Induced model structures for higher categories
IINDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES
PHILIP HACKNEY AND MARTINA ROVELLI
Abstract.
We give a new criterion guaranteeing existence of model structuresleft-induced along a functor admitting both adjoints. This works under thehypothesis that the functor induces idempotent adjunctions at the homotopycategory level. As an application, we construct new model structures on cubicalsets, prederivators, marked simplicial sets and simplicial spaces modeling ∞ -categories and ∞ -groupoids. Introduction
It is common when working in abstract homotopy theory to deal with severalequivalent Quillen model categories, each with their own strengths and weaknesses.This even extends to Quillen model structures on a single category — for instance,in categories of diagrams of a fixed shape in a model category M , there are oftenboth projective and injective model structures, each of which have their place. Itis in this spirit that we present the discovery of several new model structures for ∞ -categories and ∞ -groupoids. In particular, we give model structures supportedon categories of cubical sets, prederivators, bisimplicial sets, and marked simplicialsets. For the purposes of the introduction, we focus on the first of these.Cubical sets are presheaves on a category of cubes. But there are many possiblecategories of cubes, and there is a tension between the simplicity of the cube categoryand the expressivity of the corresponding category of cubical sets. We take thisopportunity to point the reader to the introduction of [CMS20] for an overview ofwhat is known about various choices in the context of Homotopy Type Theory andUnivalent Foundations. In our case, we consider the category of cubes as a fullsubcategory of the category of posets, which is the same rich context that Kapulkinand Voevodsky operate under in [KV20]. This is a homotopically challengingindexing category to work with, as it does not admit a natural generalized Reedystructure. Our first main result is Proposition 2.1, which is about the existence ofinduced model structures on cubical sets along the triangulation functor from thecategory of cubical sets to the category of simplicial sets. These model structures are(left- or right-) induced either from the Joyal or the Kan–Quillen model structureon the category of simplicial sets. In all cases, the triangulation functor becomesboth a left and a right Quillen equivalence. On the other hand, in [DKLS], a modelstructure for ( ∞ , -categories was given on a different category of cubical sets;in Theorem 2.6 we show that the natural comparison between the two categoriesof cubical sets is a Quillen equivalence between this one and the model structureleft-induced from the Joyal model structure.The main technical tools we use are lifting theorems for model structures inthe presence of adjoint strings. Let M be a model category, N be a (bicomplete) a r X i v : . [ m a t h . C T ] F e b PHILIP HACKNEY AND MARTINA ROVELLI category, and suppose we have a string of adjoint functors
N M F ⊥⊥ LR (which below we will write more compactly as F : N ! M : L, R ). In [DCH19],Drummond-Cole and the first author showed that if M is cofibrantly generatedand if the adjunction F L (cid:97)
F R on M is a Quillen adjunction, then there existsa right-induced model structure on N , where weak equivalences and fibrations in N are created by F . In that paper, the question was posed about whether onecould guarantee a left-induced model structure on N , that is, one where the weakequivalences and co fibrations are created by F . We give a partial answer. Theorem (Theorem 1.13 and Proposition 1.19) . Suppose that we have an adjointstring F : N ! M : L, R with N a locally presentable category and M an accessiblemodel category. If F L (cid:97)
F R is a Quillen adjunction and the adjoint string betweenhomotopy categories ho N ! ho M is an idempotent adjoint string, then N admitsa model structure left-induced along F . This occurs in the special case when L or R is fully faithful. In the theorem statement, the homotopy category ho N is obtained by inverting allmorphisms which are sent by F to weak equivalences in M . An idempotent adjointstring is an adjoint string where the two constituent adjunctions are idempotentadjunctions.We should make clear that, just as the aforementioned Drummond-Cole–Hackneyresult rests on Kan’s theorem for right-induced model structures [Hir03, Theorem11.3.2], so too does our present theorem rest on the the Acyclicity Theorem of[GKR20, HKRS17]. Acknowledgements.
We would like to thank Gabriel C. Drummond-Cole, RichardGarner, Chris Kapulkin, Emily Riehl and Christian Sattler for useful input, sug-gestions, and encouragement. This material is based upon work supported by theNational Science Foundation under Grant No. DMS-1440140 while the authors werein residence at the Mathematical Sciences Research Institute in Berkeley, California,during the Spring 2020 semester.
Contents
1. The abstract framework and results 31.1. Strings of adjoint functors 31.2. Model structures induced via adjoint strings 42. Cubical sets 82.1. New model structures on cubical sets 92.2. A comparison with cubical models of homotopy type theory 102.3. A comparison between cubical models for ( ∞ , -categories 113. Further applications 123.1. Model structures on prederivators 123.2. Model structures on marked simplicial sets 133.3. Model structures on bisimplicial sets 13References 14 NDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES 3 The abstract framework and results
Strings of adjoint functors.
We start by recalling the terminology relatedto strings of adjoint functors. A pair of adjoint functors with F left adjoint to G will be denoted by either F : M (cid:29) N : G or F (cid:97) G , depending on whether or notwe wish to emphasize the (co)domains of the functors. Definition 1.1.
