An lgebric theory of trictegories Nick Gurski Mrch 9, 2007
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i Acknowledgements First I would like to thnk Peter My nd Eugeni Cheng for helping me through ll stges of this project. I would lso like to thnk Ross Street nd John Power for nswering some technicl questions. Finlly I would like to thnk Mrtin Hylnd nd Andre Joyl for encourgement.
ii Abstrct The originl definition of trictegory given by Gordon, Power, nd Street is only prtilly lgebric. The definition is not fully lgebric since certin trnsformtions re required to be wekly invertible s 1-cells of functor bictegory, but no wek inverse is required s prt of the dt. We rectify this by replcing these equivlences with djoint equivlences. We then prove coherence by providing Yoned embedding for restricted clss of trictegories in which the trget of this embedding is functor trictegory tht is shown to be Gryctegory; in prticulr, this strtegy voids the use of the prerepresenttions in the work of Gordon, Power, nd Street. Using the fct tht the new definition of trictegory is lgebric, we compre the free trictegory on ctegory-enriched 2-grph with the free Gryctegory on the sme dt nd show tht the nturl comprison functor is strict triequivlence. This is nother sttement of coherence, nd lso gives proof tht lrge clss of digrms of constrint 3-cells commute in ny trictegory. We then produce, from ny trictegory T, Gry-ctegory GrT nd triequivlence GrT T. A similr strtegy pplied to functors yields coherence theorem for functors, nd we then produce from ny functor F : S T between trictegories Gry-functor GrF : GrS GrT.
Contents 1 Introduction 1 2 Coherence for bictegories 11 2.1 Bictegoricl conventions...................... 11 2.2 The Yoned embedding....................... 12 2.3 Coherence for bictegories...................... 13 2.3.1 Grphs nd free constructions................ 13 2.3.2 Proof of the coherence theorem............... 14 2.3.3 Using coherence: strictifiction............... 17 2.4 Coherence for functors........................ 18 2.4.1 Free functors......................... 18 2.4.2 Proof of the coherence theorem............... 20 2.4.3 Using coherence: strictifiction............... 22 3 The lgebric definition of trictegory 23 3.1 Bsic definition............................ 23 3.2 Adjoint equivlences nd trictegory xioms............ 28 3.3 Trihomomorphisms nd other higher cells............. 29 3.4 Compring definitions........................ 41 4 Bsic structure 43 4.1 Structure of functors......................... 43 4.2 Structure of trnsformtions..................... 47 4.3 Chnge of structure......................... 52 4.4 Triequivlences............................ 56 5 The Gry tensor product 59 5.1 The Gry tensor product...................... 59 5.2 Cubicl functors........................... 61 5.2.1 Defining cubicl functors.................. 61 5.2.2 The universl cubicl functor................ 66 5.3 The monoidl ctegory Gry.................... 67 iii
iv CONTENTS 6 Gry-ctegories nd the trictegory Bict 71 6.1 Cubicl trictegories......................... 71 6.2 Gry-ctegories........................... 75 6.3 The trictegory Bict........................ 77 6.3.1 Constructing Bict directly................. 77 6.3.2 The trictegories Bict nd Gry............. 80 7 Functor trictegories: Gry-structures 85 7.1 Locl structure............................ 85 7.2 Globl results............................. 87 8 The Yoned lemm nd coherence 91 8.1 The cubicl Yoned Lemm..................... 91 8.2 Coherence for trictegories...................... 100 9 Free trictegories 101 9.1 Grphs................................. 101 9.2 Free trictegories........................... 103 9.3 Free Gry-ctegories......................... 106 9.4 Preliminry results.......................... 108 10 Coherence vi free constructions 115 10.1 Coherence for trictegories...................... 115 10.2 Coherence nd digrms of constrints............... 118 10.3 Strictifying trictegories....................... 121 11 Coherence for functors 131 11.1 The coherence theorem........................ 131 11.2 Coherence nd digrms of constrints............... 136 11.3 Strictifying functors......................... 136 A Adjointness in bi- nd trictegories 141 A.1 Adjoint equivlences in bictegory................ 141 A.1.1 Definitions.......................... 142 A.1.2 The bictegory Ct..................... 142 A.1.3 The proof for bictegories.................. 144 A.1.4 Useful results......................... 146 A.2 Mtes in bictegory........................ 147 A.3 Bidjoint biequivlences....................... 149 B Unpcked definitions 157 B.1 Unpcked trictegories........................ 157 B.2 Unpcked functors.......................... 159 B.3 Unpcked trnsformtions...................... 160 B.4 Unpcked modifictions....................... 161 C Clcultions 163
CONTENTS v References 165
vi CONTENTS
Chpter 1 Introduction The study of wekened higher dimensionl structures in ctegory theory begn with the notion of bictegory, defined by Benbou in 1967 [5]. The study of bictegories now hs two eqully importnt components: one is s tool to orgnize nd generlize theorems from ctegory theory, nd nother is the study of bictegories s interesting lgebric objects in their own right. An ppliction of the first kind is the study of monds in generl 2-ctegory [35], nd n exmple of theorem of the second kind is the coherence theorem for bictegories which sttes tht every bictegory cn be mde strict in precise sense [41], [17]. There re lso importnt pplictions of this theory in physics, topology, nd representtion theory. The intense focus on understnding structures of dimension n > 2 is reltively recent phenomenon the first pper to even hint t possible definition of wek ω-ctegory ( type of wek ctegory-structure with cells of every dimension n for n 0) is Street s The lgebr of oriented simplexes in 1987 [39]. Since then, there hve been mny definitions of wek n- or ω-ctegory proposed by number of different uthors. The survey [29] by Leinster provides good ccount of mny of these proposed definitions. There is n importnt distinction to be mde between wek nd strict structures. In strict n-ctegory, ll possible xioms hold, including those for cells tht re not of the top dimension. This is not the cse for wek n-ctegories, where we only hve xioms governing cells of the top dimension nd the old xioms for lower dimensionl cells re replced by invertible or wekly invertible cells subject to their own lws. An exmple of this phenomenon occurs in the definition of bictegory, where 1-cells re only required to compose ssocitively up to 2-cell isomorphisms tht re then required to stisfy new xiom, the Mc Lne pentgon. But even the bsics of higher dimensionl ctegory theory re fr from estblished. The project of compring these different definitions is most likely yers from completion, nd for most definitions few, if ny, significnt pplictions hve been produced. There hve been few pplictions to topology, with some success in using ides from higher ctegory theory to study n-fold loop 1
