Strictly commutative models for E-infinity quasi-categories
SSTRICTLY COMMUTATIVE MODELS FOR E ∞ QUASI-CATEGORIES
DIMITAR KODJABACHEV AND STEFFEN SAGAVE
Abstract.
In this short note we show that E ∞ quasi-categories can be re-placed by strictly commutative objects in the larger category of diagrams ofsimplicial sets indexed by finite sets and injections. This complements earlierwork on diagram spaces by Christian Schlichtkrull and the second author. Introduction An E ∞ space is a space with a multiplicative structure encoded by the action ofan E ∞ operad, i.e., an operad consisting of contractible spaces with a free Σ n -action.It is shown in joint work by Christian Schlichtkrull and the second author [SS12]that E ∞ spaces can be rigidified to strictly commutative objects if one passes to alarger category of I -spaces: if I denotes the category of finite sets n = { , . . . , n } and injective maps, then the functor category sSet I has a symmetric monoidalconvolution product, and the category sSet I [ C ] of commutative monoid objects insSet I admits a model structure making it Quillen equivalent to the category of E ∞ spaces.The following construction, due to Mirjam Solberg [SS, Section 4.14], shows thatsymmetric monoidal categories give rise to commutative monoid objects in sSet I in a natural way. Example 1.1.
Let ( A , ⊗ ) be a symmetric monoidal category. We consider thefunctor Φ( A ) : I →
Cat with objects of Φ( A )( n ) the n -tuples ( a , . . . , a n ) of objectsin A and morphismsΦ( A )( n )(( a , . . . , a n ) , ( b , . . . , b n )) = A ( a ⊗ . . . ⊗ a n , b ⊗ . . . ⊗ b n ) . Functoriality in I is induced by permutation of entries and insertion of the unitobject of A . Composing with the nerve functor N gives an I -simplicial set N Φ( A ),and the symmetric monoidal structure of A makes N Φ( A ) a commutative monoidobject in sSet I , see [SS, Proposition 4.16].Equipped with the (standard or Kan) model structure, the category of simpli-cial sets sSet is Quillen equivalent to the category of topological spaces. Therefore,weak homotopy types of spaces are represented by simplicial sets. But simplicialsets also model quasi-categories up to Joyal equivalence: there is a finer Joyal modelstructure on sSet whose fibrant objects are the quasi-categories and whose weakequivalences are called Joyal equivalences (see e.g. [Lur09a] or [DS11] for publishedreferences). Simplicial sets with E ∞ structures are also interesting from this per-spective since they model symmetric monoidal ( ∞ , E ∞ objects in sSet and strictly commutative objects insSet I still holds if we regard simplicial sets as models for quasi-categories. The aimof this note is to prove that this is indeed the case: Date : October 15, 2018. a r X i v : . [ m a t h . A T ] O c t DIMITAR KODJABACHEV AND STEFFEN SAGAVE
Theorem 1.2. (i) The category sSet I [ C ] of commutative monoid objects in sSet I admits a left proper positive I -model structure where a map f is a weakequivalence if and only if hocolim I f is a Joyal equivalence.(ii) If D is an E ∞ operad, then there is a chain of Quillen equivalences relat-ing sSet I [ C ] and the category sSet[ D ] of E ∞ simplicial sets with the modelstructure lifted from the Joyal model structure. Here an E ∞ operad is an operad D in simplicial sets such that D ( n ) has afree Σ n -action and D ( n ) is contractible with respect to the Joyal model structure.Every E ∞ operad in this sense is an E ∞ operad in the classical sense since beingcontractible with respect to the Joyal model structure implies being contractiblewith respect to the Kan model structure. Moreover, every operad D with D ( n )a Σ n -free Kan complex such that D ( n ) → ∗ is a weak homotopy equivalence isan E ∞ -operad in the sense of the theorem, for example the Barratt–Eccles operadwhose n -th space is E Σ n .By [SS, Lemma 4.15], the object N Φ( A ) in sSet I [ C ] considered in Example 1.1 isfibrant in the model structure of Theorem 1.2(i). It models the E ∞ quasi-category N A associated with the symmetric monoidal category A . Therefore Example 1.1shows that