Let M and N be categories. A string of adjoint functors (or adjointstring ) F : N ! M : L, R consists of functors F : N → M and
L, R : M → N thatform adjunctions L (cid:97) F and F (cid:97) R . Remark . Given any string of adjoint functors F : N ! M : L, R , the adjunctions F : N (cid:29) M : R and L : M (cid:29) N : F can be composed to obtain an adjunction F L : M (cid:29) M : F R .The following remark gives a large source of examples of strings of adjoint functors.
Remark . Let S be a bicomplete category. Any functor f : I → J betweensmall categories induces a string of adjoint functors f ∗ : S J ! S I : f ! , f ∗ where f ∗ : S J → S I denotes the restriction functor and f ! : S I → S J (resp. f ∗ : S I → S J )denotes the left (resp. right) Kan extension along f .We consider the following two properties that adjoint strings of functors mayhave. The first is studied more deeply in [Joh11, §2]. Definition 1.4.
Let F : N ! M : L, R be a string of adjoint functors, let η and (cid:15) be the unit and counit of the adjunction L (cid:97) F , let η (cid:48) and (cid:15) (cid:48) be the unit and counitof the adjunction F (cid:97) R . We say that the adjoint string is idempotent if all of themaps in the following diagram are isomorphisms. F LFF FF RF
F (cid:15) id F F η (cid:48) ηF (cid:15) (cid:48) F This happens if and only if one of the outside maps is an isomorphism.If F happens to be fully faithful, then F (cid:15) is an isomorphism by [Mac98, The-orem IV.3.1], hence the string is idempotent. Another, different, special case ofidempotency is that of a ‘fully faithful’ string of adjoint functors, which we nowrecall.
Definition 1.5.
A string of adjoint functors F : N ! M : L, R is said to be fullyfaithful if both L and R are fully faithful. If one of L or R is fully faithful, so is theother (see [DT87, Lemma 1.3]). Remark . Any fully faithful string of adjoint functors F : N ! M : L, R isan idempotent string of adjoint functors. This is because L being fully faithful isequivalent to the unit η : id M ⇒ F L of the adjunction L (cid:97) F being an isomorphism(alternatively, R being fully faithful is equivalent to the counit (cid:15) (cid:48) of the adjunction F (cid:97) R being an isomorphism) by [Mac98, Theorem IV.3.1]. Remark . Let S be a bicomplete category. If f : I → J is a fully faithful functorbetween small categories, then the string of adjoint functors f ∗ : S J ! S I : f ! , f ∗ from Remark 1.3 is fully faithful. PHILIP HACKNEY AND MARTINA ROVELLI
Model structures induced via adjoint strings.
We discuss situations inwhich one can transfer a model structure along the middle functor of a string ofadjoint functors. In this paper, model categories will admit all small limits andcolimits (which we refer to as ‘bicomplete’) and will be assumed to come equippedwith functorial factorizations. In particular, a model category M comes equippedwith a natural transformation ( − ) c ⇒ id M so that each component X c → X is anacyclic fibration from a cofibrant object (and dually for id M ⇒ ( − ) f ). The term‘left Quillen functor’ will be synonymous with ‘left adjoint in a Quillen adjunction’,and ‘left Quillen equivalence’ will mean ‘left adjoint in a Quillen equivalence’ (andsimilarly for right adjoints). We will often be interested in the accessible modelcategories of [HKRS17, Definition 3.1.6], which are a generalization of Jeff Smith’scombinatorial model categories (see, e.g., [Lur09, §A.2.6]). Definition 1.8.