2 CHAPTER 1. INTRODUCTION spces nd homotopy n-types [4], [6], [7], [44]. The definitions lluded to bove ll hve generl nture to them. They re intended to describe wek n-ctegories for rbitrry n, sometimes including wek ω-ctegories. Some of these definitions re inductive, but some lso strt by defining wek ω-ctegory nd then specilizing to finite n. None of these definitions re wht one would cll hnds-on, though. They do not explicitly formulte the xioms involved, insted relying on complicted techniques to efficiently encode ll of the xioms t once, usully in the form of the structure of n lgebr over suitbly chosen mond or by requiring tht certin hornfilling conditions hold. There is hnds-on definition of wek 3-ctegories, which re clled trictegories, defined by Gordon, Power, nd Street in their 1995 Memoir [17]. This definition is monumentl chievement, nd s such is long nd complicted if not viewed from the proper perspective. To understnd the complexities of the definition, it is necessry to think bout the generl philosophy of ctegorifiction nd the coherence theory for bictegories. Ctegorifiction is the term used to describe the generl procedure of tking definition involving sets, functions, nd equtions between them, nd creting new definition involving ctegories, functors, nturl isomorphisms, nd equtions between those. The bsic philosophy of ctegorifiction is to replce the old xioms with new pieces of dt, nd then to construct the pproprite xioms tht this new dt is to stisfy. There re three importnt steps in the ctegorifiction process involved in the definition of trictegory. The first is ctegorifying the notion of isomorphism. Isomorphism is lredy the ctegorified version of equlity, nd the ctegorified notion of isomorphism present in [17] is tht of equivlence. The second importnt spect of the definition of trictegory is the introduction of two new pieces of dt, denoted λ nd ρ, tht do not rise s the ctegorified versions of old xioms. This is somewht misleding, s these new pieces of dt re ctegorified versions of importnt results used in the proof of the coherence theorem for bictegories [22]. The third importnt step in this ctegorifiction process is finding the correct xioms tht trictegories should stisfy. The ssocitivity xiom for trictegories is recognizble s n incrntion of the fifth ssocihedron of Stsheff [43] or the fifth orientl of Street [39]. The two unit xioms re more mysterious, however, nd in generl the unit conditions for higher ctegories re not s well understood s the ssocitivity conditions. The work of Gordon, Power, nd Street hs the primry gol of proving relevnt coherence theorem for trictegories. The coherence problem for bictegories hs strightforwrd nswer: every bictegory is biequivlent to strict 2-ctegory. Thus ll of the wekness in bictegory cn be removed by replcing the bictegory in question with biequivlent one. This is not the cse for trictegories not every trictegory is triequivlent to strict 3- ctegory, nor cn this be true for ny resonble definition of trictegory nd triequivlence s we shll see. Thus the coherence theorem for trictegories is more interesting becuse of the inherent complictions tht rise from going up dimension.
The reson tht trictegories cnnot ll be triequivlent to strict 3-ctegories is consequence of the topology of homotopy 3-types. In his fmous letter Pursuing Stcks [20], Grothendieck outlined some desidert for good theory of higher dimensionl groupoids. In prticulr, he suggested tht wek n-groupoids should be model for homotopy n-types. This gives some insight into the structure of higher ctegories, s mny topologists hve studied the problem of finding lgebric models for homotopy n-types. For exmple, homotopy 2-types re modelled by monoidl ctegories in which ech morphism is invertible nd ech object x hs tensor pseudoinverse, tht is n object y for which x y = I nd y x = I. This is just mnifesttion, in ctegoricl lnguge, of the fct tht connected homotopy 2-type is determined by its homotopy groups nd the ction of π 1 on π 2. In dimension 3, the sitution becomes slightly different. Connected, simplyconnected homotopy 3-types re clssified by their homotopy groups nd their Whitehed product π 2 π 2 π 3. In ctegoricl lnguge, this becomes the sttement tht connected, simply-connected homotopy 3-types re modelled by brided monoidl ctegories in which every morphism is invertible nd every object hs tensor pseudoinverse [21]. Since ll homotopy 3-types should be modelled by wek 3-groupoids, we cn first sk if strict 3-groupoids cn model ll connected, simply-connected homotopy 3-types. Now the Whitehed product in the nerve of strict 3-groupoid is the zero mp. (See [34] for full discussion.) Since ny resonble definition of triequivlence should induce wek equivlence between the corresponding nerves nd there re connected, simply-connected homotopy 3-types with non-trivil Whitehed product, we see tht strict 3-groupoids do not model ll homotopy 3-types. The correct coherence theorem, proved by Gordon, Power, nd Street, is tht every trictegory is triequivlent to Gry-ctegory. Here Gry denotes prticulr monoidl structure on the ctegory of strict 2-ctegories defined by Gry [18], nd Gry-ctegory is then just ctegory enriched over Gry. The reder should tke note tht we ctully use wht might be clled the strong Gry tensor product, where Gry studied the lx version. Simply put, the problem with strictifying every trigroupoid to strict 3-groupoid is the existence of briding in the trigroupoid cse (corresponding topologiclly to the Whitehed product) tht is forced to be symmetric in the strict 3-groupoid cse. The Gry tensor product of 2-ctegories builds in n pproprite interchnge isomorphism, nd the coherence theorem of [17] then sttes tht this interchnge isomorphism is the only obstruction to completely strictifying trictegory. This coherence theorem is very nturl when pproched vi the exmple of the trictegory of bictegories, functors, trnsformtions, nd modifictions, where here ll terms refer to the wek version of the notion involved. The coherence theorem for bictegories sttes tht every bictegory is biequivlent to strict 2-ctegory. Similrly, there is coherence theorem for functors tht produces the result tht, when strictifying bictegories, one cn lso strictify the mps between them to yield strict 2-functors between strict 2-ctegories. Thus we re ble to produce functor st : Bict 2Ct tht is left djoint to the inclusion of 2-ctegories into bictegories. Now both Bict nd 2Ct 3
