observation, that the left-adjoint 2-functor ş˝ : CAT{Cat Ñ CAT of Theorem 6.4, assigning to the functor Φ: B Ñ Cat the split cofibration ΠΦ : Gr˝pΦq Ñ B, actually takes values in the (non-full) sub-2-category SCoFIBsf 2 of CAT . Its objects are split cofibrations with small fibres, and its morphisms pS,T q : P Ñ Q are 2 morphisms of split cofibrations P : E Ñ B,Q : F Ñ C, i.e., CAT -morphisms that respect the cocleavages: u Su Tbu! “ pSuq!Ta and Tδ “ δ Ta, 2 for all u : a Ñ b in B, where Ta : Ea Ñ FSa is a restriction of T ; 2-cells are as in CAT (see 6.2). As a consequence, the functor T : E Ñ F must transform the designated (P -cocartesion, P - vertical)-factorization of a morphism f : x Ñ y in E into the designated (Q-cocartesian, Q-vertical)- factorization of Tf : T x Ñ Ty in F:

P f SP f δx νf δTx νT f T r x / pPfq!pxq / y s “ r T x / pSPfq!pT xq / Ty s.

Even when we consider the extension of ş˝ to CAT{{Cat as in Proposition 6.3, the values still lie in SCoFIBsf , as one confirms easily. So, we have the commutative diagram

CAT{{Cat f ş q ▼▼ ˝ qq ▼▼Inclusion qqq ▼▼ qq ▼▼▼ qx qq ▼ SCoFIBsf o CAT{Cat ş˝ in which the bottom 2-functor has a right adjoint, Diag˝, as a consequence of Theorem 6.4. We want to show that the extended Guitart adjunction factors through CAT{{Cat, leading to a non-trivial factorization of Diag˝ as a composite of right-adjoint 2-functors. To this end, we now prove:

Proposition 7.1. The inclusion 2-functor CAT{Cat Ñ CAT{{Cat has a right adjoint, Strict, given by the strictification of lax-commutative diagrams. Proof. As an ordinary functor, Strict may be described by a slight adjustment of the 2-functor Strict established in Proposition 2.12, where the ordinary category X is now taken to be the 2-category Cat. As a result, the strictification needs to account for the greater supply of 2-cells in CAT{{Cat than that in DIAG˝pCatq. The action of Strict on objects and morphisms is now visualized by

S StrictpS,τq B / C ÞÝÑ CatóΦ / CatóΨ ❇❇ ⑥ ■■ ✉ ❇❇ τ:ùñ ⑥⑥ ■■ ✉✉ ❇❇ ⑥⑥ ■ ✉ Φ ❇ ⑥ Ψ domΦ ■■ ✉✉domΨ ❇ ⑥~ ⑥ ■$ ✉z ✉ Cat Strict Cat

33 with the lax comma category CatóΦ replacing the ordinary comma category CatÓΦ that (with X instead of Cat) was considered in Proposition 2.12. We write the objects of Cat ó Φ in the form pa, I,Xq with a P B and X : I Ñ Φa in Cat and let the functor StrictpS, τq map a morphism pu,F,ϕq : pa, I,Xq Ñ pb, J , Y q as is indicated by

F F I / J ÞÝÑ I / J

X ϕ:ùñ Y τaX τbϕ:ùñ τbY  Φu   ΨpSuq  Φa / Φb StrictpS, τq ΨpSaq / ΨpSbq

Strict maps a 2-cell pσ, µq : pS, τq Ñ pS1, τ 1q to the natural transformation Strictpσ, µq : StrictpS, τq Ñ StrictpS1, τ 1q, defined at the object pa, I,XqP CatóΦ as the morphism S 1 1 p trictpσ, µqqpa,I,Xq “ pSa, IdI,µaXq : pSa, I, τaXq Ñ pS a, I, τaXq in CatóΨ. For any functors Φ, Ψ as above, the needed adjunction isomorphism

pCAT{{CatqpΦ, Ψq – pCAT{CatqpΦ, domΨq of categories associates with pS, τq :Φ Ñ Ψ the functor

7 pS, τq : B Ñ CatóΨ, pu : a Ñ bq ÞÑ ppSu, Φu, 1τbΦuq : pSa, Φa, τaq Ñ pSb, Φb, τbqq ; a 2-cell pσ, µq : pS, τq Ñ pS1, τ 1q corresponds to the natural transformation pσ, µq7, defined at every a P B by 7 1 1 pσ, µqa “ pσa, IdΦa,µaq : pSa, Φa, τaq Ñ pS a, Φa, τaq. 7 7 Note that pσ, µq is indeed a 2-cell in CAT{Cat since domΨpσ, µq “ 1Φ. We omit the details of all the lengthy, but routine verifications.

7.2. Replacing diagrams by fibres

When P : E Ñ B is a split cofibration with small fibres, considered as an object of SCoFIBsf , rather than mapping it with the 2-functor Diag˝, we may now map P to its (covariant) fibre decomposition functor

´1 Fib˝pP q“ ΦP : B Ñ Cat, u! : Ea Ñ Eb “ P pbq, considered as an object of CAT{{Cat.

Proposition 7.2. The assignment P ÞÝÑ Fib˝pP q extends to a 2-functor Fib˝ : SCoFIBsf ÝÑ CAT{{Cat.

Proof. Keeping the notation of 7.1, we map a morphism pS,T q : P Ñ Q in SCoFIBsf to the CAT{{Cat-morphism Fib˝pS,T q “ pS, τq :ΦP Ñ ΦQ, where we define the natural transformation τ : ΦP Ñ ΦQS by restricting the functor T , via τa “ Ta : Ea Ñ FSa; the naturality of τ follows from pS,T q being a morphism of split cofibrations. 1 1 For a 2-cell pα, βq : pS,T q Ñ pS ,T q in SOFIBsf , we define the 2-cell

1 1 1 1 Fib˝pα, βq “ pα, µq : pS, τq Ñ pS , τ q“ Fib˝pS ,T q

34 in CAT{{Cat by specifying the modification µ : τ Ñ τ 1, as follows: for every a P B, we define the 1 natural transformation µa : pαaq!τa Ñ τa, by letting pµaqx be the Q-vertical factor of the canonical (Q-cocartesian, Q-vertical)-factorization of βx, for every x P Ea: pα q pT xq : a ! δαa ✉ Tx ✉✉ ✉✉ pµaqx ✉✉ ✉✉  T x / T 1x F βx Q α  Sa a / S1a C

αa 1 Naturality of every µa follows easily from the naturality of β and δ ; indeed, for every f : x Ñ x in Ea one has

1 αa 1 T f ¨ pµaqx ¨ δTx “ T f ¨ βx pdefinition of pµaqxq “ βx1 ¨ Tf pnaturality of βq αa “ pµaqx1 ¨ δTx1 ¨ Tf pdefinition of pµaqx1 q αa αa “ pµaqx1 ¨ pαaq!pTfq ¨ δTx pnaturality of δ q,

1 which implies the desired equality Taf ¨ pµaqx “ pµaqx1 ¨ pαaq!pTafq in FSa. 1 For µ to qualify as a modification, we must verify that the natural transformations pS uq!µa and µbu! coincide, for all u : a Ñ b in B. Indeed, by the naturality of α and the preservation of 1 1 cocartesian liftings by T and T , they have the common domain pS uq!pαaq!Ta “ pαbq!pSuq!Ta “ 1 1 1 pαbq!Tbu! and the common codomain pS uq!Ta “ Tbu!. Hence, it remains to be shown that, for 1 all x P Ea, we have the equality pS uq!ppµaqxq “ pµbqu!x in FSa, which follows from the following sequence of equalities that may be traced by chasing around this diagram:

35 αa δTx T x ❬❬❬ / pαaq!pT xq ❖ ❬❬❬❬ 1 1 ❖ ❬❬❬❬❬ S u¨αa ❚ S u ❖❖ ❬❬❬❬ δ ❚❚ δ α Tx ❖❖ ❬❬❬❬❬❬❬ Tx ❚❚❚ p aq!p q ❖❖❖ ❬❬❬❬❬❬❬❬ ❚❚❚❚ Su u ❖ αb¨Su ❬❬❬❬❬ ❚❚ δTx“T pδx q ❖ “δ ❬❬❬❬ ❚❚ ❖❖' Tx ❬❬❬❬❬❬- ❚❚* / 1 pSuq!pT xq“ T pu!xq α pS uq!pαaq!pT xq “ pαbq!pT pu!xqq δ b T pu!xq 1Tx pµaqx

1 1 T pu!xq pS uq!ppµaqxq pµbqu!x  β  T x x / T 1x ❖ ❯❯ 1 ❖❖ Su u ❯❯❯ S u 1 u ❖❖δTx“T pδx q ❯❯ δT 1x“T pδx q ❖❖❖ ❯❯❯❯ ❖❖❖ ❯❯❯❯ ❖❖'  β ❯❯❯*  u!x 1 1 1 pSuq!pT xq“ T pu!xq / pS uq!pT xq“ T pu!xq

α Sa a / S1a PP ❯❯❯ PP ❯❯❯ 1 PP Su ❯❯❯❯S u PPP ❯❯❯❯ PPP ❯❯❯❯ PP' αb ❯❯❯ Sb ❯/* S1b

1 1 S1u µ δS u¨αa S1u µ δS u δαa p q!pp aqxq ¨ Tx “ p q!pp aqxq ¨ pαaq!pTxq ¨ Tx pcomposition of cocleavagesq 1 1 S u αa S u “ δT 1x ¨ pµaqx ¨ δTx pnaturality of δ q 1 u 1 “ T pδx q ¨ βx pT preserves cocleavages; def. of pµaqxq u “ βu!x ¨ T pδx q pnaturality of βq

αb Su “ pµbqu x ¨ δ ¨ δ pdef. of pµbqu x; T preserves cocleavagesq ! T pu!xq Tx !