Let F : N → M be a functor, and M a model category. • The left-induced model structure on N , when it exists, is the model structurein which the cofibrations and the weak equivalences are created by F . We willdenote the left-induced model structure by N (cid:96) . • The right-induced model structure on N , when it exists, is the model structure inwhich the fibrations and the weak equivalences are created by F . We will denotethe right-induced model structure by N r .The following criterion for existence of right-induced model structure along F inthe presence of a string of adjoint functors L (cid:97) F (cid:97) R was given by Drummond-Coleand the first author. Theorem 1.9 ([DCH19, Theorem 2.3]) . Let F : N ! M : L, R be a string ofadjoint functors, where M is a cofibrantly generated model category and N is abicomplete category. If F L : M (cid:29) M : F R is a Quillen adjunction, then thecategory N supports the right-induced model structure N r . Further, the functor F : N r → M is both left and right Quillen. Under the additional hypothesis that F is fully faithful, Campbell observed in[Cam, Proposition 2.2] that the model structure on N is also left-induced.The following makes no reference to the functor R , but we state it this way as itis most meaningful in the case when F L (cid:97)
F R is a Quillen adjunction on M . Definition 1.10 (Homotopy idempotent string) . Let M be a model category andlet F : N ! M : L, R be a string of adjoint functors. • For each X in N , define a map (cid:15) hX of N as the composite (cid:15) hX : L (( F X ) c ) → LF X → X whose first map is L applied to the cofibrant replacement ( F X ) c ∼ → F X in M ,and whose second map is the counit of L (cid:97) F . Notice that (cid:15) h : L (( F − ) c ) ⇒ id N is a natural transformation. • We say that the string of adjoint functors is homotopy idempotent if the map
F (cid:15) hX : F L (( F X ) c ) → F X is a weak equivalence in M for every object X in N . Remark . We have chosen this definition because it is the most convenient forthe proof of Theorem 1.13, but of course there are other formulations (see alsoProposition 1.19 below). For example, by combining the naturality square for the
NDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES 5 unit η of the adjunction L (cid:97) F with a triangle identity, we obtain the followingcommutative diagram in M : ( F X ) c F XF L (( F X ) c ) F LF X F X. η ( FX ) c ∼ η FX = F (cid:15) X By two-of-three, one sees that
F (cid:15) hX is a weak equivalence in M if and only if themap η ( F X ) c : ( F X ) c → F L (( F X ) c ) is a weak equivalence. Remark . A string of adjoint functors F : N ! M : L, R is automatically ahomotopy idempotent string in the following cases: • when the adjoint string L (cid:97) F (cid:97) R is idempotent, the adjunction F L (cid:97)
F R is aQuillen pair and every object in M is cofibrant, or • when the adjoint string L (cid:97) F (cid:97) R is fully faithful.We now give a criterion for the existence of the left-induced model structurealong F in presence of a string of adjoint functors L (cid:97) F (cid:97) R . Theorem 1.13.
Let F : N ! M : L, R be a string of adjoint functors, where M is an accessible model category and N is a locally presentable category. If F L (cid:97)
F R is a Quillen adjunction and F : N ! M : L, R is a homotopy idempotent string ofadjoint functors, then the category N supports the left-induced model structure N (cid:96) .Further, the functor F : N (cid:96) → M is both left and right Quillen. The proof of the preceding theorem uses the cylinder object argument, whosestatement we recall here for the convenience of the reader. The conclusion of Theo-rem 1.14 relies on the Garner–Hess–K¸edziorek–Riehl–Shipley Acyclicity Theorem,whose corrected proof may be found in [GKR20].
Theorem 1.14 ([HKRS17, Theorem 2.2.1]) . Let F : N (cid:29) M : G be an adjointpair, M an accessible model category and N a locally presentable category. Supposethat the following conditions hold.(1) For every object X of N , there exists a morphism φ X : QX → X such that F φ X is a weak equivalence and F ( QX ) is cofibrant in M ,(2) For each morphism f : X → Y in N there exists a morphism Qf : QX → QY satisfying φ Y ◦ Qf = f ◦ φ X , and(3) For every object X of N , there exists a factorization QX (cid:113) QX j −→ Cyl( QX ) p −→ QX of the fold map such that F j is a cofibration and
F p is a weak equivalence.Then N supports the model structure N (cid:96) left-induced along F . The following proof includes a verification of the conditions of Theorem 1.14.
Proof of Theorem 1.13.
For an object X of N , we define QX to be L (( F X ) c ) andlet φ X denote the map (cid:15) hX from Definition 1.10: φ X := (cid:15) hX : QX = L (( F X ) c ) L ( ∼ (cid:16) ) −−−→ LF X (cid:15) X −−→ X. We now verify the conditions to apply Theorem 1.14.(1) The map
F (cid:15) hX is a weak equivalence in M by assumption, and, since F L isleft Quillen,
F QX = F L (( F X ) c ) is a cofibrant object of M . Thus we haveestablished Condition (1). PHILIP HACKNEY AND MARTINA ROVELLI (2) Condition (2) holds because (cid:15) h is a natural transformation Q ⇒ id N .(3) For Condition (3), notice that we have a factorization ( F X ) c (cid:113) ( F X ) c Cyl((
F X ) c ) ( F X ) cJ ∼ P of the fold map of ( F X ) c in M . The objects Cyl((
F X ) c ) and ( F X ) c arecofibrant in M , and by Ken Brown’s lemma [Hir03, Corollary 7.7.2(1)] theleft Quillen functor F L preserves weak equivalences between cofibrant objects,hence
F LP is a weak equivalence in M . Further, F LJ is again a cofibrationsince
F L is left Quillen. Thus the a factorization QX (cid:113) QX = L [( F X ) c (cid:113) L ( F X ) c ] L Cyl((
F X ) c ) QX LJ LP of the fold map of QX has the desired properties.We conclude from Theorem 1.14 that the left-induced model structure exists. Thefunctor F is left Quillen by construction, and the proof that F is right Quillen isthe evident variation from the one appearing in [DCH19, Theorem 2.3]. That is,one uses that F L preserves (acyclic) cofibrations and F reflects them to infer that L is left Quillen. (cid:3) Remark . A dual version of [DCH19, Theorem 5.6] holds, which allows oneto efficiently lift Quillen equivalences in circumstances where Theorem 1.13 holds.As we will not need this result here we will not repeat the statement (the proof isformally dual), and instead merely note that the last two bullet points of [DCH19,Theorem 5.6] should be replaced with ‘ F (cid:48) reflects cofibrations and preserves fibrantobjects’ and ‘ F preserves cofibrations and cofibrant objects’, respectively.The following proposition guarantees that the left- and right-induced modelstructures, if they exist, have equivalent homotopy theories. Proposition 1.16.