4 CHAPTER 1. INTRODUCTION form 3-dimensionl structures in nturl wy. Two questions rise. First, cn this functor st be extended to mp of 3-dimensionl structures? Secondly, if it cn, wht properties does this extension hve? Given bictegory B, there is cnonicl comprison functor stb B tht is biequivlence. This leds one to believe tht st might be triequivlence, but this is not the cse. The problem rises when trying to understnd the composition lws for trnsformtions in Bict. In defining the horizontl composite β α of pir of trnsformtions, there re two eqully good cndidtes for the component of β α t the object, nd if β is strict trnsformtion then these two choices gree. But one quickly lerns tht it is not lwys possible to replce β by n isomorphic trnsformtion tht is strict, so we see tht Bict hs n unvoidble mount of wekness built into it. This wekness, though, is precisely the fct tht interchnge for 2-cells is n isomorphism nd not n equlity. Now Gry-ctegories re the strictest form of 3-dimensionl ctegory in which interchnge remins wek (i.e., is n isomorphism not n equlity), so the exmple of the trictegory Bict leds one to the study of Gry-ctegories. To nswer the questions posed in the previous prgrph we introduce new trictegory clled Gry which consists of 2-ctegories, 2-functors, wek trnsformtions, nd modifictions. It is now reltively simple to check tht the functor st gives triequivlence Bict Gry (here Gry denotes prticulr full sub-gry-ctegory of Gry), nd this sttement brings together the mny fcets of the coherence theory for bictegories in one simple sttement. It is worth noting tht the trictegory Gry is the trictegory obtined from the ctegory of 2-ctegories by using the closed structure given by the Gry tensor product nd its right djoint. The definition given by Gordon, Power, nd Street hs feture tht will be the focus of this work: it is not completely lgebric, nd for some pplictions this is definite drwbck. In the cse of trictegories, we men tht some of the dt is required to hve certin property but verifying this property mkes use of dditionl dt tht is not uniquely specified in the definition. This is by-product of the choice mde when ctegorifying the notion of isomorphism. The dt for bictegory include ssocitivity, left unit, nd right unit isomorphisms; these exist s invertible 2-cells in the given bictegory structure. In the definition of trictegory, nlogous 2-cells exist but now they re not top-dimensionl cells, so we require them to be wekly invertible rther thn invertible. This is where the definition given by Gordon, Power, nd Street is not fully lgebric. They choose to require the 2-cells bove to be equivlence cells. This is property of cell, but leves some dt unspecified: it requires tht there exist pseudoinverse nd invertible cells of one dimension higher exhibiting the cells s wekly invertible, but does not require choice of these cells. This is different from the definition of n isomorphism in ctegory. Since inverses re unique in ctegory, requiring tht morphism be invertible nd requiring the exhibition of n inverse re logiclly equivlent conditions. The sitution here is genuinely different, s there re mny possible pseudoinverses nd even then mny possible invertible cells exhibiting this pseudoinvertibility.
5 Giving n lgebric definition of trictegory thus requires chnging these equivlence 1-cells to n lgebric condition of wek invertibility. The rest of this work will be concerned with developing the bsic coherence theory of fully lgebric definition of trictegory long these lines. We hve tken the notion of djoint equivlence s our lgebric version of wekly invertible 1-cell in bictegory. It should be noted tht every equivlence 1-cell in bictegory is prt of n djoint equivlence, but tht there is no cnonicl choice of such extr structure. The definitions given here re of course similr to those given by Gordon, Power, nd Street, but wherever they demnd tht trnsformtion be pseudonturl equivlence, we insted require n djoint equivlence in the pproprite functor bictegory. This provides cnonicl pseudoinverses for ll of the pproprite structure constrints, s well s the necessry cells of the next dimension up to exhibit this pseudoinvertibility explicitly. There re mny choices for the notion of wek invertibility. An intermedite notion between equivlence nd djoint equivlence might be clled specified equivlence. This would require giving pseudoinverse nd the invertible cells exhibiting this pseudoinvertibility, but would not require these cells to stisfy ny xioms. The choice of djoint equivlence hs the cler dvntge over this intermedite notion tht it llows the use of mtes. A hppy by-product of the theory of mtes in bictegory llows us to refrin from introducing new set of dul xioms for these dditionl cells, s they re lredy implied. This is the phenomenon tht is responsible for the fct tht the opposite trictegory, defined by reversing the direction of the 1-cells only, stisfies the trictegory xioms. The coherence theorem for bictegories sttes tht every bictegory is biequivlent to strict 2-ctegory. The simplest wy to prove this theorem is to study the Yoned embedding for bictegories, functor B Bict(B op,ct). The trget of this functor is strict since Ct is strict 2-ctegory, nd the essentil imge of this functor is 2-ctgory biequivlent to B. The proof of the coherence theorem given by Gordon, Power, nd Street hs two prts. The first is the replcement of n rbitrry trictegory T with somewht strict kind of trictegory, clled cubicl trictegory. This is done by pplying the functor st to ll of the dt for T nd then using the fct tht this functor is lx monoidl to get composition mp st ( T(b, c) ) st ( T(, b) ) st ( T(, c) ). The second step in [17] is to construct for ny cubicl trictegory S suitbly well-behved embedding of S into Gry-ctegory. The essentil imge of S inside this new Gry-ctegory will then be smller Gry-ctegory triequivlent to S. Combining these two prts gives the desired theorem. It should be noted tht Gordon, Power, nd Street do not give n exct 3-dimensionl version of this proof. Insted of using the notion of functor trictegory (which
