LIMITS OF QUASI-CATEGORIES WITH (CO)LIMITS EMILYRIEHL JointwithDominicVerity. Introduction Thistalkconcernsquasi-categorieswhichareamodelfor(∞,1)-categories,whichare categorieswithobjects,1-morphisms,2-morphisms,3-morphisms,andsoon,withevery- thingaboveleveloneinvertible. Specifically,aquasi-categoryisasimplicialsetinwhich anyinnerhornhasafiller. Wethinkofthehornfillingasprovidingaweakcomposition lawformorphismsinalldimensions. Our project is to redevelop the foundational category theory of quasi-categories (pre- viouslyestablishedbyJoyal, Lurie, andothers)inawaythatmakesiteasiertolearn. In particular,theproofsmorecloselyresembleclassicalcategoricalproofs. TodayIwantto illustrate this by mentioning one new theorem (to appear on the arXiv on Monday) and thendescribingasmuchasIcanaboutitsproof. Theorem. Homotopy limits of quasi-categories that have and functors that preserve X- shaped(co)limitshaveX-shaped(co)limits,andthelegsofthelimitconepreservethem. HereXcanbeanysimplicialset. X-shapedcolimitsmightbepushouts,filteredcolimits, initialobjects,colimitsofcountablesequences,andsoon.Thetwotheorems(forX-shaped limitsorcolimits)aredual,soIwon’tmentioncolimitsfurther. By“homotopylimits”ImeanBousfield-Kanstylehomotopylimits,whicharedefined via a particular formula. Here there is no dual result for homotopy colimits. This has to do with the fact that the quasi-categories are the fibrant objects in a model structure on simplicialsets. Andactually,theresultthatweproveisforamoregeneralclassoflimits, includingthehomotopylimits,thatIwilldescribealongtheway. TodayI’llfocusonaspecialcaseofthetheorem: quasi-categoriesadmittingandfunc- torspreserving∅-shapedlimits, akaterminalobjects. Infact, thegeneralcasereducesto thisspecialone,thoughIwon’thavetimetoexplainhow. Warmup Towarmup,let’sprovethefollowingresult: Theorem. Thehomotopylimitofadiagramofquasi-categoriesisaquasi-category. HereadiagrammeansasimplicialfunctorD: A→qCat . HereqCat isthesimpli- ∞ ∞ ciallyenrichedcategoryofquasi-categories,definedtobeafullsubcategoryofsimplicial sets. ThedomainAiseitherasmallcategoryorasmallsimplicialcategory;wecareabout bothcases. Date:ConnectionsforWomen:AlgebraicTopology—MSRI—23January,2014. 1 2 EMILYRIEHL Projective cofibrant weighted limits. Homotopy limits are examples of projective cofi- brant weighted limits. By a weight, in the context of the diagram D above, I mean a simplicialfunctorW: A→sSet. Forinstance: Example. TakingtheweighttobeN(A/−): A → sSet,thecorrespondinglimitnotionis theBousfield-Kanhomotopylimit. Example. If A is the category • → • ← •, we might define W to be the functor with image∆0 −d→1 ∆1 ←d−0 ∆0. Theweightedlimitinthenacommaobject. Example. Thereisaweightwhoseweightedlimitdefinesthequasi-categoryofhomotopy coherentalgebrasforahomotopycoherentmonad. Someofyouheardmetalkaboutthis lastweekattheJointMeetings. A weight W is projective cofibrant if ∅ → W is a retract of a composite of pushouts of coproducts of maps ∂∆n ×A(a,−) → ∆n ×A(a,−) for n ≥ 0 and a ∈ A. These are exactlythecofibrantobjectsintheprojectivemodelstructureonthecategoryofsimplicial functorssSetA. Theweightedlimitisabifunctor weightedlimit: (weight)op×diagram−−{−−,−−→} limitobject thatiscompletelycharacterizedbythefollowingtwoaxioms: (i) {A(a,−),D} = Da, i.e., the weighted limit weighted by a representable functor justevaluatesthediagramatthatobject. (ii) {−,D}sendscolimitsintheweighttolimitsintheweightedlimit. Proofstrategy. Thesetwofactscombinetogiveusastrategyfortheproofofourwarm-up theorem,whichIwillnowrestate: Theorem. A projective cofibrant weighted limit of a diagram of quasi-categories is a quasi-category. Proof. ItsufficestoshowthatqCat isclosedunder ∞ (i) splittingsofidempotents(i.e.,retracts) (ii) limitsoftowersofisofibrations (iii) pullbacksofisofibrations (iv) products (v) cotensors(−)Y withanysimplicialsetY andmoreoverthatamonomorphismX (cid:44)→Y inducesanisofibration(−)Y →(−)X ofquasi- categories. Here an isofibration is a fibration between fibrant objects in the Joyal model structureonsimplicialsets. Allofthefacts(i)-(iii)followimmediatelyfromthefactthat thequasi-categoriesarethefibrantobjectsinthismonoidalmodelstructure. (cid:3) Quasi-categorieswithterminalobjects NowletusconsiderqCat ⊂ qCat ,thesimplicialcategoryofquasi-categoriesad- ∅,∞ ∞ mittingandfunctorspreservingterminalobjects(andallhighermorphismswhosevertices arefunctorspreservingterminalobjects). Ouraimistoprove: Theorem. AprojectivecofibrantweightedlimitofadiagraminqCat isinqCat . ∅,∞ ∅,∞ LIMITSOFQUASI-CATEGORIESWITH(CO)LIMITS 3 As before, it suffices to show that qCat is closed under the classes of limits (i)-(v) ∅,∞ and that cotensors with monomorphisms induce isofibrations that preserve terminal ob- jects. Beforegoinganyfurther,weshoulddefineaterminalobjectinthequasi-categorical context. Definition. Avertextinaquasi-categoryAisterminalifanyofthefollowingequivalent conditionsaresatisfied: (i) AnysphereinAwhosefinalvertexisthasafiller. t (cid:39)(cid:39) ∆0 (cid:47)(cid:47)∂∆n (cid:47)(cid:47)(cid:62)(cid:62) A (cid:124) [n] (cid:124) (cid:124) (cid:15)(cid:15) (cid:124) ∆n ! (cid:41)(cid:41) (ii) Thereisanadjunctionofquasi-categories A(cid:104)(cid:104) ⊥ ∆0 t (iii) Forallsimplicialsets X,theconstantfunctor X→−! ∆0→−t Aisterminalinh(AX), thehomotopycategoryofthemappingspaceAX. Definitions(ii)and(iii)referimplicitlytoqCat ,thestrict2-categoryofquasi-categories, 2 definedbyapplyingthehomotopycategoryfunctortothehom-spacesofqCat . ∞ Toconclude,I’llquicklyproveparts(iv),(i),and(v)ofthetheorem. Parts(ii)and(iii) arenomoredifficult,butrequiresomebasicfactsaboutisofibrationsandterminalobjects. (cid:81) Lemma(products). Supposet ∈ A isterminal. Then(t) ∈ A isterminal. i i i i∈I i∈I i Proof. Givenasphere ti (ti)i∈I (cid:42)(cid:42) (cid:37)(cid:37) ∆0 [n] (cid:47)(cid:47)∂∆n (cid:119)(cid:47)(cid:47)(cid:81)(cid:119)(cid:59)(cid:59) i∈IAi (cid:111)πi (cid:114)(cid:47)(cid:47)(cid:57)(cid:57) Ai (cid:119) (cid:109) (cid:119) (cid:107) (cid:15)(cid:15) (cid:119)(cid:102) (cid:103) (cid:105) ∆n the fact that the t ∈ A are terminal for each i defines the components of the filler. Note i i thateachprojectionπ prefersthisparticularterminalobject. Thisimpliesthatitpreserves i allterminalobjectsbecauseallterminalobjectsareisomorphic. (cid:3) Lemma(idempotents). Supposet ∈ Aisterminal, e: A → Aisanidempotent(e2 = e), andepreservesterminalobjects(soet∈ Aisterminal).Wesplittheidempotentbyforming theequalizer Ae (cid:47)(cid:47) (cid:47)(cid:47)eq(A e (cid:47)(cid:47)(cid:47)(cid:47) A) id Thenet∈ Aeisterminal. 4 EMILYRIEHL Proof. Observethate2 =eimpliesthatet∈ Ae. Givenasphere et (cid:40)(cid:40) ∆0 (cid:47)(cid:47)∂∆n (cid:47)(cid:47)(cid:61)(cid:61)Ae (cid:123) [n] (cid:123) (cid:123) (cid:15)(cid:15) (cid:123) ea (cid:15)(cid:15) ∆n (cid:95) (cid:95) (cid:95)(cid:47)(cid:47) A a the fact that et is terminal in A implies there exists a filler a: ∆n → A for the composite sphereinA. Onecancheckthatea: ∆n → AefillsthesphereinAe (cid:3) Products and idempotents are both conical limits. For cotensors, we’ll switch to the equivalentdefinition(iii). Lemma(cotensors). Supposet∈ AisterminalandYisasimplicialset.ThenY→−! ∆0→−t A isterminalinAY. Proof. Tosayt ∈ AisterminalistosaythatforanysimplicialsetX andanymapX → A thereisaunique2-cell (cid:47)(cid:47) X(cid:64) (cid:63)(cid:63)A (cid:64) (cid:126) (cid:64) (cid:126) (cid:64) (cid:126) (cid:64)(cid:64) ⇓∃! (cid:126)(cid:126) (cid:64) (cid:126) ! (cid:32)(cid:32) (cid:126) t ∆0 By2-cellImeanamorphisminh(AX),i.e.,anendpoint-preservinghomotopyclassof1- simplicesin AX. Thisistrueforany X soinparticular,wehaveaunique2-cellasonthe leftbelow. (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) X×Y(cid:68)(cid:68)(cid:68)(cid:68)!(cid:68)(cid:68)(cid:68)(cid:68)(cid:68)(cid:34)(cid:34) ⇓∃! (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)t(cid:0)(cid:0)(cid:0)(cid:63)(cid:63)A (cid:33) X (cid:67)(cid:67)(cid:67)(cid:67)!(cid:67)(cid:67)(cid:67)(cid:67)(cid:33)(cid:33) ⇓∃! (cid:122)(cid:122)(cid:122)(cid:122)(cid:122)tY(cid:122)(cid:122)(cid:122)(cid:60)(cid:60)AY = X(cid:63)(cid:63)(cid:63)!(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:31)(cid:31) ⇓∃! (cid:126)(cid:126)(cid:126)(cid:126)(cid:126)t(cid:126)·!(cid:126)(cid:126)(cid:62)(cid:62)AY ∆0 (∆0)Y ∆0 The2-categoryofquasi-categoriesiscartesianclosed,soapplyingthe2-adjunction−×Y (cid:97) (−)Y,thistransposestoaunique2-cellinthetriangleontheright. By(iii)thissaysexactly thattheconstantmapattisterminalinAY. (cid:3) DepartmentofMathematics,HarvardUniversity,1OxfordStreet,Cambridge,MA02138 E-mailaddress:[email protected]