αb¨Su “ pµbqu!x ¨ δTx pcomposition of cocleavagesq 1 S u¨αa “ pµbqu!x ¨ δTx pnaturality of αq.

The remaining lengthy verifications for the 2-functoriality of Fib˝ may be left to the reader.

7.3. The Grothendieck Equivalence Theorem for split cofibrations We are now ready to formulate the following “folklore” theorem, a sufficiently elaborate proof of which does not seem to be easily available in the literature, at least not for a variable base category:

Theorem 7.3. The 2-functors ş˝ % Fib˝ : SCoFIBsf ÝÑ CAT{{Cat are adjoint 2-equivalences. Proof. We first establish an invertible 2-natural transformation κ : ˝ Fib Ñ 1Id . For a split ş˝ ˝ SOFIBsf 2 cofibration P : E Ñ B with small fibres, the CAT -morphism κP “ pKP , IdBq as depicted by

KP / E ş˝ ΦP

ΠΦP P   B IdB / B

36 is given by the “composition functor” u KP : ppu,fq : pa, xq Ñ pb,yqq ÞÑ pf ¨ δx : x Ñ yq which, in the dual situation, is displyed in Theorem 10.6. By design, κP is a morphism of split cofibrations and, quite trivially, invertible. To confirm its 2-naturality, we consider, in the notation 1 1 of Section 6.2, a 2-cell pα, βq : pS,T q ùñ pS ,T q : P ÝÑ Q in SCoFIBsf and the following diagram:

pS,T˜q ˜ + ΠΦP pα,βq ó 3 ΠΦQ pS1,T˜1q κP “pKP ,IdBq pKQ,IdC q“κQ pS,T q   + P pα,βq ó 3 Q pS1,T 1q

Here, the functors T,˜ T˜1 and the natural transformation β˜ : T˜ Ñ T˜1 are obtained by applying to the 2-cell pα, βq first Fib˝ and then ş˝, with both 2-functors leaving the “base” transformation α unchanged. According to the definitions given in the proofs of Propositions 6.3 and 7.2, one has ˜ T : ş˝ΦP Ñ ş˝ΦQ, rpu,fq : pa, xq Ñ pb,yqs ÞÝÑ rpSu,Tfq : pSa,Txq Ñ pSb,Tyqs,

˜ 1 1 αa βpa,xq “ pαa, pµaqxq : pSa,Txq Ñ pS a,T xq, with pµaqx ¨ δTx “ βx. ˜ αa Φ ˜ Now, from KQpβpa,xqq “ pµaqx ¨ δTx “ βx “ βKP pa,xq for all pa, xq P ş˝ P one has KQβ “ βKP , which is the crucial ingredient to concluding the equality

κQ ¨ ş˝pFib˝pα, βqq “ pα, βq ¨ κP , i.e., the 2-naturality of κ. Λ:1 Fib Next, we establish an invertible 2-natural transformation IdCAT {{Cat Ñ ˝ ˝ ş˝ which, at Φ Φ the CAT{{Cat-object Φ: B Ñ Cat, is the morphism ΛΦ “ pIdB, λ q :Φ Ñ ΦΠΦ , where λa :Φa Ñ pş˝ Φqa is the trivial bijective functor x ÞÑ pa, xq (see Theorem 10.6 in the dual situation). We check the 2-naturality of Λ and, in the notation of 6.2, consider a 2-cell pσ, µq : pΣ, τq ùñ pΣ1, τ 1q :Φ ÝÑ Ψ in CAT{{Cat. An examination of the definitions given in the proofs of Propositions 6.3 and 7.2, Φ Ψ show that, up to the identifications λ , λ , the composite functor Fib˝ ˝ ş˝ maps pσ, µq to itself. As a consequence one obtains the needed equality

ΛΨ ¨ pσ, µq“ Fib˝pş˝pσ, µqq ¨ ΛΦ. This completes the proof. The proof of Theorem 7.3 remains intact if we drop the condition of small-fibredness and consider the 2-category SCoFIB with objects all split cofibrations P : E Ñ B. These then correspond to CAT-valued functors, rather than to Cat-valued functors. Hence, one has to define the 2-category6 CAT{{CAT just as CAT{{Cat has been defined, to obtain the Grothendieck Equivalence Theorem [18] for split cofibrations:

6The notation CAT{{CAT is to be understood as analogous to the standard lax-comma category notation CAT{{Cat. While the latter is legitimate (as Cat is an object of CAT), the former is not; rather, CAT{{CAT has to be considered as a full subcategory of CAT{{CAT, with some higher-universe CAT that contains CAT as an object.

37 Corollary 7.4. The 2-functors ş˝ % Fib˝ : SCoFIB ÝÑ CAT{{CAT are adjoint 2-equivalences. We can finally compose the 2-equivalence of Theorem 7.3 with the 2-adjunction of Proposi- tion 7.1, to obtain an alternative proof for the left-adjointness of the restricted functor ş˝ of 7.1, without recourse to the fundamental adjunction of Theorem 6.4. However, the advantage of hav- ing established the fundamental adjunction of Theorem 6.4 first is that we may conclude that the ˝ right adjoints Diag and Strict ˝ Fib˝ (with Strict as in Proposition 7.1) coincide, up to 2-natural isomorphism—a fact that is a bit cumbersome to confirm when pursued directly. Either way, we have established the following important fact: Corollary 7.5. The diagram

CAT{{Cat 6 c Fib˝ Strict

˝ ş Diag Incl # v ˝ - SCoFIBsf m J CAT{Cat

ş˝ of adjunctions of 2-functors commutes. In particular, the 2-functor Diag˝ factors through the Grothendieck equivalence Fib˝.

7.4. The Grothendieck Equivalence Theorem for split fibrations It’s time for us to dualize Theorem 7.3 and Corollary 7.4 and to consider the sub-2-category

SFIB 2 of CAT , which is defined just like SCoFIB in 7.1, except that its objects P : E Ñ B are now split fibrations (rather than split cofibrations), and that its morphisms pS,T q : P Ñ Q preserve cleavages, so that ˚ ˚ u Su Tau “ pSuq Tb and T ϑ “ ϑ Tb, 2 for all u : a Ñ b in B, where Ta : Ea Ñ FSa is a restriction of T ; 2-cells are as in CAT (see 6.2). An application of the bijective 2-functor lop : CAT co Ñ CAT to objects, morphisms and 2-cells 2 2 2 of CAT gives rise to the bijective 2-functor lop : pCAT qco Ñ CAT with

rpα, βq : pS,T q ñ pS1,T 1q : P Ñ Qs ÞÝÑ rpαop,βopq : pSop,T opq ð ppS1qop, pT 1qopq : P op Ñ Qops.

It maps morphisms covariantly but 2-cells contravariantly, and it restricts to a bijective 2-functor

lop : SFIBco ÝÑ SCoFIB.

The bijective 2-functor lop : CAT co Ñ CAT gives also rise to the bijective 2-functor

lopp´q : CAT{{CAT ÝÑ CAT{{CAT, which post-composes every object, morphism and 2-cell with the functor lop: rpσ, µq:pΣ, τq ñ pΣ1, τ 1q:Φ Ñ Ψs ÞÝÑ rplopσ, lopµq:plopΣ, lopτq ñ plopΣ1, lopτ 1q:lopΦ Ñ lopΨs.

˝ Now we may define the 2-functor Fib as the dualization of the 2-functor Fib˝ of Corollary 7.4, that is: as the composite 2-functor given by the commutative diagram

38 Fib˝ SFIBco / CAT{{CAT O lop lopp´q  Fib SCoFIB ˝ / CAT{{CAT.

Chasing a split fibration P : E Ñ B around the lower path of the diagram shows that, as expected, Fib˝ maps P to its fibre representation ΦP : Bop Ñ CAT (as in 10.1), and morphisms and 2-cells of SFIB get mapped as indicated by

op op T pS ,τ q * op op op + op E β ó F ÞÝÑ B pα ,µ q ò C 4 ❏ 3 1 ❏❏ 1 op 1 op tt T ❏❏ ppS q ,pτ q q tt P Q ❏❏ tt S ΦP ❏❏ tt ΦQ   ❏% ty t B α ó *4 E Cat S1

Here the natural transformations τ : ΦP Ñ ΦQSop, τ 1 : ΦP Ñ ΦQpS1qop, analogously to the respective definitions for Fib˝ in the proof of Proposition 7.2, are determined as the restrictions 1 1 τa “ Ta : Ea Ñ FSa, τa “ Ta : Ea Ñ FS1a of T for all a P B, while the transformations µa : ˚ 1 1 τa Ñ pαaq τa comprising the modification µ : τ Ñ τ are defined as the Q-vertical factors in the αa factorization βx “ θT 1x ¨ pµaqx, for all x P Ea. ˝ Now, since Fib˝ is an equivalence of 2-categories, its dualization, Fib is one as well. Moreover, ˝ one obtains its quasi-inverse, ş , from the 2-functor ş˝, by the same dualization procedure that has ˝ produced Fib from Fib˝. Indeed, the dualization diagram

Fib˝ SFIBco n . CAT{{CAT ş˝ O lop lopp´q

 Fib˝ SCoFIB n / CAT{{CAT. ş˝ commutes at both, the Fib- and the ş-level. Hence, with Theorem 7.3 and Corollary 7.4 we conclude: ˝ Corollary 7.6. The 2-functors ş % Fib˝ : SFIBco ÝÑ CAT{{CAT are adjoint 2-equivalences. By ˝ ˝ co restriction to the small-fibred split fibrations they give the 2-equivalences ş % Fib : pSFIBsf q ÝÑ CAT{{Cat.