6 CHAPTER 1. INTRODUCTION remins undefined using their definition), they use the Gry-ctegory of prerepresenttions of cubicl trictegory; one cn view this Gry-ctegory s the functor trictegory but with some dt nd xioms omitted. Our proof follows strtegy tht combines both tht used to prove coherence for bictegories nd tht used by Gordon, Power, nd Street. We explicitly construct the functor trictegory Trict(S, T) in the cse when T is Gryctegory, nd then show tht it is gin Gry-ctegory. The outline of the proof is s follows. First we show how to replce T with cubicl trictegory s in [17], nd then we explicitly construct Yoned embedding S Trict(S op,gry) when S is ny cubicl trictegory. Restricting to the essentil imge gives the desired triequivlence. This shows the benefit of replcing T with cubicl trictegory, s the generl Yoned embedding would be functor of the form T Trict(T op,bict) which would not yield the desired coherence result s Bict is not Gryctegory. This pth to the coherence theorem requires defining multitude of compositions for functors, trnsformtions, modifictions, nd perturbtions. These compositions re given by messy formuls, but inspecting these demonstrtes the need for fully lgebric definition of trictegory s ll prts of the definition re necessry for writing down these formuls. We see this s good indictor of wht we hve ccomplished by mking the definition fully lgebric: with ll structure in plin sight, it is possible to write down formuls nd thus mke concrete constructions tht required rbitrry choices in the originl definition. The drwbck of this pproch should lso be cler: in trying to write down explicit formuls, one needs to work with very lrge digrms. Verifying bsic xioms with these digrms becomes difficult tsk. This is solved in the cse of bictegories by proving nother kind of coherence theorem, one tht sttes tht ll digrms of constrints commute. It is, fter ll, this kind of theorem tht llows the explicit construction of the strictifiction stb for ny bictegory B. Proving n nlogue of this theorem, nd reping the ttendnt benefits, is the focus of the lst third of this work. To prove this theorem for bictegories, we first tke slight detour to prove nother kind of coherence theorem (see [22] for the sme line of proof but restricted to the cse of monoidl ctegories). Given set of objects A 0 nd for ech pir of objects ctegory A(, b), we cn construct two cnonicl 2- dimensionl structures: the free bictegory on A nd the free strict 2-ctegory on A. Ech of these hs the set A 0 s its set of objects, but the sets of 1- nd 2-cells differ. The coherence theorem here sttes tht these two structures re biequivlent by the strict functor induced by the universl property of the free bictegory. The theorem tht every digrm of constrints in bictegory commutes is now simple corollry of the universl property of the free bictegory nd this coherence theorem pplied to the cse when ech of the ctegories
A(, b) is discrete. Our first gol, then, will be to mimic this coherence theorem compring the free wek structure with the free strict structure, except tht in our cse we compre the free trictegory with the free Gry-ctegory. There is new difficulty tht rises by going up dimension. This is the fct tht there re t lest three different choices of underlying grphs for trictegory, two of which we use here. The sme is true for Gry-ctegories, but these two types of grphs re not the sme s the two types of grphs tht underlie trictegories. This leds to sitution in which we re required to use vriety of universl properties in different ctegories to produce the desired comprison. The fct tht trictegories nd functors between them do not form ctegory enters the picture s well. With these fcts in mind, we tke cre to lwys stte in wht ctegory digrm is to be interpreted. We then prove tht every free trictegory is triequivlent to the free Gryctegory on the sme underlying dt vi the strict functor given by the universl property. Using this, we re in position to prove new theorem bout digrms of constrint cells commuting. Note tht it is not true tht every digrm of constrint 3-cells in trictegory commutes; the counterexmple comes from the fct tht trictegories with one 0-cell nd one 1-cell should be the sme (in some sense, see [11] for tretment of the difficulties in mking this sttement rigorous for 2-dimensionl structures) s brided monoidl ctegories. If we tke B to be brided monoidl ctegory with briding γ, then the eqution γ 2 = 1 is the condition tht B be symmetric. There re mny brided monoidl ctegories which re not symmetric, giving exmples of trictegories for which not every digrm of constrint 3-cells commutes. The theorem for bictegories tht we re emulting hs two components, universl property nd coherence theorem pplied to prticulr kind of exmple. Focusing on the prticulr kind of exmple involved (n underlying grph in which ll the 3-cells re identities, clled 2-loclly discrete), we prove tht in the free trictegory on 2-loclly discrete grph every digrm of constrint 3-cells commutes. This relies on new result tht in the free Gry-ctegory on 2-loclly discrete grph, every digrm of 3-cells commutes. The nlogous result for free 2-ctegories on loclly discrete grph is trivil, but the proof in this cse is not. Using these results, we exhibit digrm of constrint 3- cells tht does not lwys commute. Here it is the units in the trictegory tht prevent the ppliction of the coherence theorems; see [34] for more discussion of units in higher ctegories. It should be noted tht most of the digrms encountered in this work re esily shown to commute by this theorem. Using this theorem, we re ble to construct explicitly Gry-ctegory GrT nd triequivlences GrT T nd T GrT from ny trictegory T. These constructions mimic those given for bictegories, but re by necessity much more complicted. Finlly, we give prllel tretment of the coherence theory for functors. First we prove tht the trictegory freely generted by n underlying grph nd the constrint cells for functor is triequivlent to n pproprite Gryctegory. Using this triequivlence, we prove tht certin digrms consisting of constrint cells from both functor nd its trget must commute. This provides 7
