Coalgebras in the Dwyer-Kan localization of a model category
aa r X i v : . [ m a t h . A T ] J un COALGEBRAS IN THE DWYER-KAN LOCALIZATION OF AMODEL CATEGORY
MAXIMILIEN P´EROUX
Abstract.
We show that weak monoidal Quillen equivalences induce equiv-alences of symmetric monoidal ∞ -categories with respect to the Dwyer-Kanlocalization of the symmetric monoidal model categories. The result will inducea Dold-Kan correspondance of coalgebras in ∞ -categories. Moreover it showsthat Shipley’s zig-zag of Quillen equivalences provides an explicit symmetricmonoidal equivalence of ∞ -categories for the stable Dold-Kan correspondance.We study homotopy coherent coalgebras associated to a monoidal model cat-egory and we show that these coalgebras cannot be rigidified. That is, their ∞ -categories are not equivalent to the Dwyer-Kan localizations of strict coal-gebras in the usual monoidal model categories of spectra and of connectivediscrete R -modules. Introduction
Let M be a model category and W its morphism class of weak equivalences. Re-call that the homotopy category Ho ( M ), associated to M , is an ordinary categoryobtained by inverting all weak equivalences, and can also be denoted M [ W − ], see[Hov99, 1.2.1, 1.2.10]. However, the higher homotopy information is lost in Ho ( M ).Dwyer and Kan, in [DK80], suggested instead a simplicial category L H ( M , W ) some-times called the hammock localization of M , that retains the higher information. Theidea has been translated into ∞ -categories by Lurie in [Lur17, 1.3.4.1, 1.3.4.15].Following [Hin16], we shall prefer the term of Dwyer-Kan localization instead of underlying ∞ -category of a model category (see motivation by Remark 2.3 below).If the model category is endowed with a symmetric monoidal structure compatiblewith the model structure, then the Dwyer-Kan localization is symmetric monoidalwith respect to the derived tensor product.The main result of this paper is Theorem 2.13 which shows that weak monoidalQuillen equivalences (as in [SS03]) lift to equivalences of symmetric monoidal ∞ -categories with respect to the Dwyer-Kan localizations.By [Lur17, 4.1.8.4], any A ∞ -ring spectrum is homotopic to a strictly unital andassociative ring spectrum, in some monoidal model category representing spectra,say symmetric spectra, as in [HSS00]. Similarly, any E ∞ -ring spectrum is homotopicto a strictly unital, associative and commutative ring spectrum, see [Lur17, 4.5.4.7].Associative and commutative algebras in the Dwyer-Kan localization of a symmetricmonoidal model category M are precisely the A ∞ -algebras and E ∞ -algebras of M . Mathematics Subject Classification.
Key words and phrases. homotopy, spectrum, coalgebra, ∞ -category, rigidification, Dold-KanCorrespondance. The paper is interested into the dual questions: can A ∞ -coalgebras and E ∞ -coalgebras of spectra be homotopic to strictly counital, coassociative and cocom-mutative coalgebras over the sphere spectrum? In other words, can we rigidify thecomultiplication in spectra? We show in Corollary 5.6 that it is not the case for A ∞ -coalgebras, following from a previous result in [PS19].Another consequence of our Theorem 2.13 is that it provides a Dold-Kan cor-respondance for A ∞ and E ∞ -coalgebras in ∞ -categories. At the level of modelcategories, the result has been shown to be untrue in [Sor19]. Therefore we alsoshow in Corollary 4.3 that A ∞ -coalgebras in connective modules over an Eilenberg-Mac Lane spectrum of a commutative ring do not correspond to their strict ana-logue in either simplicial modules or non-negative chain complexes. Moreover, weshow in Corollary 4.4 that the Quillen zig-zag of Shipley in [Shi07] provides Lurie’sequivalence in the stable Dold-Kan correspondance [Lur17, 7.1.2.13].In [P´er20c], we shall be interesed in comodules in the Dwyer-Kan localization ofnon-negative chain complexes over a finite product of fields and show that homotopycoherent comodules do correspond to strict comodules.The paper is constructed as follows. In Section 2 we clarify the definitions ofthe Dwyer-Kan localization in the literature in anticipation of our main result withTheorem 2.13. In Section 3, we provide a comparison map between A ∞ and E ∞ coalgebras and their strict analogue. We also provide a simple example where thismap is an equivalence in Proposition 3.3. In Section 4, we apply Theorem 2.13 tothe weak monoidal Quillen equivalences of [SS03] and [Shi07] and show the failureof rigidification for connective discrete modules in Corollary 4.3. In Section 5,we provide a model structure for strict coalgebras in symmetric spectra following[HKRS17] but show that they do not correspond to A ∞ -coalgebras in Theorem 5.4. Acknowledgment.