8. A left adjoint to the Grothendieck construction: free split cofibrations

In this section we give a novel proof of the essentially known fact7 that the composite 2-functor

ş˝ Incl 2 CAT{{CAT / SCoFIB / CAT

7See ncatlab.org, “Grothendieck construction”, for a proof of a corresponding statement with fixed base category.

39 has a left adjoint. Indeed, since ş˝ is a 2-equivalence (by Corollary 7.4), it suffices to show that the 2 inclusion 2-functor has a left adjoint, i.e., that SCoFIB is 2-reflective in CAT . This means that we must show how an arbitrary functor P can be freely “made” into a split cofibration, compatibly so with the relevant 2-categorical structures.

8.1. Free split cofibrations We define the 2-functor 2 Free : CAT Ñ SCoFIB, as follows. For every functor P : E Ñ B, the dual Grothendieck construction applied to the trivial slice functor P {l : B Ñ CAT gives us (in generalization of Example 10.6(3)) the split cofibration

l B B FreepP q :“ ΠP {l “ codP : ş˝P { “ P Ó ÝÑ of the comma category P Ó B. Here we therefore write an object in P Ó B as a pair ph, xq with x P E and h : P x Ñ a in B; a morphism pu,fq : ph, xq Ñ pk,yq is given by the commutative square on the left of the diagram

P f P 1 P f P x / Py “ P x x / P x / Py

h k h u¨h“ k¨P f k

 u   u  1b  a / b a / b / b .

The right part of the diagram describes the designated pcodP -cocartesian, codP -verticalq-factorization of pu,fq, so that one has: u u!ph, xq “ pu ¨ h, xq, δph,xq “ pu, 1xq : ph, xq Ñ pu ¨ h, xq, νpu,fq “ p1b,fq : pu ¨ h, xq Ñ pk,yq. The definition of Free on morphisms and 2-cells is also straightforward. In the notation of 6.2 , the action of Free is described by

1 1 ¯ ¯ 1 ¯1 rpα, βq : pS,T q ùñ pS ,T q : P ÝÑ Qs ÞÝÑ rpα, βq : pS, T q ùñ pS , T q : codP ÝÑ codQs, where

T¯ : P Ó B ÝÑ Q Ó C, ppu,fq : ph, xq Ñ pk,yqq ÞÝÑ ppSu,Tfq : pSh,Txq Ñ pSk,Tyqq,

¯ ¯ ¯1 1 1 βph,xq “ pαa,βxq : T ph, xq “ pSh,Txq ÝÑ T ph, xq “ pS h,T xq, ¯ for all objects ph : P x Ñ a, xq in P Ó B. Trivially, T transforms codP -cocleavages into codQ- cocleavages. Generalizing a corresponding result by [17] (in a fixedp-based and one-dimensional context), we now prove: 2 Theorem 8.1. The 2-functor Free is left adjoint to the inclusion SCoFIB ÝÑ CAT . 2 Proof. Given a CAT -object P : E Ñ B, we consider the functor

HP : E Ñ P Ó B, pf : x Ñ yq ÞÝÑ pPf,fq : p1Px, xq Ñ p1Py,yq

40 and claim that pIdB,HP q : P Ñ codP serves as the unit at P of the 2-adjunction Free % Incl. Hence, we show that, for every split cofibration Q : F Ñ C, the precomposition with pIdB,HP q provides, naturally in P and Q, a bijective functor

2 p´q ¨ pIdB,HP q : SCoFIBpcodP ,Qq ÝÑ CAT pP,Qq.

2 To establish its bijectivity on objects, we consider any CAT -morphism pS,T q : P Ñ Q and show ˜ ˜ that there is only one cocleavage-preserving functor T : P Ó B Ñ F with Q T “ S codP and ˜ T HP “ T . First, we observe that, for every morphism pu,fq : ph, xq Ñ pk,yq in P Ó B (as in the diagram above), one has the following:

u 1. pu,fq “ p1b,fq ¨ pu, 1xq“ νpu,fq ¨ δph,xq.

2. With the functor h! : pP Ó BqPx “ P {P x ÝÑ pP Ó Bqa “ P {a, the object ph, xq P P {a may be written as h!p1Px, xq, where p1Px, xqP P {P x.

3. Likewise, with the functor k! : P {Py ÝÑ P {b, the object pk,yq P P {b may be written as k!p1Py,yq, where p1Py,yq P P {Py, and the morphism νpu,fq in P {b may be written as k!p1Py,yq, with the morphism p1Py,fq : pPf,xq Ñ p1Py,yq in P {Py.

4. The morphism p1Py,fq in P {Py as in 3. may be written as p1Py,fq“ νHP f .

Consequently, for any functor T˜ : P Ó B Ñ F satisfying the above properties, one necessarily has

T˜ u,f T˜ k ν T˜ δu p q“ p !p HP f qq ¨ p h!p1Px,xqq ˜ Su “ pSkq!pT pνHP f qq ¨ δ ˜ T ph!pHP xqq Su “ pSkq!pνT˜ H f q ¨ δ ˜ p P q pShq!pT pHP xqq Sk ν δSu , “ p q!p T f q ¨ pShq!pTxq as in

δSu pShq!pTxq pSkq!pνT f q pShq!pT xq / pSuq!pShq!pT xq “ pSkq!pQpTfqq!pT xq / pSkq!pTyq F

Q 1  Sa Su / Sb Sb / Sb C Therefore, T˜ is unique. Conversely, setting

T˜ u,f Sk ν δSu , p q “ p q!p T f q ¨ pShq!pTxq one has to verify the needed properties for T˜. Showing that T˜ preserves the composition requires a careful application of the formulae for the pQ-cocartesian,Q-verticalq-factorization of composite arrows (see Section 10.3). Indeed, using the definition of T˜ for pu,fq : ph, xq Ñ pk,yq, pv,gq :

41 pk,yq Ñ pℓ,zq in P Ó B and the naturality of the transformation δSv, we obtain

T˜ v u,g f Sℓ ν δSv¨Su p ¨ ¨ q “ p q!p T g¨T f q ¨ pShq!pTxq Sℓ ν S Pg ν δSv δSu “ p q!r T g ¨ p p qq!p T f qs ¨ pSuq!pShq!pTxq ¨ pShq!pTxq Sℓ ν Sv Sk ν δSv δSu “ p q!p T gq ¨ p q!p q!p T f q ¨ pSkq!pSpP fqq!pTxq ¨ pShq!pTxq Sℓ ν δSv Sk ν δSu “ p q!p T gq ¨ pSkq!pTyq ¨ p q!p T f q ¨ pShq!pTxq “ T˜pv,gq ¨ T˜pu,fq.

The verification of the other needed properties of T˜ is straightforward. It remains to be shown that p´q ¨ pIdB,HP q is fully faithful. Given a 2-cell

pα, βq : pS,T q ùñ pS1,T 1q : P ÝÑ Q

2 in CAT as above, we should find a natural transformation β˜ : T˜ Ñ T˜1, unique with Qβ˜ “ α codP ˜ and βHP “ β. Since any object ph, xq in P Ó B gives rise to the morphism

h δx “ ph, 1xq : HP x “ p1Px, xq ÝÑ ph, xq in P Ó B, the naturality of any such β˜ and the preservation of cocleavages by T and T 1 force

˜ Sh ˜ ˜ h ˜1 h ˜ Sh βph,xq ¨ δTx “ βph, xq ¨ T pδx q“ T pδx q ¨ βHP x “ δT 1x ¨ βx .

Since, with the naturality of α, one has

˜ Sh 1 S1h Qpβph,xq ¨ δTxq“ αx ¨ Sh “ S h ¨ αPx “ QpδT 1x ¨ βxq ,

Sh ˜ we see that, by the Q-cocartesianess of δTx, the morphism βph,xq is necessarily the only F-morphism ˜ Sh S1h ˜ ˜ with βph,xq ¨ δTx “ δT 1x ¨ βx and Qpβph,xqq“ αPx. Conversely, taking this as the definition of βph,xq, one routinely shows that β˜ has the required properties. We may now compose the 2-adjunction of Theorem 8.1 with the Grothendieck equivalence of Corollary 7.4, as in

SCoFIB6 c Free Fib˝ K » Incl ş 2 w ˝ - ! CAT m K CAT{{CAT .

ş˝

2 Since the 2-functor Fib˝ ˝ Free assigns to the CAT -object P : E Ñ B the fibre representation functor of the functor codP : P Ó B Ñ B, the fibres of which are the slice categories P {b pb P Bq, we conclude: 2 Corollary 8.2. The 2-functor ş˝ : CAT{{CAT ÝÑ CAT has a left adjoint which maps the CAT- object P : E Ñ B to the CAT{{CAT-object P {l : B Ñ CAT, b ÞÑ P {b (considered in 8.1).