Let F : N ! M : L, R be a string of adjoint functors, and M a model category. If the left-induced model structure N (cid:96) and the right-inducedmodel structure N r both exist, then the adjunction id N : N r (cid:29) N (cid:96) : id N is a Quillenequivalence.Proof. As the two model structures share the same class of weak equivalences, wemerely need to show that we have a Quillen adjunction. It is enough to showthat fibrations are preserved by id N : N (cid:96) → N r , since this will imply that acyclicfibrations are preserved as well. The proof that F : N (cid:96) → M is right Quillen givenin Theorem 1.13 only used that F L is left Quillen and that F reflects (acyclic)cofibrations, so remains valid in this generality. This implies that if f is any fibrationin N (cid:96) , then F f is a fibration in M , which then implies that f is a fibration in N r .Hence the fibrations of N (cid:96) are contained in the fibrations of N r . (cid:3) In presence of a fully faithful adjoint string of functors, combining Theorems 1.9and 1.13 we obtain the following integrated result.
Corollary 1.17.
Let F : N ! M : L, R be a fully faithful string of adjoint functors, M a combinatorial model category and N a locally presentable category. Then thecategory N admits both the left-induced model structure N (cid:96) and the right-inducedmodel structure N r . Further, the left diagram below is a diagram of left Quillen NDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES 7 equivalences, and the right diagram below is a diagram of right Quillen equivalences. N r N (cid:96) N r N (cid:96) M M id N F F F F id N Proof. As M is combinatorial, it is both accessible and cofibrantly generated. Weassumed that L is fully faithful, implying that the functor F L ∼ = id M is left Quillen,so both Theorem 1.13 and Theorem 1.9 apply to show the existence of the indicatedmodel structures. By [FKKR19, Theorem 3.2], the functor F : N r → M is both aleft and right Quillen equivalence. The fact that F : N (cid:96) → M is both a left and aright Quillen equivalence now follows from Proposition 1.16 and two out of three forQuillen equivalences [Hov99, Corollary 1.3.15]. (cid:3) We devote the remainder of this section to Proposition 1.19, which gives aninterpretation of what being a homotopy idempotent adjoint string (in the senseof Definition 1.10) means at the level of homotopy categories. To give the mostnatural statement of this proposition, we make use of the language of deformableadjunctions from [Rie14, §2.2] (originally from [DHKS04, §44.2]), which will not beneeded elsewhere. The reader who prefers to stay within the world of Quillen modelcategories should be comforted to know that with some adjustments this is possible,and such a reader is advised to immediately skip down to Remark 1.20 and thestatement of Proposition 1.19.
Remark . Let F : N ! M : L, R be a string of adjoint functors and M a model category such that F L : M (cid:29) M : F R is a Quillen pair. We endow N with the class of weak equivalences created by F , and denote by ho N thecorresponding homotopy category (which is potentially not locally small). In thissituation, one can then show that F : N → M is homotopical, while L : M → N (resp. R : M → N ) preserves weak equivalences between cofibrant (resp. fibrant)objects, by Ken Brown’s lemma [Hir03, Corollary 7.7.2(1)]. Using the language from[Rie14, Definition 2.2.4], this entails that the functor L : M → N is left deformable via the cofibrant replacement natural transformation ( − ) c ⇒ id M , the functor R : M → N is right deformable via the fibrant replacement natural transformation id M ⇒ ( − ) f , and since F is homotopical, it is right and left deformable via theidentity functor id M ⇒ id M . In particular, the adjunctions L : M (cid:29) N : F and F : N (cid:29) M : R are deformable adjunctions , and by [Rie14, Theorem 2.2.11] theyinduce adjunctions L : ho M (cid:29) ho N : F and F : ho N (cid:29) ho M : R at the level ofhomotopy categories, where the values of the functors on objects are LA = L ( A c ) , RA = R ( A f ) and F X = F X . In particular, there is a string of adjoint functors atthe level of homotopy categories F : ho N ! ho M : L, R.
Proposition 1.19.
Let F : N ! M : L, R be a string of adjoint functors, and M a model category such that F L : M (cid:29) M : F R is a Quillen pair. The following areequivalent.(1) The string F : ho N ! ho M : L, R is an idempotent adjoint string.(2) The string F : N ! M : L, R is a homotopy idempotent adjoint string.
PHILIP HACKNEY AND MARTINA ROVELLI
Proof.