8 CHAPTER 1. INTRODUCTION enough informtion to construct explicitly strictifiction GrF for ny functor F. This completes the project of replcing trictegories nd functors between them with Gry-ctegories nd Gry-functors up to triequivlence. It should be noted tht mny of our results, especilly in the erlier chpters, re either similr to or the sme s those in [17], lthough with chnged definitions. We will record these differences nd similrities s they rise. There re number of plces in this work where we re required to verify xioms involving very lrge digrms built from the trictegory constrints. Some of these clcultions re not explicitly included becuse of spce issues, but the relevnt equtions hve been checked rigorously. Now we provide brief description of ech of the chpters nd the three ppendices. Chpter 2 consists of rpid tretment of the coherence theory for bictegories. We include two proofs of coherence for bictegories, one using the Yoned embedding nd the other using the universl property of the free bictegory construction. This chpter is provided both to remind the reder of necessry bictegoricl results nd to give n ide of the pth we will tke through the coherence theory for tricgories. Chpter 3 provides the lgebric definitions of trictegory nd the higher cells between them. Our definitions differ from those in [17] in tht we require djoint equivlences where Gordon, Power, nd Street require equivlences. We do not require dditionl xioms even though our definitions require dditionl dt; we explin how the theory of mtes mkes the ddition of extr xioms unnecessry nd how this leds to the definition of the opposite trictegory. Chpter 4 is devoted to proving some importnt bsic results. First we study the composition of functors between trictegories nd show why these fil to form ctegory. We provide some conditions under which n ltered composition gives ctegory structure to trictegories nd strict functors. Then we study some opertions on trnsformtions tht will be necessry lter. These first two sections focus on the structure of the puttive tetrctegory Trict. The third section is concerned with chnging known trictegory structures to obtin new ones. Finlly we study the pproprite notion of equivlence between trictegories, tht is, triequivlence. Chpter 5 gives the necessry bckground on Gry s tensor product. We define this in three wys: by giving genertors-nd-reltions definition, by giving the universl property, nd by identifying the right djoint. We then collect together the relevnt properties to describe the closed symmetric monoidl ctegory Gry, whose underlying ctegory is the ctegory of strict 2-ctegories nd strict 2-functors between them. Chpter 6 contins the first constructions of trictegory structures from scrtch s well s n importnt first step in the proof of the coherence theorem. The concept of cubicl trictegory is introduced, nd strict, cubicl trictegories re shown to be Gry-ctegories. This gives Gry-ctegories n interprettion s semi-strict version of trictegories. Additionlly, we show tht the closed monoidl ctegory Gry inherits trictegoricl structure in this wy. We define full sub-gry-ctegory Gry Gry nd show tht this structure is
triequivlent to the trictegory structure on Bict which we construct directly. Chpter 7 studies the construction of the trictegory of functors, trnsformtions, modifictions, nd perturbtions between two fixed trictegories. We show tht given trictegories S nd T nd functors F, G : S T, there is bictegory with 0-cells the trnsformtions between F nd G, 1-cells the modifictions between those, nd 2-cells the perturbtions between those. When the trget trictegory is Gry-ctegory, we give composition functor nd the rest of the required dt necessry to give trictegory structure. We dditionlly prove tht this trictegory structure is ctully Gry-ctegory. Chpter 8 contins the proof tht every trictegory is triequivlent to Gry-ctegory. This is done by first replcing the trictegory in question with triequivlent cubicl one nd then proving Yoned Lemm for cubicl trictegories. Thus we see how the coherence theorem for trictegories breks up esily into two steps, the first of which is direct consequence of coherence for bictegories nd the second of which is nlogous to the proof of coherence for bictegories. Chpter 9 contins the construction of free trictegories; this finlly brings to ber the full power of the lgebric nture of our definition of trictegory. There re mny different options for the underlying dt of trictegory, nd we construct free trictegories for the two choices tht will be most importnt for the proof of coherence. We lso construct free Gry-ctegories s well, nd prove some importnt results needed in the next chpter. We note tht these free constructions re ll left djoints to the obvious forgetful functors. Chpter 10 contins two new coherence theorems. First, we prove tht the free trictegory on grph is strictly triequivlent to the free Gry-ctegory constructed from the sme dt in cnonicl wy. Then we go on to prove tht certin free Gry-ctegories hve very restricted structure. This in turn leds to n esy proof of nother coherence theorem stting tht certin digrms of constrints in ny trictegory lwys commute. This theorem llows us to construct, from ny trictegory T, Gry-ctegory GrT nd triequivlences between these two trictegories. Chpter 11 provides coherence theorem for functors. We begin by nlyzing the free functor on mp of underlying grphs. This leds to coherence theorem for functors stting tht certin digrms consisting of both constrint cells of functor nd the constrints of its trget trictegory lwys commute. We use this theorem to produce Gry-functor GrF : GrS GrT from ny functor F : S T. Three ppendices re included. The first collects few results concerning djoint equivlences nd bidjoint biequivlences tht will be needed throughout the work. We hve lso included here brief review of the theory of mtes. The second ppendix gives unpcked versions of ll the dt in the definitions in Chpter 3. The third ppendix dels with clcultionl issues tht re present in few plces, most notbly Chpters 4, 7, nd 8. The ide of mking the definition of trictegory fully lgebric hs existed informlly for some time but the detils hve never been worked out rigorously. Even though mny of the ides behind the definitions nd proofs here re simple, 9
10 CHAPTER 1. INTRODUCTION often the clcultions re quite involved; the proof of Theorem 10.2.2 nd ll of the clcultions tht reference Appendix C re good exmples. But these clcultions, nd the coherence theory tht follows, re necessry if trictegories re to be utilized in genuine pplictions. Gordon, Power, nd Street proved n importnt coherence theorem for wek 3-ctegories. We hve ltered their definition, not becuse it is incorrect in some wy, but becuse it is not suited for mking the kinds of constructions tht we desire for future pplictions. In doing so, we were led to simple proofs of importnt coherence results tht could not be stted using the originl definition.