42 8.2. A network of global 2-adjunctions With the help of the following diagram we summarize the 2-adjunctions established in this and the previous sections:

Fib˝ SCoFIB n » . CAT{{CAT L ▲e 7 O ▲▲ ş ♣ ▲ Incl ˝ Incl ♣♣ ▲▲▲ ♣♣ ▲▲ ♣♣♣ ▲▲ Fib˝ ♣♣ . ♣ Free % Incl SCoFIBsf n » CAT{{Cat Incl r ş b S rrr ˝ trict rr J rrIncl ˝ ry r Diag Incl $ 2 . CAT nl J CAT{Cat ˝ 2 !l Diag ş˝ J J Dom ş - CAT q ˝ • The top horizontal adjunction displays the Grothendieck 2-equivalence between split cofibra- tions and CAT-valued functors (Corollary 7.4). It restricts to a 2-equivalence between split cofibrations with small fibres and Cat-valued functors (Theorem 7.3), as shown by the middle horizontal adjunction. The 2-equivalence Fib˝ decomposes a split cofibration into the “family”

of its fibres, indexed by its base category, while the Grothendieck construction ş˝ reassembles such gadgets. • The “vertical” 2-functor Free modifies a given functor by “freely adding cocartesian liftings” to it, showing that the totality of split cofibrations is 2-reflective amongst all functors (Theorem 8.1). The composition of this 2-adjunction with the top horizontal adjunction is described in Corollary 8.2. • The bottom horizontal 2-adjunction relates arbitrary functors (rather than split cofibrations)

to Cat-valued functors (Theorem 6.4). Its left adjoint, ş˝, trivially factors through the name- sakes above it. Not being able to functorially relate the fibres of an arbitrary functor with each other, the right adjoint, Diag˝, relates the totality of all small diagrams over the fibres with each other, rather than the fibres themselves. Regarding categories X as functors !X : X Ñ 1, the lower horizontal 2-adjunction reduces to the lower right diagonal 2-adjunction, as first considered in its ordinary form by Guitart (Theorem 3.1).

• The fibre-representation 2-functor, Fib˝, of the middle horizontal equivalence maps morphisms of its domain to lax-commutative diagrams over Cat, while the right adjoint of the lower horizontal adjunction, Diag˝, maps morphisms to strictly commutative diagrams over Cat. In fact, the restriction of the latter 2-functor factors through the former (up to isomorphism), by the strictification 2-functor, Strict, which is right adjoint to a (non-full) inclusion functor (Proposition 7.1, Corollary 7.5)

9. Diagram categories as 2-(co)fibred categories over Cat 9.1. Hermida-Buckley 2-fibrations In this supplementary section we pay tribute to the fact that DX : Diag˝pX q Ñ Cat and op DX : Diag˝pX q Ñ Cat are 2-functors (see Remarks 2.4) and investigate under which conditions

43 X on X (if any), D or DX may be a (co)fibration as such. For that we employ Buckley’s [9] improved version of Hermida’s [22] notion of 2-fibration. We recall the relevant definitions:

Definition 9.1. Let P : E Ñ B be a 2-functor. (1) A 1-cell f : x Ñ y in the 2-category E is P -2-cartesian if, for all objects C in E, the diagram

Epz,fq Epz, xq / Epz,yq

Pz,x Pz,y   BpPz,Pxq / BpPz,Pyq BpPz,Pfq

is a pullback in Cat.

1 (2) A 2-cell α : f Ñ f : x Ñ y in E is P -2-cartesian if it is Px,y-cartesian, with respect to the ordinary functor Px,y : Epx, yq Ñ BpPx,Pyq. (3) P is a (cloven) 2-fibration if

(a) for all 1-cells u : a Ñ b in B and y objects in Eb, there is a (chosen) P -2-cartesian lifting f : x Ñ y in E, so that P x “ a and Pf “ u;

(b) for all objects x, y P E, the ordinary functor Px,y : Epx, yq Ñ BpPx,Pyq is a (cloven) fibration; (c) P -2-cartesianness of 2-cells in E is preserved by horizontal composition.

(4) P is a 2-cofibration if P coop : Ecoop Ñ Bcoop is a 2-fibration. Remark 9.2. (1) By definition, the 1-cell f : x Ñ y in E is P -2-cartesian if, and only if, for all objects z P E, the functor

1 1 1 Epz, xq Ñ BpPz,PxqˆBpPz,Pyq Epz,yq, pτ : t Ñ t q ÞÑ pP τ : Pt Ñ Pt , fτ : ft Ñ ft q, is an isomorphism of categories. Its bijectivity on objects is equivalent to f being P -cartesian in the ordinary sense, while its full faithfulness adds the following condition to the 1-categorical notion: for all 2-cells ζ : w ñ w1 : Pz Ñ P x in B and ρ : h ñ h1 : z Ñ y in E with pPfqζ “ Pρ, there is a unique 2-cell τ : t ñ t1 : z Ñ x in E with P τ “ ζ and fτ “ ρ. (2) By definition, P -2-cartesianess of a 2-cell α : f ñ f 1 : x Ñ y in E means that, for all 1-cells k : x Ñ y, the map

1 Epx, yqpk,fq ÝÑ BpPx,PyqpPk,PfqˆBpPx,PyqpPk,Pf 1q Epx, yqpk,f q, µ ÞÑ pPµ,σ ¨ µq, is bijective, that is: for all 2-cells γ : P k ñ Pf and λ : k ñ f 1 in, respectively, B and E, with P α ¨ γ “ P λ, one has Pµ “ λ and α ¨ µ “ λ, for a unique 2-cell µ : k Ñ f in E. (3) P is a 2-fibration if, and only if,

(a) for every 1-cell u : a Ñ Py in B with y in E, there is a P -2-cartesian lifting f : x Ñ y in E with Pf “ u;

44 (b) for every 2-cell ξ : u ñ Pf 1 : P x Ñ Py in B with a (P -2-cartesian) 1-cell f 1 : x Ñ y in E, there is a P -2-cartesian lifting α : f ñ f 1 : x Ñ y with P α “ ξ; (c) for all 1-cells t : z Ñ x, s : y Ñ w and 2-cells α : f ñ f 1 : x Ñ y in E, if α is 2-cartesian, so are αt : ft Ñ f 1t and sα : sf Ñ sf 1. (Of course, since P -cartesianess of 2-cells is closed under vertical composition, closure under (horizontal) pre- and post-composition with 1-cells suffices to make the property closed also under horizontal composition.)

Remark 9.3. The definition of (in a quite obvious sense) split 2-fibration as given above is moti- vated by the fact that a 2-fibration is, via a 2-categorical Grothendieck construction, 3-equivalently represented by a 2-functor Bcoop Ñ 2Cat; see [9]. In fact, Buckley [9] proved a more general result at the bicategorical (rather than the 2-categorical) level.

9.2. How to consider diagram categories as 2-fibred or 2-cofibred over Cat op op Theorem 9.4. (1) The 2-functor DX : pDiag˝pX qq Ñ Cat is a 2-fibration, for every category X . (2) If the category X is cocomplete, then DX : Diag˝pX q Ñ Cat is a 2-cofibration.

op Proof. (1) Recall that a morphism pF, ϕq : pI,Xq Ñ pJ , Y q in pDiag˝pX qq is given by small categories I, J , functors F,X,Y , and a natural transformation ϕ, as in the triangle below on the left, and a 2-cell α : pF, ϕq ùñ pF 1, ϕ1q is given by a natural transformation α : F Ñ F 1 with ϕ “ ϕ1 ¨ Y α :

pF,ϕq F * I / J I α ó 4 J ❄❄ ⑦ ❅❅ ⑥ ❄ ðù:ϕ ⑦ ❅❅ pF 1,ϕ1q ⑥ ❄❄ ⑦⑦ ❅ ⑥⑥ X ❄ ⑦ Y X ❅❅ ⑥⑥Y ❄ ⑦~ ⑦ ❅ ⑥~ ⑥ X X

op op Now, given F : I Ñ J in Cat and pJ , Y q P pDiag˝pX qq , we have the trivial DX -cartesian lifting pF, 1Y F q : pI, Y F q Ñ pJ , Y q at the 1-category level (Proposition ??). To show that pF, 1FY q is op 1 DX -2-cartesian, it suffices to consider a natural transformation ζ : G ùñ G : K Ñ I and a 2-cell ρ : pH,γq ùñ pH1,γ1q : pK,Zq Ñ pJ , Y q with F G “ H, FG1 “ H1, F ζ “ ρ, and show that 1 1 op ζ : pG, γq ùñ pG ,γ q : pK,Zq Ñ pI, Y F q is actually a 2-cell in pDiag˝pX qq . But this is trivial: the given identity γ1 ¨ Yρ “ γ may just be restated as the needed identity γ1 ¨ pY F qζ “ γ. Next, in oder to verify property (b) of Remark 9.2, we consider a 1-cell pF 1, ϕ1q : pI,Xq Ñ pJ , Y q op 1 in pDiag˝pX qq and a 2-cell α : F ùñ F : I Ñ J in Cat and show that the emerging 2-cell 1 1 1 op 1 1 α : pF, ϕ :“ ϕ ¨ Y αq ùñ pF , ϕ q is DX -2-cartesian. Indeed, given 2-cells λ : pK,κq Ñ pF , ϕ q and op 1 γ : K Ñ F in, respectively, pDiag˝pX qq and Cat with α ¨ γ “ λ, the given identity ϕ ¨ Y λ “ κ translates to ϕ ¨ Yγ “ κ, thus making γ : pK,κq Ñ pF, ϕq a 2-cell in Diag‹pX qq, as desired. Finally, to verify property (c), for 1-cells pT, ηq : pK,Zq Ñ pI,Xq, pS,εq : pJ , Y q Ñ pL, W q and op 1 1 the DX -2-cartesian 2-cell α : pF, ϕq ùñ pF , ϕ q : pI,Xq Ñ pJ , Y q as above, we must show that the horizontal composites

αT Sα pFT,η ¨ ϕT q 3+ pF 1T, η ¨ ϕ1T q, pSF,ϕ ¨ εF q 3+ pSF 1, ϕ1 ¨ εF 1q

op 1 are DX -2-cartesian as well. Indeed, from ϕ ¨ Y α “ ϕ one obtains immediately