The results here are part of my PhD thesis [P´er20b], and assuch, I would like to express my gratitude to my advisor Brooke Shipley for her helpand guidance throughout the years. I would also like to thank Ben Antieau andTasos Moulinos for helpful discussions in the early process of writing this paper.2.
The Dwyer-Kan Localization of a Model Category
We show here our main result which is Theorem 2.13. We first provide a clarifi-cation on Dwyer-Kan localizations following [Hin16], [NS18] and [Lur17].2.1.
The General Definition.
We first start by some generality.
Definition 2.1 ([Lur17, 1.3.4.1]) . Let C be an ∞ -category and fix a collection W ⊆ Hom sSet (∆ , C ) of morphisms in C . The Dwyer-Kan localization of C withrespect to the collection W is an ∞ -category, denoted C [ W − ], together with afunctor f : C → C [ W − ] that respects the following universal property.(U) For any other ∞ -category D , the functor f induces an equivalence of ∞ -categories: Fun ( C [ W − ] , D ) Fun W ( C , D ) , ≃ where Fun W ( C , D ) is the full subcategory of functors C → D that sendsmorphisms in W to equivalences in D . OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 3
The Dywer-Kan localization C [ W − ] always exists, for any choice of C and W ,see [Lur17, 1.3.4.2], and is unique up to contractible choice. We shall be moreinterested in the case when C = N ( M ) the nerve for some model category M . Definition 2.2 ([Lur17, 1.3.4.15]) . Let M be a model category and W its classof weak equivalences. We call N ( M )[ W − ] the Dwyer-Kan localization of M withrespect to W as in Definition 2.1, where we abuse notation and let W denote theinduced class of morphisms in the nerve N ( M ).Notice that the homotopy category of N ( M )[ W − ] is precisely the category Ho ( M ). Remark 2.3.
We do not define the hammock localization L H ( M , W ), but invitethe interested reader to read the explicit definition in [DK80, 2.1]. Since simplicialcategories represents ∞ -categories, the hammock localisation simplicial category L H ( M , W ) is a model for the Dwyer-Kan localization N ( M )[ W − ]. More precisely,by [Lur09, 2.2.5.1], there is a Quillen equivalence between the category of simplicialsets sSet endowed with the Joyal model structure and the category of simplicialcategories sCat endowed with the Bergner model structure: sSet sCat . C ⊥ N The functor N : sCat → sSet is the homotopy coherent nerve, or the simplicialnerve, as in [Lur09, 1.1.5.5]. After a fibrant replacement, the functor N sends L H ( M , W ) to the equivalence class of N ( M )[ W − ], as seen in [Hin16, 1.3.1]. Remark 2.4.
As noted in [Lur17, 1.3.4.16], [Hin16, 1.3.4], and [DK80, 8.4], if themodel category M admits functorial fibrant and cofibrant replacement, in the senseof [Hov99, 1.1.1. 1.1.3], then the following ∞ -categories are equivalent: N ( M c )[ W − ] ≃ N ( M )[ W − ] ≃ N ( M f )[ W − ] , where M c ⊆ M is the full subcategory of cofibrant objects, and M f ⊆ M is the fullsubcategory of fibrant objects.2.2. Symmetric Monoidal Dwyer-Kan Localization.
