Table Of ContentAlgebra and Applications
Simona Paoli
Simplicial
Methods
for Higher
Categories
Segal-type Models
of Weak -Categories
n
Simplicial Methods for Higher Categories
Algebra and Applications
Volume 26
Serieseditors:
MichelBroué
UniversitéParisDiderot,Paris,France
AliceFialowski
EötvösLorándUniversity,Budapest,Hungary
EricFriedlander
UniversityofSouthernCalifornia,LosAngeles,CA,USA
IainGordon
UniversityofEdinburgh,Edinburgh,UK
JohnGreenlees
SheffieldUniversity,Sheffield,UK
GerhardHiß
AachenUniversity,Aachen,Nordrhein-Westfalen,Germany
IekeMoerdijk
RadboudUniversityNijmegen,Nijmegen,Utrecht,TheNetherlands
ChristophSchweigert
HamburgUniversity,Hamburg,Germany
MinaTeicher
Bar-IlanUniversity,Ramat-Gan,Israel
AlainVerschoren
UniversityofAntwerp,Antwerp,Belgium
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Simona Paoli
Simplicial Methods
for Higher Categories
Segal-type Models of Weak n-Categories
123
SimonaPaoli
DepartmentofMathematics
UniversityofLeicester
Leicester,UK
ISSN1572-5553 ISSN2192-2950 (electronic)
AlgebraandApplications
ISBN978-3-030-05673-5 ISBN978-3-030-05674-2 (eBook)
https://doi.org/10.1007/978-3-030-05674-2
MathematicsSubjectClassification(2010):18-XX,55P-XX
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To myparents
Preface
Thetheoryofhighercategoriesisaveryactiveareaofresearchwhichhaspenetrated
diversefieldsofscience.
Historically the subject was motivated by questions in algebraic topology and
mathematicalphysics,twoareaswherethemostimportantapplicationsarecurrently
found. Algebraic geometry also makes use of higher categorical notions. More
recentlyhighercategorieshavefoundtheirwayintologicandcomputerscience,and
are also starting to appear in algebra and representationtheory. Higher categories
can sometimes be used as a common language to describe complex phenomena
occurringintheseareas.
Aplethoraofdifferentapproachestohighercategorieshavebeendevelopedover
theyears.Eachonerepresentscertainrelevantaspectsoftheabstractnotionbeing
modelled,oftenwithaviewtosupportingaparticularecosystemofapplications.At
thisstagenosingleapproachsuitsallsuchcontexts,andindeedonemightdoubtthe
viabilityofauniversallyapplicablemodel.Itappearsinsteadthatthemostprudent
approachis to continuethedevelopmentofall ofthese importantstrands,relating
themwherenecessarybyexplicitcomparisons.
Thepurposeofthismonographistointroduceanewapproachtoworkingwith
highercategories:this is based on a simple higher categoricalstructure consisting
of iterated internal categories (also called n-fold categories) as well as on a
new paradigm to weaker higher categorical structures, which is the idea of weak
globularity.
We show that our new model, called weakly globular n-fold categories, is
suitably equivalent to a model of higher categories that has been studied in great
depth, the one introduced by Tamsamani [126] and further studied by Simpson
[119].
Weachievethiscomparisonbydevelopingalargercontextof‘Segal-typemodels
of weak n-categories’, based on multi-simplicial structures, of which both the
Tamsamanimodelandweaklyglobularn-foldcategoriesarespecialcases.
The use of simplicial structures to capture higher coherence phenomena has a
long history in algebraic topology,starting with the work of Graeme Segal [117],
and then the study of categories enriched in simplicial sets by Dwyer and Kan
vii
viii Preface
[48–50], Dwyer et al. [52] and others. More recently, simplicial techniques have
underpinned the development of so-called (∞,n)-categories, and several models
havebeendevelopedandstudiedbyBergner[23],BergnerandRezk[27],Barwick
andKan[9],Lurie[89],Joyal[74],Rezk[109,110]andothers.
Simplicial models of (∞,∞)-categorieshave been developedby Verity [137],
building upon insights from the study of simplicial nerves of strict n-categories
initiatedbyStreet[123].
In thiswork we concentrateon higherstructuresin the ‘truncated’case, where
there are higher morphisms only in dimensions 0 up to n. This is intimately
connected to the Postnikov tower in algebraic topology. In fact, the algebraic
modelling of the building blocks of spaces, the n-types, which are Postnikov
sections of spaces, is related to models of weak n-categories via the so-called
‘homotopyhypothesis’:agoodmodelofweakn-categoriesshouldgiveanalgebraic
modelofn-typesintheweakn-groupoidcase.Weshowthatourmodelsdosatisfy
thehomotopyhypothesis.
There are long-standing open questions about weak n-categories, both within
category theory and in its applications to homotopy theory: for instance the
comparisonbetweenthesimplicialandhigheroperadicmodelsofhighercategories
andthealgebraicdescriptionofthek-invariantsofspacesandsimplicialcategories.
Thepresentworkprovidesaplatformwheretheseandotheropenquestionscan
bestudied,asweoutlineinthelastchapter.Thesequestionshowevergobeyondthe
scopeofthiswork,whosegoalistolaythefoundationsofthistheory.
The potential of our model to tackle these open questions comes from one of
themainnoveltiesofourapproach:theuseofanentirelyrigidstructure,namelya
subcategoryofn-foldcategories,tomodelweakn-categories.