the nerve of a symmetric monoidal category can be rigidified to a com-mutative monoid object in sSet I in a natural way.More generally, it follows from Theorem 1.2 that for any E ∞ simplicial set X in the sense of the theorem, there is an A ∈ sSet I [ C ] and a chain of maps A ← B → const I X of E ∞ objects in sSet I that induces a chain of Joyal equiva-lences when applying hocolim I (compare [SS12, Corollary 3.7]). Hence the E ∞ object X can be replaced by the strictly commutative object A . Although thisrigidification of a structure up to homotopy by a strict one is in contrast to thephilosophy of quasi-categories, we think that it is valuable to observe that E ∞ quasi-categories can be expressed this way: when viewing simplicial sets as modelsfor spaces, it is often easy to write down explicit objects in sSet I [ C ] that model E ∞ spaces. This applies for example to Q ( X ) if X is connected [SS12, Example1.3] or to B GL ∞ ( R ) + [Sch04, Remark 2.2]. It is likely that besides Example 1.1above, there are more instances where interesting E ∞ quasi-categories arise fromcommutative I -functors.Theorem 1.2 and the corresponding statement about weak homotopy types of E ∞ spaces [SS12, Theorem 1.2] refer to different model structures on the same categoriesthat have the same cofibrations. Nonetheless, several arguments from [SS12] donot apply here since [SS12, Theorem 1.2] was derived from a result about diagramspaces indexed by more general categories than I , and some of the more generalarguments were based on special features of the Kan model structure. However,the Joyal model structure differs from the Kan model structure since it fails to beright proper and simplicial, and because it doesn’t have an explicit set of generatingacyclic cofibrations. In the proof of Theorem 1.2 presented here, we put emphasison the points where new arguments are required and simply cite those parts of theproof of [SS12, Theorem 1.2] that also apply here.This note is a condensed and revised version of the first author’s master’s thesis atthe University of Bonn, supervised by the second author. We thank an anonymousreferee for a quick and helpful report on an earlier version of this note.2. Model structures on I -simplicial sets The category of simplicial sets sSet admits a
Joyal model structure with cofibra-tions the monomorphisms and fibrant objects the quasi-categories, i.e., the weak or inner Kan complexes. See [Lur09a, Theorem 2.2.5.1] or [DS11, Theorem 2.13].The Joyal model structure is cofibrantly generated with generating cofibrations
TRICTLY COMMUTATIVE MODELS FOR E ∞ QUASI-CATEGORIES 3 I = { ∂ ∆ n → ∆ n | n ≥ } . We let J be a set of generating acyclic cofibrations.(There is no known explicit description of such a set J .)Let I be the category with objects the finite sets n = { , . . . , n } for n ≥ (cid:116) makes I a symmetricmonoidal category with unit and symmetry isomorphism the obvious shuffle map.Let sSet I be the functor category of I -diagrams of simplicial sets. For everyobject n of I , there is a free/forgetful adjunction F I n : sSet (cid:29) sSet I : Ev n with F I n ( K ) = I ( n , − ) × K and Ev n ( X ) = X ( n ). For X and Y in sSet I , the leftKan extension of the I × I -diagram X ( − ) × Y ( − ) along (cid:116) : I × I → I defines anobject X (cid:2) Y in sSet I . This construction defines a symmetric monoidal product (cid:2) : sSet I × sSet I → sSet I with unit F I ( ∗ ).We now start to consider model structures on sSet I . Let I + be the full sub-category of I on the objects n with | n | ≥
1. We say that a map f : X → Y insSet I is an absolute (resp. positive) level equivalence if f : X ( n ) → Y ( n ) is a Joyalequivalence for all n in I (resp. all n in I + ), and an absolute (resp. positive) levelfibration if f : X ( n ) → Y ( n ) is a fibration in the Joyal model structure for all n in I (resp. all n in I + ). A map is an absolute (resp. positive) level cofibration if ithas the left lifting property with respect to any map that is both an absolute (resp.positive) level fibration and level equivalence. Lemma 2.1.