We now construct thesymmetric monoidal structure on the Dwyer-Kan localization of a symmetric mono-idal model category M . This is a recollection of Appendix A in [NS18] and Section4.1.7 on monoidal model categories in [Lur17]. Definition 2.5 ([Lur17, 4.1.7.4], [NS18, A.4, A.5]) . Let C ⊗ be a symmetric mono-idal ∞ -category. Let W ⊆ Hom sSet (∆ , C ) be a class of edges in C that is stableunder homotopy, composition and contains all equivalences. Suppose further that ⊗ : C × C → C preserves the class W seperately in each variable. The symmetricmonoidal Dywer-Kan localization of C ⊗ with respect to W is a symmetric monoidal ∞ -category, denoted C [ W − ] ⊗ , together with a symmetric monoidal functor i : C ⊗ → C [ W − ] ⊗ which is characterized by the following universal property.(U) For any other symmetric monoidal ∞ -category D ⊗ , the functor i inducesan equivalence of ∞ -categories: Fun ⊗ ( C [ W − ] ⊗ , D ⊗ ) ≃ −→ Fun W ⊗ ( C ⊗ , D ⊗ ) , where Fun W ⊗ ( C ⊗ , D ⊗ ) is the full subcategory of symmetric monoidal func-tors C ⊗ → D ⊗ that sends W to equivalences. MAXIMILIEN P´EROUX
As noticed in [NS18, A.5], the underlying ∞ -category of the symmetric monoidalcategory C [ W − ] ⊗ is precisely the Dwyer-Kan localization of C with W in the senseof Definition 2.1, i.e.: (cid:16) C [ W − ] ⊗ (cid:17) h i ≃ C [ W − ] . Remark 2.6.
Let C ⊗ and W be as in Definition 2.5. Given the symmetric monoidalstructure C ⊗ → N ( Fin ∗ ), products of n edges in W in C correspond precisely, underthe equivalence: C × n ≃ C ⊗h n i , to morphisms lying over id h n i in N ( Fin ∗ ). This defines a class of edges W ⊗ ⊆ Hom sSet (∆ , C ⊗ ). Then the Dwyer-Kan localization of C ⊗ with respect to W ⊗ , inthe sense of Definition 2.1, denoted C ⊗ h ( W ⊗ ) − i , is equivalent to C [ W − ] ⊗ definedabove.We would like to study the case where the underlying ∞ -category of C ⊗ isthe Dwyer-Kan localization N ( M )[ W − ] of a model category M . We first recallthe induced symmetric monoidal structure on the nerve of a symmetric monoidalcategory. Definition 2.7 ([Lur17, 2.0.0.1]) . Let ( C , ⊗ , I ) be a symmetric monoidal category.Define a new category C ⊗ as follows. • Objects are sequences ( C , . . . , C n ) where each C i is an object in C , for all1 ≤ i ≤ n , for some n ≥
1. We allow the case n = 0 and thus the emptyset ∅ as a sequence. • A morphism ( C , . . . , C n ) → ( C ′ , . . . , C ′ m ) in C ⊗ is a pair ( α, { f j } ), where α is a map of finite sets α : h n i → h m i and { f j } is a collection of m -morphismsin C : f j : O i ∈ α − ( j ) C i −→ C ′ j , for all 1 ≤ j ≤ m . If α − ( j ) = ∅ , then f j is a morphism I → C ′ j . • The composition of morphisms in C ⊗ is defined using the compositionsin Fin ∗ and C together with the associativity of the symmetric monoidalstructure of C . • The identity morphism on an object ( C , . . . , C n ) is given by the identitesin Fin ∗ and C : ( id h n i , { id C j } ).We obtain a functor: C ⊗ −→ Fin ∗ , that sends ( C , . . . C n ) to h n i . The induced functor N ( C ⊗ ) → N ( Fin ∗ ) in ∞ -categories is coCartesian and defines a symmetric monoidal structure. Proposition 2.8 ([Lur17, 2.1.2.21]) . Given ( C , ⊗ , I ) a symmetric monoidal cate-gory, let C ⊗ be as Definition 2.7 . Then the nerve N ( C ⊗ ) is a symmetric monoidal ∞ -category whose underlying ∞ -category is N ( C ) . Remark 2.9.
