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Advances in Mathematics 169, 118–175 (2002) doi:10.1006/aima.2001.2056 ACellular Nerve for Higher Categories Clemens Berger Laboratoire J.-A. Dieudonn!e, Universit !e de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice, Cedex 2, France E-mail: cberger@math:unice:fr Communicated by Ross Street Received December 6, 2000; Accepted September 29, 2001 We realise Joyal's cell category Y as a dense subcategory of the category of o- categories. The associated cellular nerve of an o-category extends the well-known simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen's sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of % A-algebras for each o-operad A in Batanin's sense. Whenever A is contractible, the resulting homotopy category of % A-algebras (i.e. weak o-categories) is equivalent to the homotopy category of compactly generated spaces. _ 2002 Elsevier Science (USA) Key Words: higher categories; globular operads; combinatorial homotopy. The following text arose from the desire to establish a firm relationship between higher categories and topological spaces. Our approach combines the algebraic features of Batanin's o-operads [2] with the geometric features of Joyal's cellular sets [25] and tries to mimick as far as possible the classical construction of the simplicial nerve of a small category
Voir
- simplex category
- between weak
- cellular spaces
- nerve functors
- homotopy category
- intn ? id
- category comes
- quillen equivalence between
- insn ?
Advances in Mathematics169, 118–175 (2002) doi:10.1006/aima.2001.2056
A Cellular Nerve for Higher Categories
Clemens Berger
Laboratoire J.-A. Dieudonne!, Universit!ede Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice, Cedex 2, France E-mail: cberger@math:unice:fr
Communicated by Ross Street
Received December6, 2000; Accepted September29, 2001
We realise Joyal’s cell categoryYas a dense subcategory of the category ofo-categories. The associated cellular nerve of ano-category extends the well-known simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategoryYAof the category of%A-algebras for eacho-operadAin Batanin’s sense. WheneverAis contractible, the resulting homotopy category of%A-algebras (i.e. weako-categories) is equivalent to the homotopy category of compactly generated spaces.#2002 Elsevier Science (USA) Key Words:higher categories; globular operads; combinatorial homotopy.
The following text arose from the desire to establish a firm relationship betweenhighercategoriesandtopologicalspaces.Ourapproachcombines the algebraic features of Batanin’so-operads[2] with the geometric features of Joyal’scellular sets[25] and tries to mimick as far as possible the classical construction of the simplicial nerve of a small category. Eacho-category has an underlyingo-graph(also calledglobular set[37]) and comes equipped with a family ofcomposition lawsgoverned by Godement’sinterchange rules[21, App.1.V]. The forgetful functor from o-categories too-graphs ismonadic. The left adjoint free functor may be deduced from Batanin’s formalism ofo-operads; indeed, it turns out that o-categories are the algebras for theterminalo-operad. This leads to Batanin’s definition ofweako-categoriesas thealgebrasfora (fixed) contractibleo-operad, which may be compared with Boardman–Vogt– May’s definition ofE1-spaces [8, 29]. The main purpose here is to define a whole family ofnerve functors, one foreacho-operad, and to study under which conditions these
0001-8708/02 $35.00 #2002 ElsevierScience (USA) All rights reserved.