As observed in Remark 1.18, the adjunction L : M (cid:29) N : F induces anadjunction L : ho M (cid:29) ho N : F at the level of homotopy categories. The naturalbijections coming from the total derived adjunction L (cid:97) F and the adjunction L (cid:97) F fit into the square N ( LA, X ) M ( A, F X )ho N ( LA, X ) ho M ( A, F X )ho N ( LA, X ) ho M ( A, F X ) . ∼ = = ∼ = One sees that the counit (cid:15) : L F ⇒ id ho N of the adjunction L (cid:97) F is representedat an object X by the map (cid:15) hX : L ( F X ) c → LF X → X in N , obtained by applying L to the cofibrant replacement map of F X and by composing with (cid:15) X , the counit of theadjunction L (cid:97) F . As F is homotopical, the natural transformation F (cid:15) : F L F ⇒ F at X is then represented by F (cid:15) hX : F L ( F X ) c → F LF X → F X in M . Theequivalence now follows from Definition 1.4 and Definition 1.10. (cid:3) Remark . Suppose we are in the situation of Proposition 1.19, and additionallyassume that M is cofibrantly generated and that N is bicomplete. One then hasaccess to the right-induced model structure N r from Theorem 1.9, so that the functor F : N r → M is both left and right Quillen. Then the category ho N = ho N r isdefined as usual, and the induced adjoint string at the level at the level of homotopycategories uses that the left and right derived functors of F coincide (see [Shu11,Corollary 7.8]). In that special case, this proposition may be proved by explicitlycomparing the map from Definition 1.10 to the derived unit of the Quillen pair L (cid:97) F . 2. Cubical sets
In this section and the next, we denote by s S et ( ∞ , the Kan–Quillen modelstructure on the category s S et for ( ∞ , -categories (namely ∞ -groupoids), and by s S et ( ∞ , the Joyal model structure on s S et for ( ∞ , -categories (see e.g. [DS11,Lur09]). In these model structures, the cofibrations are precisely the monomorphismsand the fibrant objects are respectively the Kan complexes and the quasi-categories.We uniformly denote these two model structures by s S et ( ∞ ,ε ) , where ε = 0 , .The goal of the present section is to discuss several model structures on thecategory of cubical sets. As mentioned in the introduction, there are many usefulcategories of cubical sets, depending on the choice of the underlying cube category;several of these are discussed in [GM03, BM17]. For example, one can use theminimal structure where only the face and degeneracy maps are present, or onecould add in either positive or negative connections, or one can consider all posetmaps between cubes. These cube categories are Grothendieck test categories, soeach is suitable for modeling ∞ -groupoids (see [Jar06a, Cis06, Mal09, SW]). Morerecently, model structures for higher categories have been developed for cubicalsets with connection: for ∞ -categories this was done in [DKLS], and using markedcubical sets a model for ( ∞ , n ) -categories was given in [CKM]. The methods of thelast references break down when working with the full cube category (which is noteven a generalized Reedy category in the sense of [BM11]), but below we will showthat one can nevertheless obtain interesting results. We compare one of our new NDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES 9 model structures to a type-theoretical model structure from [Sat, SW] and anotherto the cubical Joyal model structure of [DKLS].2.1.
New model structures on cubical sets.
Let (cid:3) denote the full subcategoryof C at of cubes [1] n for n ≥ , and let c S et = S et (cid:3) op denote the category of cubical sets , namely presehaves X : (cid:3) op → S et . This category of cubes, and thecorresponding category of cubical sets, are those studied in [Sat, KV20] .There is a triangulation functor T : c S et → s S et , which can be defined onrepresentables by T ( (cid:3) [1] n ) := ∆[1] n , and extended cocontinuously to all cubicalsets, and the functor T has a right adjoint C : s S et → c S et . Kapulkin–Voevodkskyshow in [KV20, §1] that the functor C is fully faithful.Sattler shows in [Sat, Theorem 2.1] that for this specific choice of category ofcubical sets the functor T also admits a left adjoint L : s S et → c S et . In particular,there is a fully faithful adjoint string T : c S et ! s S et : L, C between the category of simplicial sets and this specific choice of category of cubicalsets.From Corollary 1.17, we obtain the following, which endows c S et with two Quillenequivalent model structures for ( ∞ , ε ) -categories for any fixed ε = 0 , . Proposition 2.1.
Let ε = 0 , . The category c S et admits both the left-induced modelstructure c S et (cid:96),ε and the right-induced model structure c S et r,ε along T : c S et → s S et ( ∞ ,ε ) . Further, the left diagram below is a diagram of left Quillen equivalences,and the right diagram below is a diagram of right Quillen equivalences. c S et r,ε c S et (cid:96),ε c S et r,ε c S et (cid:96),ε s S et ( ∞ ,ε ) s S et ( ∞ ,ε )id c S et T T T T id c S et Proof.
As mentioned above, T admits both adjoints. Further, as discussed above,the right adjoint C was shown to be fully faithful by Kapulkin–Voevodsky in [KV20,§1], so this is a fully faithful adjoint string. Corollary 1.17 implies the result. (cid:3) The following lemma, which we learned from Sattler, clarifies the relation betweenthe classes of cofibrations in the model structures c S et r,ε and c S et (cid:96),ε . Lemma 2.2.