Chpter 2 Coherence for bictegories In this chpter, we will give rpid tretment of the coherence theory for bictegories, including full proof for the coherence theorem for functors. The gol of this chpter is to prepre the reder for the pth we will tke through the coherence theory for trictegories, s well s to recll some crucil fcts tht will be used throughout. The overll strtegy here is dpted from the one used in [22] for monoidl ctegories. We will give two proofs tht every bictegory is biequivlent to strict 2- ctegory, ech hving its own flvor. The first proof cn be dispensed with quickly. The second proof requires some of the tools developed for the first, but lso llows us to prove the coherence theorem for functors. 2.1 Bictegoricl conventions In ny bictegory B, we shll use the letters, l, nd r to denote the ssocitivity, left unit, nd right unit isomorphisms, respectively. Verticl composition of 2-cells will be written s conctention, nd the symbol * will be used to denote horizontl composition. The terms pseudofunctor, wek functor, nd homomorphism of bictegories re ll used throughout the literture to refer to the sme concept. We will lwys write functor for this notion; ny strict or lx functor will be lbeled s such. Given functor F, we will genericlly denote its constrints by ϕ since the source nd trget of this constrint mke it cler wht kind of constrint cell it is. We follow the convention of [17] nd not of the other references ([5] nd [37] for instnce) in wht is ment by lx trnsformtion. For our purposes, lx 11
12 CHAPTER 2. COHERENCE FOR BICATEGORIES trnsformtion α : F G consists of 1-cells α : F G nd 2-cells F α G Ff α f Gf Fb subject to two xioms. A trnsformtion is lx trnsformtion such tht the cells α f re invertible for every f : b. A trnsformtion between strict 2-functors is 2-nturl trnsformtion if the cells α f re identities for ll f. Since we hve chnged the orienttion of the nturlity isomorphism in the definition of trnsformtion, it is necessry to lter the definition of modifiction by chnging its xiom. These chnges re not substntive, they merely void excessive use of the prefix op-. A numbered prefix, such s in 2-ctegory or 2-functor, will lwys refer to the strict notion. Our nming conventions for the corresponding concepts for trictegories will be the sme, s we reserve the terms functor, trnsformtion, etc., to men the wek version. Any strict or lx version of these concepts will lwys be clled such. α b Gb 2.2 The Yoned embedding This section is devoted to proving coherence theorem by first developing n pproprite Yoned lemm for bictegories. We will not provide ny proofs in this section, we insted refer the reder to [36] or [41]. Proposition 2.2.1. Let B, C be bictegories. There is bictegory Bict(B, C) whose 0-cells re the functors F : B C, whose 1-cells re the trnsformtions α : F G, nd whose 2-cells re the modifictions Γ : α β. The proof of this proposition requires identifying the constrint cells nd then checking the bictegory xioms. These constrint cells re obtined from the constrint cells in the trget, giving the following corollry. Corollry 2.2.2. If C is strict 2-ctegory nd B is ny bictegory, then the functor bictegory Bict(B, C) is strict 2-ctegory. Definition 2.2.3. Let B be bictegory. Then the bictegory B op hs the sme cells s B, the 1-cell source nd trget mps re switched, r op = l, l op = r, nd op fgh = 1 hgf. Now we re in position to define the Yoned mp y : B Bict(B op,ct) nd stte the Yoned Lemm for bictegories.
2.3. COHERENCE FOR BICATEGORIES 13 Definition 2.2.4. Let B be bictegory. Then the Yoned mp y : B Bict(B op,ct) is defined on the underlying 2-globulr set s follows. The functor y cts by sending n object to the functor which is defined on 0-cells by b B(b, ), on 1-cells by the functor which is g gf on objects, nd on 2-cells by sending α to the trnsformtion with components 1 g α. The functor y cts on the 1-cell f : by sending it to the trnsformtion with component t b given by g fg, nd for h : b c, the 2-cell yf h is 1 fgh. The functor y cts on 2-cells by sending α : f f to the modifiction with component α 1 g. Definition 2.2.5. Let P be property of functors between ctegories. A functor F : B C between bictegories is loclly P if ech functor F b hs property P. Theorem 2.2.6 (Bictegoricl Yoned Lemm). The Yoned functor y : B Bict(B op,ct) is loclly full nd fithful. Corollry 2.2.7. Every bictegory is biequivlent to strict 2-ctegory. Proof. Let I be the sub-2-ctegory of Bict(B op,ct) consisting of those 0- cells which re in the imge of y, those 1-cells which re isomorphic to some yf, nd ll 2-cells between them. It is immedite tht this is 2-ctegory. Then y : B I is loclly full nd fithful by Theorem 2.2.6, nd it is biessentilly surjective nd loclly essentilly surjective by definition. 2.3 Coherence for bictegories This section is devoted to proving coherence theorem of the form every free bictegory is biequivlent to strict free 2-ctegory vi strict functor. Using this, we obtin biequivlence stb B for every bictegory B, where stb is strict 2-ctegory. Other notions of coherence re mentioned. 2.3.1 Grphs nd free constructions Definition 2.3.1. The ctegory Gr(Ct) of ctegory-enriched grphs (which we will lso cll Ct-grphs) hs objects G consisting of set G 0 of objects nd for every pir of objects, b, ctegory G(, b). A mp f : G G of Ct-grphs consists of functions f 0 : G 0 G 0 nd functors f b : G(, b) G (f 0, f 0 b). The free bictegory on Ct-grph G, denoted FG, hs the following underlying 2-globulr set. The set of 0-cells of FG is G 0. The set of 1-cells is inductively defined to include new 1-cells I for ech G 0, 1-cells f : b for ech object f G(, b), nd 1-cells f g if f, g re both 1-cells of FG. The source nd trget functions re defined in the obvious fshion. The set of 2-cells of FG is defined in three steps. The first is to define bsic 2-cell. These re built inductively from the rrows in ll of the G(, b) nd new isomorphism 2-cells fgh, l f, r f by binry horizontl composition. Secondly,
14 CHAPTER 2. COHERENCE FOR BICATEGORIES we form composble strings of these bsic 2-cells. Finlly, we quotient out by the equivlence reltion generted by nturlity of the 2-cells fgh, l f, r f, the middle-four interchnge lw, the rule tht the composition α β in FG grees with tht of G if α, β re rrows in some G(, b), nd the two bictegory xioms. Note tht there is n obvious inclusion i : G FG of ctegory-enriched grphs. Proposition 2.3.2. 1. The dt bove stisfy the necessry xioms to constitute bictegory. 2. Let B be bictegory. Then given mp f : G B of ctegory-enriched grphs, there is unique strict functor of bictegories f : FG B such tht fi = f in Gr(Ct). Proof. The first sttement is obvious by the definition. The second sttement follows by defining f using induction nd strictness. Now we define the free 2-ctegory on Ct-grph G, denoted F s G. The set of 0-cells is the set G 0. The set of 1-cells is the set of composble strings of length 0, where the unique string of length zero will be the identity 1-cell. The set of 2-cells from one string f n f n 1 f 1 to nother g m g 1 is empty if n m, nd otherwise consists of the strings α n α n 1 α 1 where α i : f i g i in some G(, b). Composition of 1-cells is by conctention, nd composition of 2-cells is given by (α n α 1 ) (β n β 1 ) = (α n β n ) (α 1 β 1 ). It is simple mtter to verify the following proposition, where here j denotes the inclusion of G into F s G. Proposition 2.3.3. 1. The dt bove stisfy the necessry xioms to constitute 2-ctegory. 2. Let X be 2-ctegory. Then given mp f : G X of ctegory-enriched grphs, there is unique 2-functor f : F s G X such tht fj = f in Gr(Ct). Thus the sttement of the coherence theorem for bictegories becomes the following. Theorem 2.3.4 (Coherence for bictegories). The functor Γ : FG F s G induced by j : G F s G is strict biequivlence. 2.3.2 Proof of the coherence theorem Definition 2.3.5. Let G, G be ctegory-enriched grphs, nd let S, T : G G be mps between them. The ctegory-enriched grph Eq(S, T) is defined to hve objects those G 0 such tht S 0 = T 0. The ctegory Eq(S, T)(, b) hs objects pirs (h, α) where h : b in G nd α : Sh Th is n isomorphism in G (S 0, S 0 b). The morphisms β : (h, α) (h, α ) re those β : h h in G such tht α S(β) = T(β) α.