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nerve functors define a well-behavedhomotopy theoryforthe undelrying algebras. Nerve functors are induced by suitablesubcategories. The simplicial nerve, for instance, is defined by embedding the simplex categoryDin the category of small categories. By analogy, we construct for eacho-operadA ofAe induced nerve adensesubcategoryYAof the category%-algebras. Th NAis then afully faithfulfunctorfrom%A-algebras to presheaves onYA:Its image may be characterised by a certainrestricted sheaf condition. Even in the case ofo-categories, the existence of such a fully faithful nerve functor is new, cf. [13, 36, 38]. We denote the corresponding dense subcategory byY and call presheaves onYcellular sets. This terminology has been suggested to us by the remarkable fact that theoperator categoryYcoincides with Joyal’scell categoryYalthough the latterhas been defined quite differently. Indeed, Joyal’sYplays the same role foro-operads and weako-categories as Segal’sGfor symmetric operads andE1-spaces, cf. [34, App. B]. According to Joyal [25], cellular sets have a geometric realisation in which simplexandballgeometry are mixed through the combinatorics ofplanar level trees. It follows thato-categories realise via their cellular nerve the same way as categories do via their simplicial nerve.Weako-categoriesalso have a geometric realisation by means of theleft Segal extension[34, App. A] of theirA-cellular nerve along the canonical functor fromYAtoY:This realisation induces a natural concept ofweak equivalencebetween weako-categories. Cellular sets carry aclosed model structurein Quillen’s sense [31]. Like for simplicial sets, thefibrationsare defined byhorn fillerconditions. There is a wholetowerof Quillen equivalent model categories beginning with simplicialsetsandendingwithcellularsets.Indeed,thecellcategoryYis filtered by full subcategoriesYðnÞsuch thatYð1Þequals the simplex category Dand such thatYðnÞis a Cauchy-complete extension of Simpson’s [35] quotientYn¼Dn=: Thehomotopy categoryof cellularsets is equivalent to the homotopy category of compactly generated spaces. The cellular nerve, however, does not ‘‘create’’ a model structure foro-categories, mainly because the left adjointo-categorificationdoes not yield the correct homotopy type forall cellularsets. In odrerto solve this difficulty, we consider cellularsets as thediscrete objectsamongcellular spacesand construct a convenient model structure for cellular spaces. Here, theo-categorification yields a Quillen equivalence between cellularspaces andsimplicial o-categories. Both homotopy categories are determined by the discrete objects so that we end up with an equivalence between the homotopy categories of cellular sets and ofo-categories. More generally, for eachcontractibleo-operadA;there exists a model structure forA-cellular
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spaces such that theA-categorification induces a Quillen equivalence betweenA-cellularspaces and simplicial%A-algebras. Again, the discrete objects span the entire homotopy categories. Moreover, base change along YA!Yinduces a Quillen equivalence betweenA-cellularspaces and cellularspaces. Each topological spaceXdefines a fundamentalo-graphPXwhosen-cells are the continuous maps from then-ballBntoX:There is a contractible o-operad acting onPX;constructed by Batanin [2], so that viainductively the above-mentioned Quillen equivalences, the homotopy type ofXis entirely recoverable from this algebraic structure. In what sense the fundamentalo-graph is aweako-groupoidand to what extent weako-groupoids recover all homotopy types among weako-categories will be the theme of subsequent papers.
0. NOTATION AND TERMINOLOGY
We shall follow as closely as possible the expositions of Borceux [9], Gabriel-Zisman [20] and Quillen [31] concerning categorical, simplicial and model structures, respectively. Below, a summary of the most frequently used concepts. A functorFis called (co)continuousifFpreserves small (co)limits. A functorF preserves(resp.detects) a propertyPif, wheneverthe morphismf (resp.Ff) has propertyP;then alsoFf(resp.f). Thecategory of sets(resp.simplicial sets) is denoted byS(resp.sS).
0.1. Tensor Products ForfunctosrF:Cop!SandG:C!E;thetensor product FCG is an object ofEsubject to theadjunctionformulaEðFCG;EÞ ffi HomCðF;EðG;EÞÞ;where HomCðF;F0Þdenotes the set ofnatural transfor-mations F!F0;and whereEðG;EÞdenotes the presheaf defined by EðG;EÞðÞ ¼EðGðÞ;EÞ: If the categoryEis cocomplete, the tensorproductFCGis the so-called coendof the bifunctorðC0;CÞ/FðC0Þ GðCÞ:¼‘FðC0ÞGðCÞand can thus be identified with the coequaliser aFðC0Þ GðCÞ4aFðCÞ GðCÞ!FCG: f:C!C0C