η ¨ ϕ1T ¨ Y pαT q“ η ¨ ϕT, ϕ1 ¨ εF 1 ¨ W pSαq“ ϕ1 ¨ Y α ¨ εF “ ϕ ¨ εF,

45 as desired. (2) We now consider pI,Xq P Diag˝pX q and F : I Ñ J in Cat and form the DX -cocartesian lifting pF, ϕq : pI,Xq Ñ pJ , Y q at the 1-categorical level, so that ϕ : X Ñ Y F presents Y as a left Kan extension of X along F (Proposition 2.6). To show that pF, ϕq is DX -2-cocartesian, given any 2-cells τ : G ùñ G1 : J Ñ K and ρ : pH,γq ùñ pH1,γ1q : pI,Xq Ñ pK,Zq with GF “ H, G1F “ H1, τF “ ρ, we let β : Y Ñ ZH, β1 : Y Ñ ZH1 be determined by the identities βF ¨ ϕ “ γ, β1F ¨ ϕ “ γ1 and must then confirm that τ : pG, βq ùñ pG1,β1q : pJ , Y q Ñ pK,Zq is a 2-cell in Diag˝pX q. But this is straightforward, since from

pZτ ¨ βqF ¨ ϕ “ ZτF ¨ βF ¨ ϕ “ ZτF ¨ γ “ Zρ ¨ γ “ γ1 “ β1F ¨ ϕ one deduces the desired identity Zτ ¨ β “ β1. Finding a DX -2-cocartesian lifting for a 2-cell α : F ùñ F 1 in Cat that comes with a 1-cell pF, ϕq : pI,Xq Ñ pJ , Y q in Diag˝pX q proceeds as in (1): one just puts ϕ1 :“ Y α¨ϕ and easily shows that, given 2-cells λ : pF, ϕq ùñ pK,κq in Diag˝pX q and χ : F 1 Ñ K in Cat with χ ¨ α “ λ, then χ : pF 1, ϕ1q Ñ pK,κq actually lives in Diag˝pX q. Likewise, also the easy proof that pre- and post- composition with 1-cells in Diag˝pX q preserves the DX -2-cocartesianess of α : pF, ϕq ùñ pF 1, ϕ1q proceeds as in (1).

op Remark 9.5. We note that the 2-fibration DX is split, in the obvious sense that the induced functor

op ˚ op op Π op : Cat Ñ CAT, pF : I Ñ J in Catq ÞÝÑ pF : rJ , X s ÑrI, X s , Y ÞÑ Y F q, DX

op is actually a 2-functor. It assigns to a small category I its fibre in pDiag˝pX qq , which is precisely op op the category rI, X s . Furthermore, for all objects pI,Xq, pJ , Y q in pDiag˝pX qq , the fibration

op pDiag˝pX qq pX, Y q ÝÑrI, J s, pF, ϕq ÞÑ F is actually discrete.

10. Appendix 1: Grothendieck fibrations and the Grothendieck construction

10.1. Cartesian morphisms Given a functor P : E Ñ B, a morphism f : x Ñ y in E is a lifting (along P ) of a morphism u : a Ñ b in B if Pf “ u. The lifting f is P -cartesian8 if every diagram of solid arrows below can be filled uniquely, as shown:

h z ❴❴❴❴❴ / x )/ y E t f Ph P )  Pz / P x / Py B w P f

8Following [18], the older literature, such as [17], uses strong cartesian instead.

46 Thus, if h: z Ñ y in E and w : Pz Ñ P x in B satisfy Pf ¨ w “ Pg, then there is exactly one morphism t: z Ñ x in E with f ¨ t “ g and Pt “ v; equivalently, for every object z in E, the square

Epz,fq Epz, xq / Epz,yq

Pz,x Pz,y   BpPz,Pxq / BpPz,Pyq BpPz,Pfq is a pullback diagram in Set. We note that, for any functor P ,a P -cartesian lifting f of u is an isomorphism if, and only if, u is an isomorphism. The class CartpP q of P -cartesian morphisms in E contains all isomorphisms of E, is closed under composition, and satisfies the cancellation condition pg ¨ f P CartpP q ùñ f P CartpP qq whenever g is monic or P -cartesian. Moreover, the class CartpP q is stable under those pullbacks in E which P transforms into monic pairs; in particular, CartpP q is stable under the pullbacks that are preserved by P .

10.2. Grothendieck fibrations

For an object b in B we denote by Eb the fibre of P : E Ñ B at b; this is the (non-full) subcategory of E of all morphisms in E that are liftings of 1b. Hence, for the inclusion functor Jb : Eb Ñ E the functor PJb “ ∆b is constant. The morphisms in the fibres of P are also called P -vertical. The functor P is a (Grothendieck) fibration if, for every morphism u: a Ñ b in B and every object y in Eb, there is a P -cartesian lifting f : x Ñ y in E. Since such a lifting is unique up to isomorphism when considered as an object in the slice category E{y, one may call f the P -cartesian lifting of u at y. In fact, we will assume throughout that our fibrations are cloven; this means, that a choice of P -cartesian liftings, also called a cleavage, has been made for all u and y. We denote the chosen u ˚ P -cartesian lifting of u : a Ñ b in B at y P Eb by θy : u pyq Ñ y. With this notation one sees immediately that a functor P is a fibration if, and only if, for every object y in E, the induced functor Py : E{y ÝÑ B{Py of the slice categories has a right adjoint right inverse (rari), namely θy (see [17]). For a fibration P : E Ñ B, one also calls E fibred over B. Every morphism f : x Ñ y in E then has a (P -vertical, P -cartesian)-factorization, as in x ✤ ❋❋ ❋❋ f ǫf ✤ ❋❋ ✤ ❋❋  ❋❋ ˚ / " u pyq u y , θy where u “ Pf, and where the P -vertical morphism ǫf is uniquely determined by f. In fact, there is, for all morphisms u : a Ñ b and objects x P Ea, y P Eb a natural bijective correspondence

˚ Eupx, yq– Eapx, u pyqq,

´1 where Eupx, yq“ Epx, yqX P puq. This correspondence means precisely that, for all a P B, the full embedding Ia : Ea ãÑ aÓP, x ÞÑ p1a : a Ñ P xq,

47 ˚ has a right adjoint; it sends u : a Ñ Py, seen as an object of aÓP , to u y. (An Ia-universal arrow at u : a Ñ Py is also called a P -precartesian lifting of u and y; obviously, when P is a fibration, P -precartesian liftings are P -cartesian; see [5].) We call Ia the comma insertion of the fibre Ea. The P -vertical morphisms and, more generally, the E-morphisms which are mapped by P to isomorphisms, are orthogonal to P -cartesian morphisms. As a consequence one obtains that a functor P : E Ñ B is a fibration if, and only if, P is an iso-fibration (that is:, if every isomorphism ´1 u : a Ñ b in B admits a P -cartesian lifting at every y P Eb), and if pP pIso Bq, CartpP qq is an orthogonal factorization system of E; the second property means equivalently that P is a Street fibration [40]. For every morphism u : a Ñ b in B, the domains of the P -cartesian liftings of u at the objects ˚ u ˚ of Eb give the object assignment of a functor u : Eb Ñ Ea that makes θ : Jau Ñ Jb a natural 1 ˚ : E u transformation; for a morphism j y Ñ y in b one has u pjq“ ǫj¨θy .

u ˚ θ u ˚ y Ea o Eb u pyq / y ❄ ❄❄ θu:ùñ ⑧ ❄ ⑧ ˚ ❄ ⑧⑧ u pjq j Ja ❄ ⑧ Jb ❄❄ ⑧ u ⑧  θy1  E u˚py1q / y1

The commutative diagram below shows that the object assignment b ÞÑ Eb leads to a pseudofunctor

P op ˚ Φ : B ÝÑ CAT, pu : a Ñ b in Bq ÞÝÑ pu : Eb Ñ Eaq and, thus, presents the fibration P as an indexed category [28].

x ❘ ❘❘❘ ❘❘❘ f ǫf ❘❘❘ ❘❘❘  ❘❘❘ ˚ ❘❘)/ u pyq u y P θy PP PP g ˚ PP u pǫgq ǫg PP PPP u v P 1  θ ˚  P b v pzq θz PP θy ˚ ˚ ˚ / ˚ P/( ˚ / pv ¨ uq pzq– u pv pzqq v pzq 5 z y – 1b pyq y

v¨u θz u v 1b a / b / c “ Pz b / b

If ΦP is actually a functor, with the above canonical isomorphisms becoming identities, so that

˚ ˚ ˚ ˚ v¨u v u ˚ 1b pv ¨ uq “ u v , p1 q “ IdE , and θ “ θ ¨ θ v , θ “ 1 b b IdEb for all composable morphisms u, v and objects b in B, then P is called a split fibration. A functor P : E Ñ B is small-fibred if all of its fibres are small; in case of a fibration P , this means that ΦP takes its values in Cat. A fibration P is discrete if all of its fibres are discrete, that is: if ΦP takes its values in SET. Clearly a functor P is a discrete fibration if, and only if there is, for every u : a Ñ b in B and y P Eb, exactly one lifting with codomain y; the fibration is necessarily split. Here is how some elementary properties manifest themselves for a fibration P : E Ñ B: P is faithful (full; essentially surjective on objects) if, and only if, for every b P B, the fibre Eb is a

48 preordered class (has all of its homs non-empty; is non-empty, respectively). When E has a terminal object, a fibration P is an equivalence of categories if, and only if, P preserves the terminal object and reflects isomorphisms. (In the last statement, the preservation of the terminal object is essential: for a monic arrow f : x Ñ y in a a category C, the discrete fibration f ¨ p´q : C{x Ñ C{y is fully faithful, but does not preserve the terminal object 1x of C{x, unless f is an isomorphism in C.)