In particular, given ( C , . . . , C n ) in C ⊗ , and α : h n i → h m i a mapin Fin ∗ , the associated coCartesian lift is induced by defining C ′ j as follows: C ′ j := O i ∈ α − ( j ) C i , OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 5 for each 1 ≤ j ≤ m . Define C ′ j = I if j is such that α − ( j ) = ∅ . This defines amorphism ( C , . . . , C n ) → ( C ′ , . . . , C ′ m ) in C ⊗ as desired.If the symmetric monoidal category ( C , ⊗ , I ) happens to be endowed with a modelstructure, the bifunctor ⊗ : C × C → C need not preserve weak equivalences ineither variable. We need to restrict to (symmetric) monoidal model categories as in[Hov99, 4.2.6]. In any monoidal model category ( M , ⊗ , I ), the tensor ⊗ : M × M → M preserves weak equivalences in each variable, if we restrict to cofibrant objects M c ⊆ M . Moreover. the tensor product of cofibrant objects is again cofibrant. Inmodel categories, this allows us to define a derived tensor product for the homotopycategory Ho ( M ) = M [ W − ], see [Hov99, 4.3.2]. In higher category, the transitionbetween the tensor product and the derived tensor product is exactly through theDwyer-Kan localization of a symmetric monoidal ∞ -category as in Definition 2.5.If we suppose in addition that I is cofibrant, then, as in Definition 2.7, we can define M ⊗ c ⊆ M ⊗ from the full subcategory of cofibrant objects M c ⊆ M , since ( M c , ⊗ , I )is symmetric monoidal. Proposition 2.10 ([Lur17, 4.1.7.6], [NS18, A.7]) . Let ( M , ⊗ , I ) be a symmetricmonoidal model category. Suppose that I is cofibrant. Then the Dwyer-Kan lo-calization N ( M c )[ W − ] of M can be given the structure of symmetric monoidal ∞ -category via the symmetric monoidal Dwyer-Kan localization of N ( M ⊗ c ) in thesense of Definition 2.5 , N ( M ⊗ c ) N ( M c )[ W − ] ⊗ , where W is the class of weak equivalences restricted to cofibrant objects in M . Remark 2.11.
The inclusion of cofibrant objects M c ⊆ M induces a lax symmetricmonoidal functor N ( M ⊗ c ) → N ( M ⊗ ). From Remark 2.4, Proposition 2.10 implieswe can also construct a symmetric monoidal ∞ -category N ( M )[ W − ] ⊗ whose fiberover h i is precisely N ( M )[ W − ]. However, cofibrant replacement induces a functor N ( M ⊗ ) → N ( M )[ W − ] ⊗ that is only lax symmetric monoidal and does not sharethe same properties of universality as in Definition 2.5. We invite the interestedreader to look at [NS18, A.7] for more details.2.3. Weak Monoidal Quillen Equivalence.
Given C and D model categories,denote W C and W D their respective class of weak equivalences. Let: L : C D : R, ⊥ be a Quillen adjunction. Then as the left adjoint functor L preserves weak equiv-alences between cofibrant objects and the right adjoint functor R preserves weakequivalences between fibrant objects, we get, by [Hin16, 1.5.1], a pair of adjointfunctors in ∞ -categories between the Dwyer-Kan localizations of C and D : L : N ( C )[ W − C ] N ( D )[ W − D ] : R , ⊥ where L and R represent the derived functors of L and R . If C and D are symmetricmonoidal model categories, we investigate when the derived functors are symmetricmonoidal functors of ∞ -categories. Definition 2.12 ([SS03, 3.6]) . Let ( C , ⊗ , I ) and ( D , ∧ , J ) be symmetric monoidalmodel categories. A weak monoidal Quillen pair consists of a Quillen adjunction: L : ( C , ⊗ , I ) ( D , ∧ , J ) : R, ⊥ MAXIMILIEN P´EROUX where L is lax comonoidal such that the following two conditions hold.(i) For all cofibrant objects X and Y in C , the comonoidal map: L ( X ⊗ Y ) L ( X ) ∧ L ( Y ) , is a weak equivalence in D .(ii) For some (hence any) cofibrant replacement λ : c I ∼ −→ I in C , the compositemap: L ( c I ) L ( I ) J , L ( λ ) is a weak equivalence in D , where the unlabeled map is the natural comono-idal structure of L .A weak monoidal Quillen pair is a weak monoidal Quillen equivalence if the under-lying Quillen pair is a Quillen equivalence. Theorem 2.13.