The class of monomorphisms of c S et contains the cofibrations of c S et r,ε and is contained in the class of cofibrations of c S et (cid:96),ε .Proof. The generating cofibrations of c S et r,ε are given by applying L to a set ofgenerating cofibrations for s S et ( ∞ ,(cid:15) ) , and Sattler shows in [Sat, Proposition 3.3]that the functor L preserves monomorphisms. Since c S et is a presheaf category, theclass of monomorphisms of c S et is saturated. It follows that the class of cofibrationsof c S et r,ε is contained in the class of monomorphisms of c S et .Further, being a right adjoint the functor T preserves monomorphisms. It follows,using the description of the cofibrations in c S et (cid:96),ε , that the class of monomorphismsof cubical sets is contained in the class of cofibrations of c S et (cid:96),ε . (cid:3) This category is slightly different than that in [CCHM18], but according to Kapulkin andVoevodsky is still sufficient for giving a constructive interpretation of the univalence axiom independent type theory.
Proposition 2.3.
Let ε = 0 , . There is a model structure on cubical sets, denotedby c S et m,ε , in which the cofibrations are the monomorphisms and the weak equiv-alences are created by T : c S et → s S et ( ∞ ,ε ) . Further, the left diagram below is adiagram of left Quillen equivalences, and the right diagram below is a diagram ofright Quillen equivalences. c S et r,ε c S et m,ε c S et (cid:96),ε s S et ( ∞ ,ε )id T T id T c S et r,ε c S et m,ε c S et (cid:96),ε s S et ( ∞ ,ε ) T id T id T Proof.
By a theorem of Jardine [Jar06b] (following the interpretation from [HKRS17,Remark 2.3.4]), we know that if N be a category endowed with two model structures N and N having the same class of weak equivalences, and if K is a class of mapswhich contains the cofibrations of N and is contained in the cofibrations of N ,then N supports a model structure in which the class of weak equivalences is thesame as for N and N , and the class of cofibrations is K . By Lemma 2.2, we canapply this when K is the class of monomorphisms of c S et to obtain the indicatedmodel structures. (cid:3) The model structure c S et m, should be related to the test model structure (see[KV20, Remark 1.2]) which has the same cofibrations and also has weak equivalencescreated in ∞ -groupoids. The model structures c S et r,ε for ε = 0 , were known toSattler, while the model structures c S et (cid:96),ε for ε = 0 , , as well as the model structure c S et m, are new.2.2. A comparison with cubical models of homotopy type theory.
Theother model structure on the category c S et considered in the literature, motivatedby homotopy type theory, is the minimal Cisinski model structure on c S et from[SW, §3], which was also studied in [Sat, §3.3]. Theorem 2.4.
There is a model structure on c S et , which we denote by c S et HoTT ,in which the cofibrations are the monomorphisms, and the fibrations are the mapsthat have the right lifting property with respect to the class of maps ( (cid:3) [0] × B ) (cid:113) (cid:3) [0] × A ( (cid:3) [1] × A ) → (cid:3) [1] × B where A → B is a monomorphism of cubical sets and (cid:3) [0] → (cid:3) [1] is either of thetwo canonical inclusions. Further, the functor T : c S et HoTT → s S et ( ∞ , is left andright Quillen. It is mentioned in [Sat18] and in [SW, §5] that, unlike for other categories ofcubical sets, it is an open problem whether the homotopy theoretic model structure c S et HoTT is Quillen equivalent to the model structure s S et ( ∞ , for ∞ -groupoids.We now explore how the model structure c S et HoTT compares with c S et m, . Proposition 2.5.
The identity is a left Quillen functor c S et HoTT → c S et m, .Proof. The two model structures have the same class of cofibrations, so it sufficesto show that pushout-products of monomorphisms and inclusions (cid:3) [0] (cid:44) → (cid:3) [1] of c S et HoTT are weak equivalences in c S et m, , namely are sent by T to weakequivalences of s S et ( ∞ , . To this end, we consider a generic acyclic cofibration of c S et HoTT , namely the pushout-product in c S et ( (cid:3) [0] × B ) (cid:113) (cid:3) [0] × A ( (cid:3) [1] × A ) → (cid:3) [1] × B NDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES 11 of a monomorphism
A (cid:44) → B and an inclusion (cid:3) [0] (cid:44) → (cid:3) [1] . Since the functor T isboth a left and a right adjoint, it preserves pushouts, products and monomorphisms,and therefore it sends this map to the pushout product in s S et (∆[0] × T B ) (cid:113) ∆[0] × T A (∆[1] × T A ) → ∆[1] × T B of the monomorphism
T A (cid:44) → T B (which is a cofibration of s S et ( ε, ) and theinclusion ∆[0] (cid:44) → ∆[1] (which is an acyclic cofibration of s S et ( ε, ). Since s S et ( ∞ , is cartesian closed, the desired map is a weak equivalence of s S et ( ∞ , . (cid:3) However, understanding whether the identity functor from Proposition 2.5 is a leftQuillen equivalence is a non-trivial matter. Indeed, it is equivalent to the functor T : c S et HoTT → s S et ( ∞ , being a left Quillen equivalence, which as mentionedearlier is an open question.2.3. A comparison between cubical models for ( ∞ , -categories. In [DKLS],a different model structure was constructed on cubical sets which also serves as amodel for ( ∞ , -categories. They use smaller cube categories than we have usedabove, and they give a much more explicit description of their model structure thanwe have given; this process is helped along by the fact that their cube categories areEZ-Reedy categories in the sense of [BR13].For concreteness, write (cid:3) (cid:48) for the wide subcategory of (cid:3) which is generated byfaces, degeneracies, and both positive and negative connections, and let k : (cid:3) (cid:48) → (cid:3) be the inclusion functor. The main theorem of [DKLS] is that there is a Quillen modelstructure c S et (cid:48) cJ , dubbed the cubical Joyal model structure , on c S et (cid:48) := S et ( (cid:3) (cid:48) ) op sothat the triangulation functor T (cid:48) : c S et (cid:48) cJ → s S et ( ∞ , is a left Quillen equivalence. Theorem 2.6.
