Higher Segal Spaces PDF
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Lecture Notes in Mathematics 2244 Tobias Dyckerhoff Mikhail Kapranov Higher Segal Spaces Lecture Notes in Mathematics 2244 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryEditors: KarinBaur,Leeds MichelBrion,Grenoble CamilloDeLellis,Princeton AlessioFigalli,Zurich AnnetteHuber,Freiburg DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Cambridge AngelaKunoth,Cologne ArianeMézard,Paris MarkPodolskij,Aarhus SylviaSerfaty,NewYork GabrieleVezzosi,Florence AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Tobias Dyckerhoff (cid:129) Mikhail Kapranov Higher Segal Spaces 123 TobiasDyckerhoff MikhailKapranov DepartmentofMathematics KavliInstituteforthePhysics UniversityofHamburg andMathematicsoftheUniverse Hamburg,Germany Kashiwa,Japan ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-030-27122-0 ISBN978-3-030-27124-4 (eBook) https://doi.org/10.1007/978-3-030-27124-4 MathematicsSubjectClassification(2010):19D10,18G30,55U10,55U40,55U35,05E10,05E05 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Since the first draft of this text, which appeared in 2012 on the arXiv, the theory of higher Segal spaces has been further developed in various directions by several groups of authors. For the sake of avoiding confusions related to references to the currenttext,thesecontributionswillnotbementionedinthemainbodyofthiswork. However,wewouldliketopresentashortdescriptionofsomeofthesemorerecent developments. Our notion of a unital 2-Segal space has been introduced independently by Gálvez-Carrillo,Kock,andTonksunderthenamedecompositionspace[GCKT18]. While the two notions are precisely equivalent to one another, the perspective on these structures in loc. cit. is rather different from the one presented in this work: whileourpointofviewislargelyinspiredbythetheoryofHallalgebras,thecore goalofloc.cit.istoprovide asystematicstudyofincidence algebras and Möbius inversion. Structured 2-Segal spaces turned out to play an interesting role in categorified state sum constructions for 2-dimensional topological field theories. The founda- tions for this approach were laid in [DK15] and further developed by Stern in [Ste16]. In [DK18], these techniques were applied to implement a 2-dimensional proposalofKontsevichtodescribethetopologicalFukayacategoryofaSteinman- ifoldvialocalizationtoasingularspine[Kon09].Kapranov-Schechtmanproposeto studycategorifiedversionsofperversesheaves,so-calledperverseSchobers,which are locally described by 2-Segal spaces. As explained in [DKSS19], these objects canbeusedtodefinetopologicalFukayacategorieswithcoefficients. In[Dyc18],a2-SegalperspectiveonGreen’stheoremisprovided.Theoperadic characterization of unital 2-Segal sets in § 3.6 was clarified and generalized to ∞-categories by Walde [Wal17]. Relative versions of the 2-Segal condition, designed to produce modules over Hall algebras, were introduced and studied in [Wal16,You18].Thedescriptionof2-Segalspacesasalgebrasinspans(§10,§11) wasdescribedinmorenaturaltermsbyPenney[Pen17a]whoalsoconstructedlax bialgebra structures in [Pen17b], leading to categorified versions of Green’s theo- + rem.IntheworkofBergner-Osorno-Ozornova-Rovelli-Scheimbauer[BOO 18a],it isshownthatanyunital2-Segalsetcanbeconstructedbyasuitablegeneralizationof v vi Preface theS-construction,andtheforthcomingworkbythesamegroupofauthorsshows that the statement generalizes to simplicial spaces. A new characterization of 2- + Segalspacesbythe1-Segalpropertyoftheirsubdivisionwasprovenin[BOO 18b]. In[EJS18],theK-theoryandHallalgebraofmatroidsarestudied,basedonthe notionofaproto-exactcategoryintroducedinthiswork. The first series of examples of d-Segal spaces beyond d > 2 were provided byPoguntke [Pog17].They ariseasnaturalhigher-dimensional analogs ofSegal’s and Waldhausen’s constructions and underline the relevance of the higher Segal conditions. In the context of stable ∞-categories, these constructions organize into a categorified Dold-Kan correspondence [Dyc17] exhibiting the higher Segal conditionsastruncationconditionsoncategorifiedcomplexes[DJ19]. Acknowledgements We would like to thank A. Goncharov, M. Groechenig, P. Lowrey, J. Lurie, I. Moerdijk, P. Pandit, and B. Toën for useful discussions which influenced our understanding of the subject. We further thank P. James, S. Mozgovoy,T.Walde,andM.Youngforpointingoutinaccuraciesinearlierversions of the draft. The first author was a Simons Postdoctoral Fellow while part of this work was carried out. The research of the second author was partially supported by an NSF grant, by the Max-Planck-Institut für Mathematik in Bonn, and by the UniversitéParis13.ItwassupportedbytheWorldPremierInternationalResearch CenterInitiative(WPI)andtheMEXT,Japan. Hamburg,Germany TobiasDyckerhoff Kashiwa,Japan MikhailKapranov June2019 Contents 1 Preliminaries................................................................ 1 1.1 LimitsandKanExtensions .......................................... 1 1.2 SimplicialObjects.................................................... 2 1.3 HomotopyLimitsofDiagramsofSpaces........................... 4 2 Topological1-Segaland2-SegalSpaces.................................. 9 2.1 Topological1-SegalSpacesandHigherCategories................ 9 2.2 MembraneSpacesandGeneralizedSegalMaps.................... 12 2.3 2-SegalSpaces........................................................ 