Let ( C , ⊗ , I ) and ( D , ∧ , J ) be symmetric monoidal model categorieswith cofibrant units. Let W C and W D be the classes of weak equivalence in C and D respectively. Let: L : ( C , ⊗ , I ) ( D , ∧ , J ) : R, ⊥ be a weak monoidal Quillen pair. Then the derived functor of L : C → D induces asymmetric monoidal functor between the Dwyer-Kan localizations: L : N ( C c ) (cid:2) W − C (cid:3) N ( D c ) (cid:2) W − D (cid:3) , where C c ⊆ C and D c ⊆ D are the full subcategories of cofibrant objects. If L and R form a weak monoidal Quillen equivalence, then L is a symmetric monoidalequivalence of ∞ -categories.Proof. Let C ⊗ c and D ⊗ c be as Definition 2.7. Denote the symmetric monoidal Dwyer-Kan localizations (Definition 2.5) by: i C : N ( C ⊗ c ) N ( C c )[ W − C ] ⊗ , i D : N ( D ⊗ c ) N ( D c )[ W − D ] ⊗ , and denote their coCartesian fibrations by: p : N ( C c )[ W − C ] ⊗ N ( Fin ∗ ) , q : N ( D c )[ W − D ] ⊗ N ( Fin ∗ ) . The functor L : C → D , as a left Quillen functor, defines N ( C c ) → N ( D c ), andhence a functor L ⊗ : N ( C ⊗ c ) → N ( D ⊗ c ) that is compatible with the coCartesianstructures: N ( C ⊗ c ) N ( D ⊗ c ) N ( D c )[ W − D ] ⊗ N ( Fin ∗ ) . L ⊗ i D q We show that the composite: N ( C ⊗ c ) N ( D ⊗ c ) N ( D c )[ W − D ] ⊗ L ⊗ i D is a symmetric monoidal functor that sends W C to equivalences, i.e., belongs tothe ∞ -category Fun W C ⊗ ( N ( C ⊗ c ) , N ( D c )[ W − D ] ⊗ ), as in Definition 2.5. The latter isclear as L is a left Quillen functor. We are left to show that the composite sends p -coCartesian lifts to q -coCartesian lifts. Let ( C , . . . , C n ) be an object of C ⊗ c , and OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 7 let α : h n i → h m i be a morphism of finite sets. A p -lift of ( α, ( C , . . . , C n )) is givenin Remark 2.9 by a certain sequence ( C ′ , . . . , C ′ m ) in C ⊗ c , i.e., the induced map( C , . . . , C n ) → ( C ′ , . . . , C ′ m ) is sent to α via the coCartesian functor p . Since L isweak monoidal functor, from (i) of Definition 2.12, we get that: ^ i ∈ α − ( j ) L ( C i ) L O i ∈ α − ( j ) C i = L ( C ′ j ) , ∼ is a weak equivalence in D , for all 1 ≤ j ≤ m . In the case α − ( j ) = ∅ , we apply (ii)of Definition 2.12 to obtain a weak equivalence: J L ( I ) = L ( C j ) . ∼ Applying the localization i D and Remark 2.6, we get that ( L ( C ′ ) , . . . , L ( C ′ m )) de-fines the desired q -coCartesian lift.By the universal property (U) of the symmetric monoidal Dwyer-Kan localizationin Definition 2.5, the composite functor i D ◦ L ⊗ represents a symmetric monoidal ∞ -functor: L ⊗ : N ( C c )[ W − C ] ⊗ N ( D c )[ W − D ] ⊗ . Fiberwise over N ( Fin ∗ ), the functor L ⊗ is precisely the product of the derived leftadjoint functor L : N ( C c )[ W − C ] → N ( D c )[ W − D ]. In particular, if L is a Quillenequivalence, then L is an equivalence of ∞ -category, and hence L ⊗ is an equivalenceof symmetric monoidal ∞ -categories. (cid:3) Remark 2.14.