10.3. Grothendieck cofibrations and bifibrations For a functor P : E Ñ B, a morphism f : x Ñ y in E is P -cocartesian if f is P op-cartesian in Eop, with P op : Eop Ñ Bop. This means that every solid-arrow diagram below on the left can be filled uniquely as shown.

h δu ) x x / y ❴❴❴❴❴ / z x / u!pxq“ y E f s Ph P )  P x / Py / Pz u P f v a “ P x / b B

P is a (cloven Grothendieck) cofibration if P op : Eop Ñ Bop is a fibration9. This means that for every morphism u : a Ñ b in B and every object x in Ea one has a (chosen) P -cocartesian lifting, u u which we denote by δx : x Ñ u!pxq; this fixes the cocleavage δ : Ja Ñ Jbu!. Every morphism u f : x Ñ y in E now admits the (P -cocartesian, P -vertical)-factorization f “ νf ¨ δx , with u “ Pf.

u! Ea / Eb 7 y ❄ ♣♣ O ❄ δu: ⑧ f ♣♣ ❄❄ ùñ ⑧ ♣♣ ν ❄ ⑧⑧ ♣♣ f Ja ❄❄ ⑧ Jb ♣♣ ❄ ⑧ ♣♣ δu ⑧ ♣♣ x E x ♣ / u!pxq

One obtains a pseudofunctor

ΦP : B ÝÑ CAT, pu : a Ñ bq ÞÝÑ pu! : Ea Ñ Ebq, and the cofibration P is split if ΦP is a functor; more precisely, if

v¨u v u 1b pv ¨ uq! “ v! u!, p1 q! “ IdE , and δ “ δ u! ¨ δ , δ “ 1Id , b b Eb for all composable morphisms u, v and objects b in B. For all b P B we have the dual comma insertion of the fibre Eb: b I : Eb ãÑ P Ób, y ÞÑ p1b : Py Ñ bq.

When P is a cloven cofibration, it has the left adjoint pu : P x Ñ bq ÞÝÑ u!pxq. A functor P is a bifibration if it is simultaneously a fibration and a cofibration. The following criterion is certainly known but is not easily found and clearly spelled out in the literature: Theorem 10.1. The following assertions are equivalent for a functor P : E Ñ B : (i) P is a bifibration;

9Grothendieck cofibrations are now commonly referred to as opfibrations: see Footnote 1 of the Introduction.

49 ˚ (ii) P is a fibration, and the functor u has a left adjoint u!, for all u : a Ñ b in B;

˚ (iii) P is a cofibration, and the functor u! has a right adjoint u , for all u : a Ñ b in B.

u u ˚ For a bifibration P , the units η and counits ε of the adjunctions u! % u are determined by the commutative diagrams J J a ❍ : Ob ❍❍ u u ✉✉ u ❍ δ θ ✉ u Jaη ❍❍ ✉✉ Jbε ❍❍ ✉✉  ❍$ ✉✉ J u˚u / J u J u˚ / J u u˚ a ! u b ! a u ˚ b ! θ u! δ u Corollary 10.2. For a bifibration P : E Ñ B and every morphism u : a Ñ b in B, the functor ˚ u : Eb Ñ Ea preserves all limits and u! : Ea Ñ Eb preserves all colimits.

10.4. Limits and colimits in a fibred or cofibred category It is certainly known how to form ordinary (co)limits of a specified type in a bifibred category E from given (co)limits of the same type in the base category B and the fibres of the fibration. (For notions of, and criteria for, fibrational completeness, see [5, Section 8.5].) We sketch here a detailed but compact proof of this fact in a more general form, as we trace its steps in our main application in Section 4. Theorem 10.3. Let P : E Ñ B be a fibration. If limits of shape D exists in B and in all fibers of P , such that their comma insertions preserve them, then also E has D-limits, and P preserves them. Proof. We choose a cleavage θ for P . For a diagram F : D Ñ E, let b – limpP F q in B, with limit cone β : ∆b Ñ P F . For every object d in D we have the P -cartesian lifting of βd at F d,

βd ˚ pαd : Ld Ñ F dq – pθFd : βd pF dq Ñ F dq , to obtain a functor L: D Ñ Eb, together with a natural transformation α: JbL Ñ F . By design, PJbL “ ∆b and P α “ β. Now let z – limpLq, with limit cone λ: ∆z Ñ L. in Eb. We claim that the composite transformation

Jbλ α ∆z / JbL / F is a limit cone in E. Consider any cone µ: ∆x Ñ F in E. Its P -image factors as β ¨ ∆u “ Pµ, for a unique B- morphism u: P x Ñ b. As αd is P -cartesian, for every d P D one has a morphism γd : x Ñ Ld, unique with αd ¨ γd “ µd and Pγd “ u. This gives a cone γ : ∆x Ñ JbL in E with α ¨ γ “ µ and b Pγ “ ∆u. With the comma insertion I : Eb ÝÑ b Ó P preserving the limit cone λ : ∆z Ñ L, we can view γ as a cone ∆pu : P x Ñ bq ÝÑ IbL in b Ó P and, hence, factor it uniquely through Ibλ : ∆Ibz Ñ IbL. This means that there is unique morphism f : x Ñ z in E with Pf “ u and λd ¨ f “ γd for all d P D, and we obtain the factorization pα ¨ Jbλq ¨ ∆f “ α ¨ γ “ µ. If g : x Ñ z is any E-morphism with pα¨Jbλq¨∆g “ µ, we must confirm g “ f. An application of P to the given identity shows β ¨ ∆Pg “ Pµ “ β ¨ ∆u and, hence, Pg “ u. Now the P -cartesianness of αd shows λd ¨ g “ µd for every d P D, so that Ibλ ¨ ∆g “ γ. This forces g “ f.

50 An application of the theorem to P op instead of P produces the dual statement:

Corollary 10.4. Let P : E Ñ B be a cofibration. If colimits of shape D exist in B and in all fibers of P , such that the dual comma insertions preserve them, then D-colimits exist in E, and P preserves them. With Corollary 10.2 we conclude from Theorem 10.3 and Corollary 10.4:

Corollary 10.5. Let P : E Ñ B be a bifibration. If (co)limits of shape D exist in B and in all fibers of P , then also E has all (co)limits of shape D, and P preserves them.

10.5. The Grothendieck construction for indexed categories The indexed category ΦP : Bop Ñ CAT of a fibration P : E Ñ B preserves all information about P the base category B and the fibres Eb “ Φ b pb P Bq, including their pseudo-functorial interaction. The Grothendieck construction shows how one can rebuild the category E from that information. Below (on the left) is the definition of the Grothendieck category (also called total category) of Φ, ˝ usually denoted by ş Φ, in the split (=strict) case, that is, for a genuine functor Φ: Bop Ñ CAT. On the right we describe the dual construction, i.e., give the definition of the dual Grothendieck B E B category, ş˝ Φ, for a functor Φ: Ñ CAT. In the case Φ “ ΦP where P : Ñ is a cofibration, it recovers the category E.

˝ The Grothendieck category ş Φ of a functor The dual Grothendieck category ş˝ Φ of a Φ: Bop Ñ CAT is the category with functor Φ: B Ñ CAT is the category with ‚ objects pairs pb,yq, for b P B and y P Φb; ‚ objects pairs pa, xq, for a P B and x P Φa; ‚ morphisms pu,fq: pa, xq Ñ pb,yq, for ‚ morphisms pu,fq: pa, xq Ñ pb,yq, for u: a Ñ b in B and f : x Ñ pΦuqy in Φa; u: a Ñ b in B and f : pΦuqx Ñ y in Φb;

pa, xq 6 pb,yq ◗◗◗ ♠♠ O ◗◗◗pu,fq pu,fq ♠♠♠ p1a,fq ◗◗◗ ♠♠♠ p1b,fq ◗◗◗ ♠♠♠  ◗◗( ♠♠♠ pa, pΦuqyq / pb,yq pa, xq / pb, pΦuqxq pu,1pΦuqy q pu,1pΦuqxq / / a u b a u b

‚ composition pv,gq¨pu,fq “ pv¨u, pΦuqg¨fq. ‚ composition pv,gq¨pu,fq “pv¨u,g¨pΦvqfq. ˝ B B ş Φ is fibred over , with split fibration ş˝ Ψ is fibred over , with split cofibration Φ ˝ B B Π : ş Φ Ñ , pu,fq ÞÑ u, ΠΦ : ş˝ Φ Ñ , pu,fq ÞÑ u, ˚ u pb,yq “ pa, pΦuqyq, u!pa, xq “ pb, pΦuqxq, u u θpb,yq “ pu, 1pΦuqyq, εpu,fq “ p1a,fq. δpa,xq “ pu, 1pΦuqxq, νpu,fq “ p1b,fq.

One can make precise in which sense the construction on the right is dual to the construction on the left, as follows. Given Φ: B Ñ CAT , dualize the “base" B and every “fibre” Φb pb P Bq, that is: form the indexed category

51 op Φ p´q Φ˝ :“rB “ pBopqop / CAT / CATs.

Then there is a trivial bijective functor mapping objects and morphisms identically and making

˝ ˝ – / op ş pΦ q pş˝ Φq ●● ✈ ●● ✈✈ ˝ ● ✈ op Φ ●● ✈✈pΠΦq Π ●# ✈{ ✈ Bop commute. (We note that there is also the Borceux-Kock dualization of a fibration which dualizes the fibres but not the base, turning the (vertical, cartesian) factorization for a fibration into a (cartesian, vertical) factorization for a “dual fibration"; see [29] for details). An elementary (and quite obvious) rendition of the equivalence of split fibrations and strictly functorial indexed categories reads as follows; as a 2-categorical equivalence it is formulated as Corollary 7.6 .