The terminology‘rigidstructure’referstothe factthatn-foldcategories,being
iteratedinternalcategories,haveassociativeandunitalcompositionsinndifferent
simplicialdirections.Inthissense, n-foldcategoriesare a stricthighercategorical
structure, though they are not the same as strict n-categories, since the higher
morphismsindimensions0toupndonotformjustaset.Inourmodel,thehigher
morphismsindimensionk(foreach0≤k ≤n−2)havethemselvesan(n−1+k)-
foldcategoricalstructureofaspecialtypewhichissuitablyequivalenttoadiscrete
structure(thatis,aset):wecallthispropertythe‘weakglobularitycondition’.
n-FoldstructureswereusedinhomotopytheorybyLoday[85]forthemodelling
ofconnected(n+1)-typesviacatn-groups.Theideaofweakglobularitywasfirst
introduced by the author in [102] in an internal setting for the category of catn-
groups:weakly globular catn-groupswere shown in [102] to be algebraic models
ofconnected(n+1)-types;weakglobularitywasextendedandfurtherstudiedby
Blancandtheauthorin[29]inthecontextofgeneraln-types,forwhichananalogue
ofLoday’smodelwasnotavailable.
However, none of these workscaptured the generalcategoricalcase. This case
necessitates many novel ideas and techniques, such as the use of pseudo-functors
tomodelhigherstructuresandtheconstructionofarigidificationfunctorfromthe
Tamsamanimodeltoweaklyglobularn-foldcategories.
Preface ix
This work uses a blend of techniques from category theory and simplicial
homotopytheory,reviewedinPartI.Wethereforehopeitwillbeaccessiblebothto
categorytheoristsandtoalgebraictopologists.
Thisbookisorganizedintofourparts:
PartIHigherCategories:IntroductionandBackground.
This Part aims to provide the reader with a guide to the rest of the book. It
containsabroadintroductiontohighercategories,somehistoricaldevelopment
of the notion of weak globularity and a non-technical overview of the main
ideas and results we shall encounter. It also covers the main techniques that
willbeusedfromcategorytheoryandsimplicialhomotopytheory.
PartIITheThreeSegal-typeModelsandSegalicPseudo-Functors.
In this Part we introduce the three Segal-type models, that is, the categories
Tan of weakly globular Tamsamani n-categories and its subcategories Tan
wg
(Tamsamanin-categories)andCatn (weaklyglobularn-foldcategories).We
wg
establish the relation between the category Catn and a class of pseudo-
wg
functorswhichwecallSegalicpseudo-functors.
PartIIIRigidificationofWeaklyGlobularTamsamaninnn-Categories.
ThemaingoalofthisPartistheconstructionoftherigidificationfunctorfrom
weaklyglobularTamsamanin-categoriestoweaklyglobularn-foldcategories.
PartIVWeaklyGlobularnnn-FoldCategoriesasaModelofWeaknnn-Categories.
This Part contains the construction of the discretization functor from weakly
globularn-foldcategoriestoTamsamanin-categories,andthefinalresults:the
equivalenceafterlocalizationofCatn andTan,exhibitingCatn asamodelof
wg wg
weakn-categories,andtheproofofthehomotopyhypothesis.Thelastchapter
ofthisPartcontainsanoutlineoffurtherdirectionsofapplicationsandofopen
questionsarisingfromthiswork.
Leicester,UK SimonaPaoli
Acknowledgements
This work was supported by the EU International Reintegration Grant HOMAL-
GHIGHNo256341,ofwhichIwasprincipalinvestigator.Italsoreceivedfinancial
supportfromtheUniversityofLeicester,whereIhaveworkedsince2011andwhich
supportedmystudyleavein2015;fromtheCentreofAustralianCategoryTheoryat
MacquarieUniversity,whichhostedmeduringAugust–December2015;andfrom
theUniversityofChicago,whereIvisitedin2016.
This work was presented in several talks, where I could gather invaluable
feedbackfrom colleagues.In particular I thank the Centre of Australian Category
TheoryfortheopportunitytogivealongseriesoftalksattheAustralianCategory
Seminar, and I am particularly indebted for useful feedback from its members,
especiallyMichaelBatanin,SteveLack,RichardGarner,RossStreet,MarkWeber,
andDominicVerity.
IwouldalsoliketothankPeterMayforgivingmethechancetogiveaseriesof
talksattheUniversityofChicagoandformanyusefulcomments.
Iamgratefulfortheopportunitytopresentthismaterialatinternationalconfer-
encesandatseminarsandforthediscussionswithexpertsinthefieldthatfollowed.I
thankinparticularJoachimKock,MartinHyland,andDorettePronkforinteresting
discussions and feedback. I also thank David Blanc, Frank Neumann, Alexander
KurzandDominicVerityforreadingtheintroductorypartofthemanuscriptandfor
theirhelpfulcomments,andNicolaGambinoforsomebibliographicalsuggestions.
I thank my colleagues in the Department of Mathematics of the University of
Leicesterfortheirongoingsupportandencouragement.
I expressmydeepgratitudetothe fourreviewersof thisbook,fortheircareful
readingandnumeroushelpfulcommentsandsuggestions.
I finally thank my parents for their invaluable moral support throughout my
mathematicallife.
xi