In [SS03, 3.12], Schwede and Shipley show that given a weakmonoidal Quillen pair L : ( C , ⊗ , I ) ( D , ∧ , J ) : R, ⊥ with cofibrant units, thenthe right adjoint R induces Quillen equivalences between the category of monoids Mon ( D ) and Mon ( C ), and also their categories of modules. Our Theorem 2.13strenghten the results when we worked with ∞ -categories. In particular, given any ∞ -operad O ⊗ , we get an equivalence of ∞ -categories: Alg O (cid:0) N ( C c )[ W − C ] (cid:1) ≃ Alg O (cid:0) N ( D c )[ W − D ] (cid:1) , which has been challenging to prove in the case of O = E ∞ in the past, see forinstance [RS17] and [Man03, 1.3, 1.4]. We also obtain an equivalence on the coal-gebras (see [P´er20a, 2.1]): CoAlg O (cid:0) N ( C c )[ W − C ] (cid:1) ≃ CoAlg O (cid:0) N ( D c )[ W − D ] (cid:1) . Such a result on coalgebras has been showed to be untrue in model categories, seefor instance [Sor19] and Section 4.2 below.3.
The Rigidification Problem
In this section, we want to compare homotopy coherent coalgebras and comoduleswith their strict analogue. One one hand, given a nice enough symmetric monoidalmodel category M , we can obtain its Dwyer-Kan localization which is a symmetricmonoidal ∞ -category. We can then apply [P´er20a, 2.1], and define A ∞ or E ∞ -coalgebras and their comodules. Alternatively, we can consider coalgebras in M inthe classical sense (see [P´er20c, 1.1.10] for instance), and then take their Dwyer-Kanlocalization as in Definition 2.1. MAXIMILIEN P´EROUX
There are classical rigidification results that compare A ∞ -algebras with theirstrict associative analogue, see [Lur17, 4.1.8.4]. There is also a comparison betweenthe E ∞ -case with the commutative case in [Lur17, 4.5.4.7]. However, there is noreason to expect that these results dualize in general. In particular, if A ∞ -algebrascorrespond to strict associative algebras in a model category M , there is no reasonto expect that A ∞ -coalgebras correspond to strict coassociative coalgebras in M aswe shall see in Corollary 4.3 and Theorem 5.4 below.3.1. Rigidification Properties.
Let C be a symmetric monoidal category. Let C ⊗ be as in Definition 2.7. Let p : C ⊗ → ∆ op be its associated Grothendieck opfibration(see [Gro15, 4.5]) that determines the monoidal structure of C , and induces thecoCartesian fibration N ( C ⊗ ) → N (∆ op ). There is a correspondance betweenmonoids in C and sections of p that sends convex morphisms to p -coCartesianarrows (see [Gro15, 4.21]). In particular, we obtain the following identification in ∞ -categories: N ( Mon ( C )) Alg A ∞ ( N ( C )) . By using opposite categories, we obtain therefore an identification: N ( CoAlg ( C )) CoAlg A ∞ ( N ( C )) . Let M be a symmetric monoidal model category with cofibrant unit. Consider M c ⊆ M the full subcategory of cofibrant objects. Apply the above identificationto C = M c to obtain the following functor in ∞ -categories: N (cid:16) CoAlg ( M c ) (cid:17) CoAlg A ∞ ( N ( M c ))Let W be the class of weak equivalences in M . By Proposition 2.10, there is asymmetric monoidal functor N ( M ⊗ c ) → N ( M c ) (cid:2) W − (cid:3) ⊗ , which thus provides amap of ∞ -categories: CoAlg A ∞ ( N ( M c )) CoAlg A ∞ (cid:0) N ( M c ) (cid:2) W − (cid:3)(cid:1) , and therefore we obtain a functor of ∞ -categories: α : N (cid:16) CoAlg ( M c ) (cid:17) CoAlg A ∞ (cid:0) N ( M c ) (cid:2) W − (cid:3)(cid:1) . Denote W CoAlg the class of morphisms in
CoAlg ( M c ) that are weak equivalencesas underlying morphisms in M . Notice that the above functor α sends W CoAlg toequivalences. By the universal property of Dwyer-Kan localizations as in Definition2.1, we obtain the following natural functor of ∞ -categories: α : N (cid:16) CoAlg ( M c ) (cid:17) h W − CoAlg i CoAlg A ∞ (cid:0) N ( M c ) (cid:2) W − (cid:3)(cid:1) . Similarly, for the cocommutative case we obtain the natural functor of ∞ -categories: β : N (cid:16) CoCAlg ( M c ) (cid:17) h W − CoCAlg i CoAlg E ∞ (cid:0) N ( M c ) (cid:2) W − (cid:3)(cid:1) . Definition 3.1.