Theorem 10.6. (i) For every split fibration P : E Ñ B with cleavage θ, there is a bijective P functor KP , satisfying PKP “ ΠΦ and preserving the cleavages, given by

˝ KP ş ΦP / E pb,yq ✤ / y u P ✤ / ΠΦ P rpu,fq: pa, xq Ñ pb,yqs rθy ¨ f : x Ñ ys   B B

Φ (ii) For every functor Φ: Bop Ñ CAT, there is a natural isomorphism ΛΦ : Φ Ñ ΦΠ whose component at b P B is the bijective functor

Φ ˝ f 1 p1b,fq 1 Λb : Φb ÝÑ pş Φqb, py ÝÝÑ y q ÞÑ rpb,yq ÝÝÝÝÝÑpb,y qs .

Under the above dualization principle one concludes from the theorem that split cofibrations cor- respond equivalently to functors Φ: B Ñ CAT. Furthermore, Theorems 10.1, 10.3 and Corollaries 10.4, 10.2 may now be formulated in indexed-category form, as follows.

Corollary 10.7. (1) A functor Φ: Bop Ñ CAT has the property that every functor Φu (with u a ˝ morphism in B) has a left adjoint if, and only if, ΠΦ : ş Φ Ñ B is a bifibration. In that case, ˝ if B and all categories Φb pb P Bq have (co)limits of a specified diagram type D, so does ş Φ. (2) A functor Φ: B Ñ CAT has the property that every functor Φu (with u a morphism in B) has a B B right adjoint if, and only if, ΠΦ : ş˝ Φ Ñ is a bifibration. In that case, if and all categories B D Φb pb P q have (co)limits of a specified diagram type , so does ş˝ Φ.

10.6. Standard examples (1) For any category C, the functors Id : C Ñ C and !: C Ñ 1 (where 1 is terminal in CAT) are split bifibrations. Every morphism in C is Id-(co)cartesian and !-vertical; the !-(co)cartesian morphisms are the isomorphisms in C. The indexed categories induced by Id and ! have (up to isomorphism) constant value 1 and C, respectively.

52 (2) For a fixed object A in a category C, consider its hom-functor Cp´, Aq : Cop Ñ Set as having ˝ discrete-category values. Then ş Cp´, Aq is the slice category C{A, presented as a discretely- fibred category over C. (3) The slice categories of (2) define a functor C{p´q : C Ñ CAT, A ÞÑ C{A whose dual Grothendieck C C2 category ş˝ {p´q is the arrow category (where the only non-identical morphism in the 2 category 2 is 0 Ñ 1), equipped with its codomain functor cod “ ΠC{p´q : C Ñ CAT. Hence, cod is a split cofibration, and it is a (cloven) fibration precisely when C has (chosen) pullbacks. A morphism pf,uq : x Ñ y in C2, represented by the commutative square

f ‚ / ‚

x y  u  a / b in C, is cod-cocartesian precisely when f is is an isomorphism, and it is cod-cartesian precisely when it is a pullback diagram in C. (4) Bijective functors (isomorphisms in CAT) are split bifibrations. The composite of two (split) fibrations is again a split fibration, and so is any pullback in CAT of a (split) fibration; likewise for (split) cofibrations. (5) A left action of a group G on a group N is described by a homomorphism φ : G Ñ AutpNq or, equivalently, by a functor φ : G Ñ Cat which maps the only object of G (seen as a category) to

N (seen as a category and, hence, as an object in Cat). The dual Grothendieck category ş˝ φ is (up to switching coordinates) precisely the semidirect product N ¸ G. A right action of G on N is given by a functor Gop Ñ Cat with value N or, equivalently by its Grothendieck category ˝ ş φ. op (6) There is a functor Φ: Rng Ñ CAT which assigns to a ring R the category ModR of (left) ˚ R-modules; every homomorphism ϕ : R Ñ S gives the functor ϕ : ModS Ñ ModR which considers every S-module N as an R-module, via ra “ ϕprqa for all r P R, a P N. The category ˝ ş Φ is the category Mod of all modules; its objects are pairs pR,Mq where R is a ring and M is an R-module, and its morphisms pϕ, fq : pR,Mq Ñ pS,Nq are given by a morphism ϕ : R Ñ S in Rng and an R-linear map f : M Ñ ϕ˚pNq. The projection ΠΦ : Mod Ñ Rng is a split fibration. (7) The functor O : Set Ñ Cat assigns to every set X the set of topologies on X, ordered by Ě and, as such, considered as a small category; for a map f : X Ñ Y one has the monotone ˝ map f ˚ : OpY q Ñ OpXq, defined by taking inverse images. The Grothendieck category ş O is the category Top of topological spaces, with underlying Set-functor ΠO, which is in fact a split bifibration. More generally, one may characterize topological functors with small fibres (see [1]) as those fibrations P for which the indexed category ΦP takes values in the category of complete lattices and their inf-preserving maps (see [45, 42]), making P in fact a split bifibration. (8) Considering the functor IdSet : Set Ñ Set as having discrete-category values, one obtains the Set category Set‚ of pointed sets as its dual Grothendieck category ş˝ Id . Writing the domain of op op ˝ IdSet in the form pSet q we also have the Grothendieck category ş IdSet, which is precisely op the category Set‚ . The “categorification” of the last (rather trivial) example leads to an important fact, which we describe next.

53 10.7. The classifying split (co)fibration

Cat Cat The dual Grothendieck category ş˝ Id of the functor Id : Cat Ñ Cat (with its codomain to be embedded into CAT) is the category Cat‚ of small lax-pointed categories. Its objects pC, xq are given by a small category C equipped with an object x P C, and a morphism pF,fq : pC, xq Ñ pD,yq is given by a functor F : C Ñ D and a morphism f : F x Ñ y in D. The forgetful functor

Π‚ :“ ΠIdCat : Cat‚ Ñ Cat is a small-fibred split cofibration, called classifying, since one has the following rather obvious fact:

Theorem 10.8. Every small-fibrerd split cofibration is a pullback (in CAT) of the classifying split cofibration, as shown in the diagram

E / Cat‚ rf :x Ñ ys ÞÝÑ rppPfq!,νf q:pEPx,xq Ñ pEPy,yqs .

P Π‚   B / Cat ΦP

op op ˝ Writing the domain of IdCat as pCat q , we can also form the Grothendieck category ş IdCat, which is the category Cat‚ of small oplax-pointed categories. Its objects are the same as those of Cat‚, but its morphisms pF,fq : pC, xq Ñ pD,yq are now given by functors F : D Ñ C equipped with a morphism f : x Ñ Fy in C. The forgetful functor Π‚ :“ ΠIdCat : Cat‚ Ñ Catop classifies the small-fibred split fibrations:

Corollary 10.9. Every small-fibred split fibration is a pullback (in CAT) of the classifying split fibration, as shown in the diagram

‚ ˚ E / Cat rf :x Ñ ys ÞÝÑ rppPfq ,νf q:pEPx,xq Ñ pEPy,yqs .

P Π‚   B / Catop pΦP qop

Of course, nothing prevents us from dropping the restriction of P being small-fibred.: Theorem ‚ 10.8 and Corollary 10.9 remain true verbatim if we delete “small-fibred” and replace Cat, Cat‚, Cat ‚ by CAT, CAT ‚, CAT , respectively, and CAT by the colossal category CAT which contains CAT as an object, with the last exchange only formally needed for the provision of a legitimate home of the amended pullback diagrams.

11. Appendix 2: Regular epimorphisms in CAT and a confinality criterion

11.1. Characterizing regular epimorphisms in CAT First we characterize regular epimorphisms in CAT, i.e., functors F : C Ñ D which serve as coequalizers of parallel pairs of functors, in terms of suitable relations on the classes of objects and morphisms of their domains. Although parts of the characterization we give may be found, at least in some similar form, in the older literature (such as [11, 35, 42]), we give here a compact description of it, also since these sources may not be easily accessible to the reader. We begin by describing the quotient of a category C with respect to a suitable pair p«, „q of relations. The “object branch” of the pair is a mere equivalence relation « on the class of objects

54 of C; we denote the «-equivalence class of c P C by rcs. These equivalence classes are the vertices of the directed graph C«; an edge u : rcsÑrds in C« is simply a C-morphism u : a Ñ b with c « a and b « d. We denote by ΩpC«q 10 the category freely generated by the graph C«; hence, its objects are the vertices of the graph, and its morphisms pun, ..., u1q : rcsÑrds are given by strings of C-morphisms

u1 u2 un c « a1 ÝÝÝÑ b1 « a2 ÝÝÝÑ b2 «¨¨¨ ÝÝÝÑ bn « d, including the empty strings pq : rcsÑrcs (for n “ 0). We may write the string puq of length 1 simply as u; composition in ΩpC«q proceeds by juxtaposition. Hence, in ΩpC«q the morphism pun, ..., u1q is the composite un...u1 of strings of length 1. The “morphism branch” of our pair is simply a relation „ on the class of morphisms of ΩpC«q which allows α „ β only if α and β belong to the same hom-set of ΩpC«q. We then consider the least equivalence relation » on the class of morphisms of ΩpC«q containing „ and satisfying the following compatibility conditions for all morphisms u : c Ñ d, v : d Ñ e in C (whose composite in 1 C is denoted by v ¨ u), and all α : rasÑrbs, β,β : rbsÑrcs, γ : rcsÑrks in ΩpC«q:

1. 1c » pq : rcsÑrcs ; 2. v ¨ u » pv,uq : rcsÑres ; 3. β » β1 ùñ γβα » γβ1α .