19 2.4 Proto-ExactCategoriesandtheWaldhausenS-Construction ...... 21 2.5 Unital2-Segalspaces ................................................ 25 2.6 TheHecke-WaldhausenSpaceandRelativeGroup Cohomology.......................................................... 27 3 Discrete2-SegalSpaces .................................................... 31 3.1 Examples:Graphs,Bruhat-TitsComplexes......................... 31 3.2 TheTwistedCyclicNerve ........................................... 34 3.3 TheMultivaluedCategoryPointofView........................... 36 3.4 TheHallAlgebraofaDiscrete2-SegalSpace...................... 43 3.5 TheBicategoryPointofView ....................................... 46 3.6 TheOperadicPointofView ......................................... 52 3.7 Set-TheoreticSolutionsofthePentagonEquation ................. 58 3.8 Pseudo-HolomorphicPolygonsasa2-SegalSpace ................ 63 3.9 Birationally1-and2-SegalSemi-SimplicialSchemes............. 67 4 ModelCategoriesandBousfieldLocalization........................... 71 4.1 ConceptsfromModelCategoryTheory............................. 71 4.2 EnrichedModelCategories.......................................... 73 4.3 EnrichedBousfieldLocalization..................................... 78 4.4 HomotopyLimitsinModelCategories ............................. 81 vii viii Contents 5 The1-Segaland2-SegalModelStructures .............................. 85 5.1 YonedaExtensionsandMembraneSpaces ......................... 85 5.2 1-Segaland2-SegalObjects......................................... 90 5.3 1-Segaland2-SegalModelStructures .............................. 92 6 ThePathSpaceCriterionfor2-SegalSpaces............................ 95 6.1 AugmentedSimplicialObjects ...................................... 95 6.2 PathSpaceAdjunctions.............................................. 96 6.3 ThePathSpaceCriterion ............................................ 100 6.4 ThePathSpaceCriterion:Semi-SimplicialCase................... 102 7 2-SegalSpacesfromHigherCategories.................................. 107 7.1 Quasi-Categoriesvs.Complete1-SegalSpaces .................... 107 7.2 Exact∞-Categories.................................................. 109 7.3 TheWaldhausenS-ConstructionofanExact∞-Category......... 111 7.4 Application:DerivedWaldhausenStacks........................... 115 7.5 TheCyclicBarConstructionofan∞-Category.................... 122 8 HallAlgebrasAssociatedto2-SegalSpaces ............................. 125 8.1 TheorieswithTransferandAssociatedHallAlgebras ............. 125 8.2 Groupoids:ClassicalHallandHeckeAlgebras..................... 128 8.3 Groupoids:GeneralizedHallandHeckeAlgebras ................. 133 8.4 ∞-Groupoids:DerivedHallAlgebras .............................. 142 8.5 Stacks:MotivicHallAlgebras....................................... 148 9 Hall(∞,2)-Categories..................................................... 153 9.1 HallMonoidalStructures ............................................ 153 9.2 SegalFibrationsand(∞,2)-Categories............................. 157 9.3 TheHall(∞,2)-Categoryofa2-SegalSpace...................... 161 10 An(∞,2)-CategoricalTheoryofSpans ................................. 169 10.1 SpansinKanComplexes............................................. 170 10.2 VerticalSpans ........................................................ 172 10.3 HorizontalSpans ..................................................... 194 10.4 Bispans................................................................ 198 11 2-SegalSpacesasMonadsinBispans .................................... 201 11.1 TheHigherHallMonad.............................................. 201 A Bicategories.................................................................. 209 References......................................................................... 213 Introduction ThetheoryofSegalspaces,asintroducedbyC.Rezk[Rez01],hasitsrootsinthe classical work of G. Segal [Seg74] where the notion of a (cid:2)-space is introduced andusedtoexhibitvariousclassifyingspacesasinfiniteloopspaces.Rezk’swork analyzes the role of Segal spaces as a model for the homotopy theory of (∞,1)- categories. The concept of a Segal space can be motivated as follows. Given a simplicialsetX,wehave,foreachn≥1,anaturalmap f :X −→X × X × ···× X (0.1) n n 1 X0 1 X0 X0 1 where the right-hand side is an n-fold fiber product. The condition that all maps f be bijective is called Segal condition and a simplicial set which satisfies this n conditioniscalledSegal.Therelevanceofthisconditioncomesfromthefactthatit characterizestheessentialimageofthefullyfaithfulfunctor N:Cat →Set (cid:3) which takes a small category to its nerve. Given a Segal simplicial set X, we can recover the corresponding category C: the set of objects is formed by the vertices of X, and morphisms between a pair of objects are given by edges in X between thecorrespondingpairofvertices.Theinvertibilityoff allowsustointerpretthe 2 diagram (0.2) as a composition law for C, while the bijectivity of f implies the associativity of 3 thislaw.OnecanviewthetheoryofSegal(simplicial)spacesasadevelopmentof thisideainahomotopytheoreticframework,wheresimplicialsetsarereplacedby simplicialspaces,fiberproductsbytheirhomotopyanalogs,andbijectionsbyweak equivalences.Thisleadstoaweakernotionofcoherentassociativitywhichcanbe usedtodescribecompositionlawsinhighercategories. ix
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