Let M be a symmetric monoidal model category with cofibrantunit. Let α and β be the functors described above. If α is an equivalence of ∞ -categories, we say that the model category M (or its Dwyer-Kan localization) OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 9 satisfies coassociative rigidification . If β is an equivalence of ∞ -categories, we saythat M (or its Dwyer-Kan localization) satisfies cocommutative rigidification .In general there is no reason to expect that if a model category M respects theassociative rigidification then it also respects the coassociative rigidification, see forinstance Corollary 4.3 below. Remark 3.2.
If we inspect the dual case of algebras [Lur17, 4.1.8.4, 4.5.4.7], wesee that we should have considered the ∞ -category N ( CoAlg ( M )) h W − CoAlg i and notthe ∞ -category N ( CoAlg ( M c )) h W − CoAlg i . There are several issues with that. • In general, these ∞ -categories are not equivalent unless for instance M admits a functorial lax comonoidal cofibrant replacement. This means thereis a functor Q : M → M c such that there is a natural map: Q ( X ⊗ Y ) → Q ( X ) ⊗ Q ( Y ) , for any X and Y in M . The main issue is that in general the functor N ( M ⊗ c ) → N ( M ⊗ ) is only lax symmetric monoidal, see Remark 2.11. Ofcourse, if all objects in M are cofibrant, no such issues appear. • There is no good guarantee to have a model structure on
CoAlg ( M ) whoseweak equivalences are W CoAlg , even when using the left-induced methodsfrom [HKRS17]. Even though we do not need a model category to de-fine N ( CoAlg ( M )) h W − CoAlg i , this would help relate if there was some kindof compatibility with M . For instance, if we suppose M is combinatorialmonoidal model category and there exists a model category on coalgebraso that the forgetful-cofree adjunction: U : CoAlg ( M ) M : T ∨ , ⊥ is a Quillen adjunction, then there exists a functorial cofibrant replacement CoAlg ( M ) → CoAlg ( M c ) that induces an equivalence of ∞ -categories: N ( CoAlg ( M c )) h W − CoAlg i ≃ N ( CoAlg ( M )) h W − CoAlg i . • In the cases where
CoAlg ( M ) does admit a model structure it is in generalleft-induced by a model category that is not a monoidal model category,see [HKRS17].All the above also applies to the cocommutative case.3.2. The Cartesian Case.
We provide here a simple case of model categoriessatisfying the coassociative and cocommutative rigidification in the sense of Defi-nition 3.1. Let ( M , × , ∗ ) be a symmetric monoidal model category with respect toits Cartesian monoidal structure. Let W be the class of weak equivalences in M .Suppose the terminal object ∗ is cofibrant. Suppose also that M admits a functorialcofibrant replacement. Proposition 3.3.
Let ( M , × , ∗ ) be as above. Then, M satisfies the coassociativeand cocommutative rigidification, i.e. the following natural maps are equivalencesof ∞ -categories: N ( CoAlg ( M )) h W − CoAlg i CoAlg A ∞ (cid:0) N ( M c ) (cid:2) W − (cid:3)(cid:1) , ≃ N ( CoCAlg ( M )) h W − CoCAlg i CoAlg E ∞ (cid:0) N ( M c ) (cid:2) W − (cid:3)(cid:1) , ≃ and all four of the ∞ -categories above are equivalent to the Dwyer-Kan localization N ( M ) (cid:2) W − (cid:3) .Proof. For any Cartesian monoidal ∞ -category C , we have the equivalence: CoAlg A ∞ ( C ) ≃ CoAlg E ∞ ( C ) ≃ C , see [Lur17, 2.4.3.10]. For any Cartesian monoidal (ordinary) category C , we havethe isomorphism of categories: CoAlg ( C ) ∼ = CoCAlg ( C ) ∼ = C . Apply Remark 2.4 to conclude. (cid:3) Dold-Kan Correspondance For Coalgebras
We now apply our Theorem 2.13 to the weak monoidal Quillen equivalence ap-pearing in [SS03], all missing details can be found there. Let R be a commutativediscrete ring subsequently. Let sMod R denote the category of simplicial R -modules,and let Ch ≥ R denote the category of non-negative chain complexes. The Dold-Kancorrespondance says that the normalization functor : N : sMod R Ch ≥ R , ∼ = (4.1)is an equivalence of categories. Its inverse functor is denoted Γ : Ch ≥ R → sMod R .We show in Corollary 4.2 that if we derive the Dold-Kan correspondance, thenwe obtain a correspondance between the coalgebraic objects. Moreover, Theorem2.13 clarifies the equivalence of the stable Dold-Kan correspondance, see Theorem4.4. Comparing a result of [Sor19], we obtain that A ∞ -coalgebras of connectivemodules over a discrete commutative ring cannot be rigidified in the Dold-Kancontext, see 4.3.4.1. The Derived Dold-Kan Equivalence.