The restriction functor k ∗ : c S et (cid:96), → c S et (cid:48) cJ is a right Quillenequivalence.Proof. For the proof, we show that left Kan extension k ! : c S et (cid:48) cJ → c S et (cid:96), is a leftQuillen equivalence. The triangulation functors are given by sending the object [1] n to ∆[1] n in s S et , and then extending using that the category of presheaves isthe free cocompletion [Kel05, Theorem 4.51]. In particular, we have the followingcommutative diagram, (cid:3) (cid:48) (cid:3) s S etc S et (cid:48) c S et kk ! T (cid:48) T whose vertical morphisms are Yoneda embeddings. If we knew that k ! was a leftQuillen functor, then we would have a diagram c S et (cid:48) cJ c S et (cid:96), s S et ( ∞ , k ! T (cid:48) T As these model structures share the same class of cofibrations, this is a left Quillen equivalenceif and only if the two model structures are equal. of left Quillen functors with T and T (cid:48) left Quillen equivalences by Proposition 2.1and [DKLS, Theorem 6.1]. By two out of three for Quillen equivalences [Hov99,Corollary 1.3.15], this would imply that k ! is a left Quillen equivalence as well.It remains to check that k ! is a left Quillen functor. But this is automatic, since T (cid:48) preserves cofibrations and acyclic cofibrations, and T reflects cofibrations andweak equivalences. (cid:3) Remark . It is also true that k ∗ : c S et m, → c S et (cid:48) cJ is a right Quillen equivalence.To establish this, it is enough to show that k ! preserves monomorphisms. This usesthat (cid:3) (cid:48) is an EZ-Reedy category by [DKLS, Corollary 1.17], so that monomorphismsare generated by boundary inclusions of representables. These generators are sentto monomorphisms by k ! , hence the same is true for all monomorphisms.3. Further applications
In this section we give model structures, induced from the Joyal and Kan–Quillenmodel structures on simplicial sets, on several other categories. We will study threefully faithful strings of adjoint functors of the form F : N ! s S et ( ∞ ,ε ) : L, R , andapply Corollary 1.17 to obtain new model structures on N for ( ∞ , ε ) -categories.3.1. Model structures on prederivators.
Let cat denote the -category of ho-motopically finite categories , namely those categories whose nerve has only finitelymany nondegenerate simplices, and let p D er denote the category of small pred-erivators , namely -functors D : cat op → C at , and strict natural transformations.As discussed in [FKKR19, §1], this category is a locally presentable category ofprederivators, as opposed to the more traditional (large and not locally presentable)category of -functors D : C at op → C AT .There is an underlying functor U : p D er → s S et , defined by ( U D ) n := D ([ n ]) .It is shown in [FKKR19, §1.15, 1.16] that the functor U admits both a left and aright adjoint, and it is shown in [FKKR19, Proposition 1.18] that the left adjointis fully faithful. In particular, there is a fully faithful string of adjoint functors U : p D er ! s S et between the category of simplicial sets and the category of smallprederivators.From Corollary 1.17, we obtain the following, which endows p D er with twoQuillen equivalent model structures for ( ∞ , ε ) -categories for any fixed ε = 0 , . Proposition 3.1.
Let ε = 0 , . The category p D er admits both the left-inducedmodel structure p D er (cid:96),ε and the right-induced model structure p D er r,ε along U : p D er → s S et ( ∞ ,ε ) . Further, the left diagram below is a diagram of left Quillen equivalences,and the right diagram below is a diagram of right Quillen equivalences. p D er r,ε p D er (cid:96),ε p D er r,ε p D er (cid:96),ε s S et ( ∞ ,ε ) s S et ( ∞ ,ε )id p D er U U U U id p D er The model structure p D er r, is the one considered in [FKKR19, §3], while theothers are new. NDUCED MODEL STRUCTURES FOR HIGHER CATEGORIES 13
Model structures on marked simplicial sets.