Fortwo functosr the tensorporduct FðÞ GðÞ:
F:C!SandG:D!Sof the same variance, FG:CD!Sis defined byðFGÞðÞ ¼
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0.2. Higher Graphs and Higher Categories The globe categoryGhas one object%nforeach integern50:The reflexive % globe categoryGhas same objects asG: Theglobular operatorsare generated by cosource/cotarget operators sn;tn:%n4nþ1 and in the reflexive case also by coidentitiesin:nþ1!n % subject to the relationssnþ1sn¼tnþ1sn;snþ1tn¼tnþ1tn;insn¼intn¼idn%; n50: % A presheaf onG(resp.G) is called ano-graph(resp. reflexiveo-graph). Street [37] callso-graphsglobular sets. Ano-graphX:Gop!Swill often be denoted as anN-graded family of setsðXnÞn50which comes equipped with source/target operations:
4Xnþ14Xn4 4X14X0:
Anois empty in degrees strictly greater than-graph which n;is called an n-graph. The operations induced bysn=tnare calledsource=targetmaps. The operations induced byinare calledidentitymaps. The representable functor Gð;n%Þis thestandard n-cell. A 2-categoryis a small Cat-enriched category, where Cat denotes the category of small categories. The objects of a 2-categoryCare the 0-cells, the objects (resp. morphisms) of the categorical hom-setsCð;Þare the 1-cells (resp. 2-cells) ofC:Thesource=targetandidentitymaps define a reflexive2-graphunderlyingC:A 2-category comes equipped with three composition laws8ji:CjiCj!Cj;04i5j42;subject toGodement’s interchange rules[21]. Ano-categoryC a reflexive[3, 36] iso-graph which comes equipped with composition laws8ji:CjiCj!Cj;i5j;lpirtyna-nonfoesuchthat,for negativeintegerwsitih5rjes5pke;teheytli(afirmttcheotðaCtie;dC)j;Cukr;c8eji/;t8akir;g8ekjtÞsrpueytamehnetsitnrduicdtahast of a 2-category so of the underlying reflexiveo-graph. The category ofo-categor ies is denoted byo-Cat orAlgo;cf. Theorem 1.12. %
0.3. Monads and their Algebras
Amonadon the categoryEis a monoidðT;Z;mÞin the category of endofunctors ofE:AT-algebrais a pairðX;mXÞconsisting of an objectX ofEand aT-action mX:TX!Xwhich isunital(mXZX¼idX) and associative(mXmX¼mXTmX). Thecategory of T -algebrasis denoted by AlgT: We shall use (slightly abusively) the same symbol to denote as well the monadTas well thefree functor T:E!AlgTsince the freeT-algebra on an objectXofEis given byðTX;mXÞ:
122CLEMENS BERGER ApairofadjointfunctorsF:E.E0:Gwith left adjointFinduces a monadðGF;Z;mÞonEwhereZis the unit of the adjunction andm¼GeFis induced by the couniteof the adjunction. A functorG:E0!EismonadicifGhas a left adjointF:E!E0such hatY/ðGY;GeYÞnduces an equivalence of categoriesE0 ! t i AlgGF: 0.4. Categories of Elements, Filtered Colimits and Finite Objects
Foraset-valuedpresheafFonC;thecategory of elementselðFÞhas as objects the pairsðC;xÞwithx2FðCÞand as morphismsf:ðC;xÞ ! ðC0;x0ÞtheC-morphismsf:C!C0withFðfÞðx0Þ ¼x: Every set-valued presheaf is the colimit of representable presheaves Cð;CÞaccording to the formula: lim!ðC;xÞ2elðFÞCð;CÞ !Fwhere the componentsx:Cð;CÞ !Fof the colimit cone are induced by the Yoneda-lemma. A categoryCisfilteredif the following three properties hold:Cis non-empty; forany two objectsA;BofCthere is an objectCofCsuch that the morphism-setsCðC;AÞandCðC;BÞare non-empty; for any parallel pair ofC-morphismsf;g:A4Bthere is aC-morphismh:C!Asuch that fh¼gh: Acolimitlim!CFisfilteredif theoppositecategoryCopof the indexing category is filtered. An objectAof a categoryEis calledfiniteif the representable diagram EðA;Þpreservesfilteredcolimits. Quillen [31] calls finite objectssmall, whence hissmall object argument. Finite objects are often called finitely presentable oro-presentable whereois the first infinite cardinal, cf. [9, II.5]. The finite objects of the category of sets are precisely the finite sets; the finite objects of a category of set-valued presheaves are precisely the quotients of finite coproducts of representable presheaves. The finite objects of an algebraic category are the objects of finite presentation, i.e., those having finitely many generators and finitely many relations, cf. [9, II.3.8.14].
0.5. Compactly Generated Spaces
The category Top of topological spaces has certain drawbacks among which the lack of acartesian closedstructure and thenon-finitenessof compact (i.e. quasi-compact Hausdorff) spaces. These disadvantages disappear when we restrict to the full subcategory Topcofcompactly generatedspaces, which is the largest subcategory of Top with the property that the categoryKof compact spaces isdensein Topc:According to Day [15], the category Topcembeds as areflectivesubcategory in a categoryDof special presheaves onK;which is cartesian closed and in which the compact spaces are finite; the reflector fromDto Topcpreserves finite products and