There is then only one way of making the »-equivalence classes of morphisms in ΩpC«q the morphisms of a category ΩpC«q{ » with the same objects as ΩpC«q , so that the projection ΩpC«q Ñ ΩpC«q{ » that maps objects identically and assigns to morphisms their »-equivalence classes, becomes a functor. We note that, by condition 1, the relation « is actually determined by „, since a « b ðñ 1a » 1b, for all objects a,b P C. Moreover, because of conditions 1 and 2, the composite graph morphism

P : C Ñ ΩpC«q Ñ ΩpC«q{», pu : a Ñ bq ÞÑ pu : rasÑrbsq ÞÑ pPu : rasÑrbsq, becomes a functor, where Pu is the »-equivalence class of the morphism u : rasÑrbs in ΩpC«q. We call P the projection of the quotient structure on C as given by the relations « and „. The quotient structure is locally small if the category ΩpC«q{ » has small hom-sets. We now prove that the projection of a locally small quotient structure is the prototypical regular epimorphism in CAT. Proposition 11.1. The following assertions for a functor F : C Ñ D are equivalent: (i) F is a regular epimorphism in CAT; (ii) every morphism in D may be written as a composite morphism Fun ¨ ... ¨ Fu1 with morphisms u1, ..., un in C, n ě 1; furthermore, if Fun ¨...¨Fu1 “ F vm ¨...¨F v1 with v1, ..., vm in C, m ě 1, then also Gun ¨ ... ¨ Gu1 “ Gvm ¨ ... ¨ Gv1, for any functor G : C Ñ B satisfying the condition pFu “ F v ùñ Gu “ Gvq for all morphisms u, v in C ;

10 It is important to note that the category ΩpC«q will generally fail to have small hom-sets. Indeed, since the equivalence classes rcs may be large, the graph C« may have a proper class of edges between two fixed vertices, causing its generated category to have large hom-sets. As Remark 11.2 shows, an unwelcome consequence of this fact is that, unlike Cat, the (huge) category of CAT of locally small categories fails to have coequalizers.

55 (iii) there is a locally small quotient structure p«, „q on C with projection P and a bijective functor J :ΩpC«q{» ÝÑ D, such that F “ JP .

Proof. (iii)ñ(ii): Every morphism in ΩpC«q has the form Pun ¨ ... ¨ Pu1, including P p pq : rcs Ñ rcsq“ P 1c ), which confirms the first statement of (ii), by bijectivity and functoriality of J. For any functor G : C Ñ B satisfying pFu “ F v ùñ Gu “ Gvq for all morphisms u, v in C, one has in particular pa « b ùñ P 1a “ P 1b ùñ Ga “ Gbq for all objects a,b P C. Consequently, G induces a morphism C« Ñ B of graphs, which extends uniquely to a functor G : ΩpC«q Ñ B, via Gpun...u1q “ Gun ¨ ... ¨ Gu1. By functoriality of G and G, the equivalence relation „G on ΩpC«q with pα „G β ðñ Gα “ Gβq satisfies conditions 1-3, and it contains „ by hypothesis on G:

u „ v ùñ Pu “ P v ùñ Fu “ F v ùñ Gu “ Gv .

Hence, » is contained in „ , making G factor through ΩpC q{», by a unique functor G :ΩpC q{»Ñ G « r « B. Consequently, assuming Fun ¨...¨Fu1 “ F vm ¨...¨F v1, one first has Pun ¨...¨Pu1 “ P vm ¨...¨P v1 and therefore u ...u » v ...v . Now the application of G to the »-equivalence classes of these n 1 m 1 r morphisms gives the desired equality Gun ¨ ... ¨ Gu1 “ Gvm ¨ ... ¨ Gv1. (ii)ñ(i): The conditions given allow one to show routinely that any functor G : C Ñ B that coincides on the projections C ˆB C Ñ C of the kernel pair of F factors uniquely through F , so that F is in fact the coequalizer of its kernel pair. (i)ñ(iii): As a regular epimorphism, F is the coequalizer of its kernel pair. Let «F “« be the equivalence relation induced by F on the class of objects of C. As in (iii)ñ(i), with G traded for F , one obtains the functor F : ΩpC«q Ñ B. The equivalence relation „F “» induced by F via pα „ β ðñ F α “ Fβq actually satisfies the conditions 1-3, so that „ equals ». Hence, in the notation of (iii)ñ(ii), F factors through P , via J :“ F :ΩpC q{„ ÝÑ B. Since for every morphism r « pu, vq in C ˆD C one has Fu “ F v, the functor P coincides on the projections of the kernel pair of F and must therefore factor through the coequalizer F , by what then turns out to be the inverse of J, thanks to the uniqueness part of the universal property of a coequalizer. The bijection J makes the category ΩpC«q{ „ inherit the local smallness from the given category D, so that the quotient structure p«, „q is indeed locally small. Remark 11.2. (1) As a quotient structure p«, „q on any given category C, one may make the extreme choice and let « be the all-relation on the class of objects of C and „ be empty, so that » will be the least relation on the class of morphisms of ΩpC«q satisfying the conditions 1-3 above. Then T pCq “ ΩpC«q{ » has been called the totalizer of (the partially defined composition rule for the morphisms of) C; its projection P : C Ñ T pCq serves as a reflection of C into the (colossal) category MON of large monoids and their homomorphisms. Even for finite C, T pCq may be infinite: T pt0 Ñ 1uq – pN, `q. Remarkably, as shown in [6], while identifying all objects, P preserves the morphism structure of C to the largest extent possible: for any morphism f,g in C, one has Pf “ Pg if, and only if, f “ g or both, f and g, are identity morphisms. (2) The classical (and most important) instance of the formation of a quotient category arrises when one forms the category of fractions CrS´1s for a class S of morphisms in a category C: formally adjoin to C a designated inverse s1 of each morphism s in S and then impose the relations that insure that s1 becomes an actual inverse of s in CrS´1s. Again, it is important to note that, local smallness of C does not guarantee at all local smallness of CrS´1s.

56 11.2. Confinality vs. conneted fibres for regular epimorphisms and fibrations We begin with a sufficient condition for the recognition of confinal11 regular epimorphisms in CAT. We did not find this criterion in the literature. Theorem 11.3. A regular epimorphism in CAT is confinal if each of its fibres is a connected category. Proof. By Proposition 11.1 we may assume that the given regular epimorphism is the projection P of a quotient structure p«, „q on C and, when P has connected fibres, must show that, for every object c P C, the comma category rcs Ó P is non-empty and connected. As we have the object p1rcs “ P 1c, cq in rcs Ó P , it suffices to show that every object pxαy, dq in rcs Ó P is connected to pP 1c,cq by a zig-zag of morphisms in rcs Ó P ; here xαy denotes the »-equivalence class of a morphism α “ pun, ..., u1q : rcsÑrds in ΩpC«q, given by C-morphisms

u1 u2 un c « a1 ÝÝÝÑ b1 « a2 ÝÝÝÑ b2 «¨¨¨ ÝÝÝÑ bn « d .

Then u1 becomes a morphism p1Pc, a1q ÝÑ pPu1, b1q in rcsÓ P , and every ui with i ě 2 gives a morphism pPui´1 ¨ ... ¨ Pu1, aiq ÝÑ pPui ¨ ... ¨ Pu1, biq, as shown in

rcs ❙ ♦♦✇ ✺◆◆❙❙❙ ♦♦✇♦✇✇✠ ✺ ◆◆◆❙❙❙ ♦♦♦✇✇✇✇ ✠ ✺ ◆◆❙❙❙ ♦♦♦✇✇✇ ✠✠ ✺✺ ◆◆ ❙❙❙ ♦♦♦ ✇✇ ✺ ◆◆ ❙ P un¨...¨P u1“xαy ♦♦ ✇✇✇ ✠ P u2¨P u1 ◆ ❙❙ ♦♦♦♦ ✇✇✇ ✠✠ ✺✺ ◆◆ ❙❙❙ ♦♦ ✇✇✇ ✠ P u1 ✺ ◆◆ ❙❙ ♦♦♦♦ ✇✇✇ ✠ ✺ ◆◆ ❙❙❙ ♦♦♦ ✇✇✇✇ ✠ ✺ ◆◆◆ ❙❙❙ ♦♦♦ ✇✇✇ ✠✠ ✺✺ ◆◆ ❙❙❙ ♦♦♦ ✇✇✇✇ Ô ✠  ✺ ◆◆ ❙❙❙ ♦♦♦ ✇✇ ✠ ◆' ❙❙) Pc Pa1 / Pb1 Pa2 / Pb2 ... / Pbn P d . P u1 P u2 P un It therefore suffices to show that the horizontal equalities of the diagram may be replaced by fitting zig-zags of morphisms in rcsÓ P . In fact, it is easy to see that any two objects pt,aq, pt,bq in rcsÓ P are connected by a zig-zag of morphisms in that category; it is provided by any zig-zag connecting the objects a,b in the connected fibre of P at Pa “ Pb. This completes the proof. A confinal regular epimorphism in Cat (or CAT) may have disconnected fibres. In deed, if we consider the category ta Ñ c Ð bu and identify its two non-identical arrows, we obtain the category 2 “ t0 Ñ 1u as its quotient category; the projection is confinal, but its fibre at 0 is the discrete category ta,bu. For fibrations, rather than regular epimorphisms, we obtain a stronger recognition criterion for confinality: Proposition 11.4. A fibration is confinal if, and only if, all of its fibres are non-empty and connected.

Proof. Given a fibration P : E Ñ B and an object a P B, the embedding Ea ãÑ aÓP is full and has a right adjoint, by 10.2. Since P is confinal if and only if every category a Ó P is non-empty and connected, the claim follows from the fact that a reflective or coreflective subcategory is connected if, and only if, its parent category is connected.

11In order to avoid ambiguity, we follow Gabriel and Ulmer [14] (Definition 2.12) and use the traditional term confinal for functors F : C Ñ D for which the comma categories z Ó F pz P Dq are (non-empty and) connected. More recent authors use “cofinal”, “final”, or even “initial” for the same notion.

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