These classes of maps define two cofibrantly generated left propermodel structures on sSet I , called the absolute and the positive level model struc-tures.Proof of Lemma 2.1. The absolute case follows from [Hir03, Theorem 11.6.1], andthe positive case works as in [SS12, Proposition 6.7]. The sets(2.1) I levelabs = { F I n ( i ) | i ∈ I, n ∈ I} and I levelpos = { F I n ( i ) | i ∈ I, n ∈ I + } provide the generating cofibrations. The generating acyclic cofibrations J levelabs and J levelpos are defined similarly with J in place of I . (cid:3) Since the Joyal model structure fails to be simplicial, the usual Bousfield-Kanformula does not provide a homotopy invariant homotopy colimit functor. In thefollowing, hocolim I : sSet I → sSet denotes the functor constructed in [Hir03, § I X → colim I X . Lemma 2.2. If X is absolute or positive level cofibrant in sSet I , then the map hocolim I X → colim I X is a Joyal equivalence.Proof. This is analogous to [Hir03, Theorem 19.9.1], with the absolute level modelstructure replacing the Reedy model structure in that reference. (cid:3)
We say that a map f : X → Y in sSet I is an I -equivalence if hocolim I f is aJoyal equivalence of simplicial sets, and an absolute (resp. positive) I -cofibrationif it is an absolute (resp. positive) level cofibration. A map is an absolute (resp.positive) I -fibration if it has the right lifting property with respect to any map thatis both an absolute (resp. positive) I -cofibration and an I -equivalence. Proposition 2.3.
These classes of maps define two cofibrantly generated left propermodel structures on sSet I , called the absolute and the positive I -model structures. We write sSet I abs and sSet I pos for these model categories. These (Joyal) I -modelstructures have the same cofibrations as the corresponding (Kan) I -model struc-tures constructed in [SS12, Proposition 6.16] by a different technique. DIMITAR KODJABACHEV AND STEFFEN SAGAVE
Proof.
Since I has an initial object, its classifying space is contractible. Hence theexistence of the absolute I -model structure follows from [Dug01, Theorem 5.2], andwe recall from [Dug01] that it is constructed as the left Bousfield localization of theabsolute level model structure at S = { α ∗ : F I n ( ∗ ) → F I m ( ∗ ) | α : m → n ∈ I} .The positive I -model structure is defined to be the left Bousfield localization ofthe positive level model structure with respect to T = { α ∗ : F I n ( ∗ ) → F I m ( ∗ ) | α : m → n ∈ I + } . It exists and is left proper by [Hir03, Theorem 4.1.1]. Hence it remains to showthat its weak equivalences, the T -local equivalences , are the I -equivalences. Since T ⊂ S , every T -local equivalence is an I -equivalence. Let f be an I -equivalence.Passing to fibrant replacements, we may assume that f is a map of T -local objects.Restricting f along the inclusion I + → I and applying [Dug01, Theorem 5.2]to sSet I + , it follows that hocolim I + f is a Joyal equivalence. Since I + → I ishomotopy cofinal [SS12, Proof of Corollary 5.9], this implies the claim. (cid:3) Corollary 2.4.
There is a chain of Quillen equivalences sSet I pos id (cid:47) (cid:47) sSet I absid (cid:111) (cid:111) colim I (cid:47) (cid:47) sSet const I (cid:111) (cid:111) relating sSet I equipped with the positive and absolute I -model structures and sSet equipped with the Joyal model structure.Proof. It is clear that (id , id) is a Quillen equivalence. The adjunction (colim I , const I )is a Quillen equivalence by [Dug01, Theorem 5.2(b)]. (cid:3) The next lemma and the subsequent proposition are analogous to [SS12, Propo-sition 7.1(iii)-(v) and Proposition 8.2]. The proofs given here avoid using featuresof the Bousfield-Kan formula for homotopy colimits.
Lemma 2.5. (i) The gluing lemma for levelwise monomorphisms and level equiv-alences holds.(ii) The gluing lemma for levelwise monomorphisms and I -equivalences holds.(iii) For any ordinal λ and any λ -sequence ( X α ) α<λ of levelwise monomorphisms,the canonical map hocolim α<λ X α → colim α<λ X α is a level equivalence.Proof. Part (i) follows from the gluing lemma in left proper model categories [Hir03,Proposition 13.5.4]. Using (i) and the absolute level cofibrant replacement, it isenough to show (ii) for a diagram of absolute cofibrant objects. This special casefollows from the gluing lemma in the Joyal model structure by applying colim I .Part (iii) follows from [Hir03, Theorem 19.9.1]. (cid:3) Proposition 2.6. If X is absolute cofibrant in sSet I , then X (cid:2) − preserves I -equivalences between not necessarily cofibrant objects.Proof. We first assume that X = F I k ( L ) with L ∈ sSet and k ∈ I . Let Y → Z be an I -equivalence. If Y and Z are absolute cofibrant, then the claim follows byapplying the strong symmetric monoidal functor colim I and using Corollary 2.4 andthe pushout-product axiom for the Joyal model structure [DS11, 2.15 Proposition].If Y c → Y is an absolute level cofibrant replacement, then [SS12, Lemma 5.6]implies that ( X (cid:2) ( Y c → Y ))( m ) is isomorphic to(2.2) L × colim k (cid:116) l → m Y c ( l ) → L × colim k (cid:116) l → m Y ( l ) . Since each connected component of the comma category k (cid:116) − ↓ m has a termi-nal object [SS12, Corollary 5.9], the colimits in (2.2) are Joyal equivalent to thecorresponding homotopy colimits and (2.2) is a level equivalence. It follows that TRICTLY COMMUTATIVE MODELS FOR E ∞ QUASI-CATEGORIES 5 X (cid:2) ( Y → Z ) is an I -equivalence since X (cid:2) ( Y c → Z c ) is. With Lemma 2.5 replac-ing those parts of [SS12, Proposition 7.1] that involve weak equivalences, the caseof general X follows as in the proof of [SS12, Proposition 8.2]. (cid:3) Corollary 2.7.