2.3. COHERENCE FOR BICATEGORIES 15 Note tht there is mp π : Eq(S, T) G defined by π() = π(h, α) = h π(β) = β. Lemm 2.3.6. Let B, C be bictegories, nd F, G : B C be functors between them. Then Eq(F, G) supports bictegory structure such tht π cn be extended to strict functor Eq(F, G) B. Furthermore, there is trnsformtion σ : Fπ Gπ whose components re ll identity mps. Proof. For the first clim, we must define composition, identity 1-cells, constrint 2-cells, nd check the bictegory xioms. To fix nottion, the constrint cells for F will be ϕ fg nd ϕ 0, while those for G will be ψ fg nd ψ 0. Composition of 1-cells is then defined by the formul (g, β) (f, α) = (gf, ψ fg (β α) ϕ 1 fg ). The identity 1-cell for the object is (id, ψ 0 ϕ 1 0 ). It is simple to check tht the ssocitivity nd unit constrints from B re 2-cells in Eq(F, G) with the pproprite sources nd trgets; from this the bictegory xioms follow immeditely. It is trivil to check tht π cn be extended to strict functor. Finlly, we define the trnsformtion σ : Fπ Gπ. The component t is id. The component t (f, α) is r 1 α l; this is nturl trnsformtion by the definition of morphisms in Eq(F, G) nd the nturlity of both l nd r. The trnsformtion xioms follow esily. Proposition 2.3.7. Let F : FX B be functor from free bictegory into ny bictegory. Then there is strict functor G : FX B nd trnsformtion α : F G such tht α = id F for every object. Proof. Since FX is free, there is unique strict functor G : FX B such tht Fi = Gi s mps X B. We lso hve mp ι : X Eq(F, G) which is the identity on objects, sends f to (f, id Ff ), nd sends β to β. Note tht πι = i nd the trnsformtion σ 1 ι is the identity. This produces, by the universl property of FX, unique strict functor ι : FX Eq(F, G) such tht ιi = ι. This gives the equlity π ιi = i, nd since π ι is strict, it must be the identity functor on FX. Then the trnsformtion σ 1 ι is trnsformtion from Fπ ι = F to Gπ ι = G, nd it hs s its component t the 1-cell id G by the definition of σ 1 ι.
16 CHAPTER 2. COHERENCE FOR BICATEGORIES It should be noted tht we hve used tht functors of bictegories compose in strictly ssocitive nd unitl fshion in this proof. Let f : X B be mp of ctegory-enriched grphs into bictegory B. Then we cn extend f to mp of ctegory-enriched grphs ˆf : F s X B which is defined s follows. The object function ˆf 0 grees with f 0. The identity 1-cell on gets mpped to the identity 1-cell on f 0, nd ˆf(h) = f(h) where h : b is n object of X(, b). If h n h 1 : b in F s X, then ˆf(h n h 1 ) = ( (fh n fh n 1 ) fh n 2 ) fh 2 ) fh 1. Similrly, ˆf(α n α 1 ) is the 2-cell ( (fα n fα n 1 ) fα 2 ) fα 1. Lemm 2.3.8. Let G be ctegory-enriched grph, nd let F : FG X be strict functor into 2-ctegory X. Then there exists unique strict functor F s : F s G B such tht F = F s Γ. Proof. This is n immedite consequence of the universl properties of F, F s, nd the fct tht Γi = j. Lemm 2.3.9. Let F, G : B C be functors between bictegories, nd let α : F G be trnsformtion between them. Assume tht F nd G gree on objects, nd tht α = id F for ll objects. Then F is loclly fithful (loclly full) if nd only if G is loclly fithful (loclly full). Proof. We need only show tht F loclly fithful implies G loclly fithful since there is trnsformtion α 1 : G F tht hs ll its components identity mps defined by tking (α 1 ) f = l 1 r (α f ) 1 r 1 l. Using the nturlity of r nd the nturlity of the 2-cells α f, we get Gα = r α f (1 Fα) α 1 f r 1, where α : f f. Thus G is loclly fithful since the the composite on the right is loclly fithful function of α. The sme proof shows locl fullness. Proof of 2.3.4. It is cler tht Γ is surjective on objects, so we need only show tht it is loclly n equivlence of ctegories. We hve the mp î : F s G FG, nd it is simple to check tht the composite mp of ctegory-enriched grphs F s G î FG Γ F s G is the identity, so Γ is loclly essentilly surjective. From this it lso follows tht Γ is loclly full. To show tht Γ is loclly fithful, first note tht there is loclly fithful functor T : FG X into strict 2-ctegory X by the Yoned Lemm. There is strict functor S : FG X nd trnsformtion α : S T with α = id by Proposition 2.3.7. By the universl property of the mp Γ, there is unique strict functor R : F s G B such tht RΓ = S. Now S is loclly fithful since T is, hence Γ must be loclly fithful s well.