Let s S et + denote the cate-gory of marked simplicial sets, namely simplicial sets endowed with a specified setof marked -simplices, and maps that preserve the marking, as in [Lur09, §3.1].There is an underlying functor U : s S et + → s S et which just forgets the marking.This functor admits both a left adjoint and a right adjoint which are given by theminimal and maximal marking respectively, ( − ) (cid:91) , ( − ) (cid:93) : s S et → s S et + . The minimaland maximal marking functors are fully faithful, since the unit of the adjunction ( − ) (cid:91) (cid:97) U is an identity. In particular, there is a fully faithful string of adjointfunctors U : s S et + ! s S et : ( − ) (cid:91) , ( − ) (cid:93) between the category of simplicial sets and the category of marked simplicial sets.From Corollary 1.17, we obtain the following, which endows s S et + with twoQuillen equivalent model structures for ( ∞ , ε ) -categories for any fixed ε = 0 , . Proposition 3.2.
Let ε = 0 , . The category s S et + admits both the left-inducedmodel structure s S et + (cid:96),ε and the right-induced model structure s S et + r,ε along U : s S et + → s S et ( ∞ ,ε ) . Further, the left diagram below is a diagram of left Quillen equivalences,and the right diagram below is a diagram of right Quillen equivalences. s S et + r,ε s S et + (cid:96),ε s S et + r,ε s S et + (cid:96),ε s S et ( ∞ ,ε ) s S et ( ∞ ,ε )id s S et + U U U U id s S et + These model structures on s S et + are all new, and seemingly different from themodel structure on s S et + that models ∞ -categories constructed by Lurie in [Lur09]. Remark . Recall from [Lur09, §3.1.3] the
Cartesian model structure on s S et + ,which we denote by s S et +qcat where the cofibrations are the monomorphisms and thefibrant objects are the naturally marked quasi-categories. Lurie shows in [Lur09,Proposition 3.1.5.3] that the functor U : s S et +qcat → s S et ( ∞ , is a right Quillenequivalence. Although the model structures s S et + (cid:96), and s S et + r, do not seem to becomparable with s S et +qcat via the identity functor, we have the following composablechain of Quillen equivalences. s S et + r, s S et + (cid:96), s S et ( ∞ , s S et +qcatidid U ( − ) (cid:93) ( − ) (cid:91) U Model structures on bisimplicial sets.
Let ss S et denote the category ofbisimplicial sets, and i ∗ : ss S et → s S et the zeroth row functor, defined by takingthe first row of a bisimplicial set as in [JT07, §4]. The functor i ∗ can be seen as thefunctor induced by the fully faithful inclusion i : ∆ (cid:44) → ∆ × ∆ , given by [ n ] (cid:55)→ [ n ] × [0] .In particular, i ∗ admits a left and a right adjoint, ( i ) ! , ( i ) ∗ : s S et → ss S et obtainedas left and right Kan extension along the fully faithful functor i , and they aretherefore fully faithful. In particular, there is a fully faithful string of adjoint functors i ∗ : ss S et ! s S et : ( i ) ! , ( i ) ∗ between the category of simplicial sets and the category of bisimplicial sets.From Corollary 1.17, we obtain the following, which endows ss S et with twoQuillen equivalent model structures for ( ∞ , ε ) -categories for any fixed ε = 0 , . Proposition 3.4.
Let ε = 0 , . The category ss S et admits both the left-inducedmodel structure ss S et (cid:96),ε and the right-induced model structure ss S et r,ε along i ∗ : ss S et → s S et ( ∞ ,ε ) . Further, the left diagram below is a diagram of left Quillen equivalences,and the right diagram below is a diagram of right Quillen equivalences. ss S et r,ε ss S et (cid:96),ε ss S et r,ε ss S et (cid:96),ε s S et ( ∞ ,ε ) s S et ( ∞ ,ε )id ss S et i ∗ i ∗ i ∗ i ∗ id ss S et These model structures on ss S et are all new, and seemingly different from themodel structure on ss S et that models ∞ -categories constructed by Rezk in [Rez01]. Remark . Recall from [Rez01, Theorem 7.2] the complete Segal space modelstructure on ss S et , which we denote by ss S et css where the cofibrations are themonomorphisms and the fibrant objects are the (injectively fibrant) complete Segalspaces. Joyal–Tierney show in [JT07, Theorem 4.11] that the functor i ∗ : ss S et css → s S et ( ∞ , is a right Quillen equivalence. Although the model structures ss S et (cid:96), and ss S et r, do not seem to be comparable with ss S et css via the identity functor, wehave the following composable chain of Quillen equivalences. ss S et r, ss S et (cid:96), s S et ( ∞ , ss S et cssidid i ∗ ( i ) ∗ ( i ) ! i ∗ References [BM11] Clemens Berger and Ieke Moerdijk,
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Department of Mathematics, University of Louisiana at Lafayette
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