The absolute and positive I -model structures on sSet I satisfy thepushout-product axiom and the monoid axiom.Proof. The part of the pushout-product axiom involving only cofibrations resultsfrom [SS12, Proposition 8.4]. As in [SS12, § (cid:3) The following lemma is analogous to [SS12, Lemma 8.1].
Lemma 2.8.
Let G be a finite group and let f : X → Y and Y → E be morphismsin (sSet I ) G such that hocolim I f is a Joyal equivalence. If G acts freely on E ( m ) for every object m in I , then f /G : X/G → Y /G is an I -equivalence.Proof. Since there is a G -map Y → E , the G -action on Y ( m ) is also free. Hencehocolim G Y ( m ) → colim G Y ( m ) ∼ = ( Y /G )( m ) is a Joyal equivalence. Using thesame argument for X , it follows thathocolim G hocolim I f (cid:39) hocolim I hocolim G f (cid:39) hocolim I ( f /G )is a Joyal equivalence. (cid:3) The use of the positive model structure is motivated by the positive model struc-ture for symmetric spectra discovered by Jeff Smith. The next lemma highlightsone of its key features.
Lemma 2.9. If X is positive I -cofibrant, then the Σ n -action on the simplicial set ( X (cid:2) n )( m ) is free for every object m of I .Proof. Let f : U → V and U → Y be maps in sSet I . By a cell induction argument,it is enough to show that if f is a generating cofibration and Σ n acts freely on( Y (cid:2) n )( m ) for every m in I , then Z = Y (cid:96) U V has this property. By [SS12,Lemma A.8], Y (cid:2) n → Z (cid:2) n has a filtration by maps that are cobase changes of mapsof the form Σ n × Σ n − i × Σ i Y (cid:2) n − i (cid:2) f (cid:3) i where f (cid:3) i is the i -fold iterated pushoutproduct map in (sSet I , (cid:2) ). Hence it suffices to show that ( Y (cid:2) n − i (cid:2) f (cid:3) i )( m ) is a(Σ n − i × Σ i )-projective cofibration of simplicial sets with (Σ n − i × Σ i )-action. Since f = F I k ( ∗ ) × g with g a generating cofibration for sSet and k ∈ I + , it follows from[SS12, Lemma 5.6] that there is an isomorphism(2.3) ( Y (cid:2) n − i (cid:2) f (cid:3) i )( m ) ∼ = (colim k (cid:116) i (cid:116) l → m Y (cid:2) n − i ( l )) × g (cid:3) i where g (cid:3) i is the i -fold iterated pushout-product map of g in (sSet , × ). By [SS12,Corollary 5.9], each connected component of the indexing category k (cid:116) i (cid:116) − ↓ m has a terminal object, and Σ i acts freely on the set of connected components.Hence colim k (cid:116) i (cid:116) l → m Y (cid:2) n − i ( l ) is a (Σ n − i × Σ i )-free simplicial set, and (2.3) is a(Σ n − i × Σ i )-projective cofibration. (cid:3) Model structures on structured diagrams of simplicial sets
In the following, an operad D denotes a sequence of simplicial sets D ( n ) with Σ n -action such that D (0) = ∗ , there is a unit map ∗ → D (1), and there are structuremaps D ( n ) ×D ( i ) ×· · ·×D ( i n ) → D ( i + · · · + i n ) satisfying the usual associativity,unit and equivariance relations. It is called Σ -free if Σ n acts freely on D ( n ) for all n .Let sSet I [ D ] be the category of D -algebras in (sSet I , (cid:2) ). We say that a modelstructure on sSet I lifts to sSet I [ D ] if sSet I [ D ] admits a model structure where amap is a weak equivalence or fibration if the underlying map in sSet I is. DIMITAR KODJABACHEV AND STEFFEN SAGAVE
Theorem 3.1.