2.3. COHERENCE FOR BICATEGORIES 17 2.3.3 Using coherence: strictifiction Let B be bictegory. We use the coherence theorem to construct strictifiction stb of B, long with biequivlence e : stb B. The 2-ctegory stb will hve the sme objects s B. A 1-cell from to b will be string of composble 1-cells of B, where there is unique empty string which will be the identity 1-cell. Before defining 2-cells, we define e on 0- nd 1-cells. On 0-cells, e is the identity. On 1-cells, we define e(f n f n 1 f 1 ) = ( (f n f n 1 )f n 2 ) f 2 )f 1 ; for the empty string :, we set e( ) = I. The set of 2-cells between the strings f n f n 1 f 1 nd g m g m 1 g 1 is defined to be the set of 2-cells between e(f n f n 1 f 1 ) nd e(g m g m 1 g 1 ) in B. It is now obvious how e cts on 2-cells. The 2-ctegory structure of stb is defined s follows. Composition of 1-cells is given by conctention of strings, with the empty string s the identity. It is immedite tht this is strictly ssocitive nd unitl. Verticl composition of 2-cells is s in B, nd this is strictly ssocitive nd unitl since verticl composition of 2-cells in bictegory is lwys strict in this wy. Let A be the sub-ctegory-enriched grph of B with ll the sme objects but with A(, b) the discrete ctegory with oba(, b) = obb(, b). By coherence, the strict functor Γ : FA F s A is biequivlence, nd it is esy to see tht the 2-ctegory F s A is loclly discrete. Thus, in FA, the set of 2-cells between ny two 1-cells is either empty or singleton, depending on whether these 1-cells re mpped to the sme 1-cell by Γ. (Note tht this is one wy to prove the ll digrms of constrint cells commute form of coherence for bictegories.) In prticulr, we hve unique coherence isomorphism e(f n f 1 )e(g m g 1 ) = e(f n f 1 g m g 1 ). Thus we cn now define the horizontl composition α β in stb s the composite e(f n f 1 g m g 1 ) = e(fn f 1 )e(g m g 1 ) α β e(f n f 1)e(g m g 1) = e(f n f 1 g m g 1 ) in B, where the unlbeled isomorphisms re induced by the strict mp FA B. The uniqueness of these isomorphisms ensures tht this definition stisfies the middle-four interchnge lws s well s being strictly ssocitive nd unitl. By definition, e is functoril on verticl composition of 2-cells. The constrint cells for e re induced by the strict mp FA B in similr fshion s bove. The uniqueness of these cells immeditely forces the functor xioms to hold. Finlly, it is trivil to see tht e is biequivlence s it is surjective on objects, loclly surjective on 1-cells, nd 2-locl isomorphism on 2-cells by definition. Thus we hve completed the tsk of producing, for ech bictegory B, strict 2-ctegory stb nd biequivlence e : stb B.
18 CHAPTER 2. COHERENCE FOR BICATEGORIES It will be useful lter to note tht there exists biequivlence f : B stb defined s follows. The mp f is the identity on objects, includes ech 1-cell s the string of length 1, nd then is the identity on 2-cells s well. This is functoril on 2-cells, nd we cn tke both constrint cells to be represented by identity 2-cells in B (lthough they re not identities in stb). The functor xioms re then esy to check. The only thing to check to show tht f is biequivlence is tht it is loclly essentilly surjective, but this is esy s every 1-cell f n f 1 is clerly isomorphic to 1-cell of length 1, nmely e(f n f 1 ); the empty string is isomorphic to the identity mp viewed s 1-cell of stb, so f is loclly essentilly surjective. It should be noted tht ef = 1 B, nd fe is biequivlent to 1 stb in Bict(stB, stb) by trnsformtion whose components on objects cn ll be tken to be identities nd whose components on 1-cells ll come from coherence. Remrk 2.3.10. The previous prgrph contins ll of the informtion needed to conclude tht every bictegory is equivlent to strict 2-ctegory inside of the 2-ctegory NHom studied by Lck nd Poli in [27]. 2.4 Coherence for functors In this section, we prove coherence result for functors of bictegories. This theorem is nlogous to Theorem 2.3.4 in tht it sttes tht free functors re biequivlent to free strict functors. The sttement is slightly more delicte, but it produces similr results to those in Section 2.3.3. 2.4.1 Free functors Let ϕ : G G be mp in Gr(Ct). Our gol is to produce the free functor generted by ϕ; the source of this functor will be the free bictegory generted by G, but the trget is more complicted object. The ide is tht the trget will be the free bictegory generted by G nd new 2-cells tht will ply the role of constrint cells. We define the bictegory F(G, ϕ) s follows. The 0-cells of F(G, ϕ) re the sme s the objects of G. The 1-cells re generted (using binry composites) by new 1-cells I :, the 1-cells of G, nd new 1-cells ϕ(r) for every 1-cell r in FG. These re subject to the requirement tht ϕ(r) = s in F(G, ϕ) if r is n object G(, b) nd ϕ(r) = s in G, nd we extend this over composition. The 2-cells re defined in sequence of steps nlogous to how we defined the 2-cells of FG. The first step is to form bsic 2-cells from the 2-cells of G, 2-cells ϕ(α) with α 2-cell of FG (subject to the sme kind of condition tht we imposed on the 1-cells ϕ(r)), nd isomorphism constrint cells fgh, l f, r f, ϕ, ϕ fg by binry horizontl composition. Then we form strings of verticlly composble bsic cells, nd finlly we quotient out by the equivlence reltion formed by the necessry nturlity conditions long with the xioms for bictegory nd those required of the 2-cells ϕ, ϕ fg to force ϕ to extend to wek functor