Let D be an operad. The positive I -model structure lifts to sSet I [ D ] ,and the absolute I -model structure lifts to sSet I [ D ] if D is Σ -free. Since the generating cofibrations coincide, these model structures have the samecofibrations as the corresponding Kan I -model structures [SS12, Proposition 9.3]. Proof.
As in the analogous statement about the Kan I -model structure [SS12,Proposition 9.3], the claim reduces to showing that for a generating acyclic cofibra-tion f : U → V in sSet I , the bottom map in a pushout square (cid:96) n ≥ D ( n ) × Σ n U (cid:2) n (cid:47) (cid:47) (cid:15) (cid:15) (cid:96) n ≥ D ( n ) × Σ n V (cid:2) n (cid:15) (cid:15) X (cid:47) (cid:47) Y in sSet I [ D ] is an I -equivalence. Replacing [SS12, Propositions 8.4 and 8.6] byCorollary 2.7 and [SS12, Lemma 8.1] by Lemma 2.8, the argument given in theproof of [SS12, Lemma 9.5] applies verbatim with one exception: we need to showthat for any n ≥ m in I , the group Σ n acts freely on V (cid:2) n ( m ). Using thatthe generating cofibrations have cofibrant domains and codomains, we may assumethat this also holds for the generating acyclic cofibrations [Bar10, Corollaries 2.7and 2.8]. Hence the last claim follows from Lemma 2.9. (cid:3) We recall that a morphism of operads Φ :
D → E induces an adjunctionΦ ∗ : sSet I [ D ] (cid:29) sSet I [ E ] : Φ ∗ . Proposition 3.2.
Let
Φ :
D → E be a morphism of operads with Φ n : D ( n ) → E ( n ) a Joyal equivalence for each n ≥ . Then (Φ ∗ , Φ ∗ ) is a Quillen equivalence withrespect to the positive I -model structures. If D and E are Σ -free, then it is also aQuillen equivalence with respect to the absolute I -model structures.Proof. Again the proof of the analogous statement about the Kan I -model struc-ture [SS12, Proposition 9.12] applies almost verbatim: in the key ingredient [SS12,Lemma 9.13], Lemma 2.5 replaces those parts of [SS12, Proposition 7.1] that involveweak equivalences, Corollary 2.7 replaces [SS12, Proposition 8.4], Proposition 2.6replaces [SS12, Proposition 8.2], and Lemma 2.8 replaces [SS12, Lemma 8.1]. (cid:3) Proof of Theorem 1.2.
Part (i) follows from Theorem 3.1 applied to the commuta-tivity operad C with C ( n ) = ∗ for every n . Left properness follows by the argumentsfrom [SS12, Lemma 11.8 and Proposition 11.9], where again the results from Sec-tion 2 replace the corresponding statements in [SS12].If D is an E ∞ operad, then there is a canonical morphism Φ : D → C , and weobtain a chain of Quillen adjunctionssSet I pos [ C ] Φ ∗ (cid:47) (cid:47) sSet I pos [ D ] id (cid:47) (cid:47) Φ ∗ (cid:111) (cid:111) sSet I abs [ D ] id (cid:111) (cid:111) colim I (cid:47) (cid:47) sSet[ D ] const I (cid:111) (cid:111) The first adjunction is a Quillen equivalence by Proposition 3.2. The last twoadjunctions are Quillen equivalences by Corollary 2.4 and the fact that cofibrantobjects in sSet I abs [ D ] are cofibrant in sSet I abs if D is Σ-free [SS12, Corollary 12.3]. (cid:3) References [Bar10] C. Barwick,
On left and right model categories and left and right Bousfield localizations ,Homology, Homotopy Appl. (2010), no. 2, 245–320.[DS11] D. Dugger and D. I. Spivak, Mapping spaces in quasi-categories , Algebr. Geom. Topol. (2011), no. 1, 263–325.[Dug01] D. Dugger, Replacing model categories with simplicial ones , Trans. Amer. Math. Soc. (2001), no. 12, 5003–5027 (electronic).
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