We can endow each category with amodel structure. For sMod R , the weak equivalences and fibrations are the underly-ing weak equivalences and fibrations in simplicial sets, i.e., they are weak homotopyequivalences and Kan fibrations. In other words, the model structure of sMod R isright-induced from sSet via the forgetful functor, in the sense of [HKRS17]. For Ch ≥ R , we use the usual projective model structure. The weak equivalences arethe quasi-isomorphisms, and the fibrations are the positive levelwise epimorphisms.Both categories can be endowed with their usual symmetric monoidal structureinduced by the tensor product of R -modules. However, the Dold-Kan equivalence(4.1) does not preserve the monoidal structure. Nonetheless, with respect to theabove choice of model structures, the categories sMod R and Ch ≥ R are both symmet-ric monoidal model categories with cofibrant units. The isomorphism of categoriesfrom (4.1) can be regarded now as two Quillen equivalences, depending on thechoice of left and right adjoints: Ch ≥ R sMod R , Γ N ⊥ (4.2) OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 11 and: sMod R Ch ≥ R . N Γ ⊥ (4.3)If we choose the normalization functor N to be the right adjoint as in (4.2), then itcan be considered as lax symmetric monoidal via the shuffle map. If we choose N to be the left adjoint as in (4.3), then the Alexander-Whitney formula gives a laxcomonoidal structure which is not symmetric.Nevertheless, this shows that both Quillen equivalences form a weak monoidalQuillen equivalence with cofibrant units, which is symmetric in the case where N is a right adjoint (4.2). We can therefore apply our Theorem 2.13 to obtain thefollowing. Corollary 4.1 (The Derived Dold-Kan Correspondance) . Let R be a commutativediscrete ring. Then the Dwyer-Kan localizations of sMod R and Ch ≥ R are equivalentas symmetric monoidal ∞ -categories: N ( sMod R ) (cid:2) W − (cid:3) ≃ N (cid:16) Ch ≥ R (cid:17) h W − dg i , via the right Quillen derived functor of N : sMod R → Ch ≥ R from the Quillen equiv-alence of (4.2) , where W ∆ is the class of weak homotopy equivalences between sim-plicial R -modules, and W dg is the class of quasi-isomorphisms between non-negativechain complexes over R . In particular, applying our Remark 2.14, we get the following result.
Corollary 4.2 (Dold-Kan Correspondance For Coalgebras) . For any ∞ -operad O ,the normalization functor induces an equivalence of ∞ -categories: CoAlg O (cid:0) N ( sMod R ) (cid:2) W − (cid:3)(cid:1) CoAlg O (cid:16) N (cid:16) Ch ≥ R (cid:17) h W − dg i(cid:17) . ≃ Rigidification Failure of Connective Coalgebras.
The above result by-passes a difficulty on the level of model categories and strict coalgebras. If we choosethe second adjunction (4.3) as a weak Quillen monoidal pair, then the normalizationfunctor, being lax comonoidal, lifts to coalgebras: N : CoAlg R ( sMod R ) → CoAlg R ( Ch ≥ R ) , but its inverse Γ, being only lax monoidal, does not lift to coalgebras. Nevertheless,a right adjoint exists on the level of R -coalgebras, either by presentability, or usingdual methods as in section 3.3 of [SS03]. We shall denote it by Γ CoAlg . Then, usingleft-induced methods of [HKRS17], we can endow model structures such that weget a Quillen adjunction:
CoAlg R ( sMod R ) CoAlg R (cid:16) Ch ≥ R (cid:17) . N ⊥ Γ CoAlg
The weak equivalences are the underlying weak equivalences and every object iscofibrant, in both model categories. However, it was shown in [Sor19, 4.16] thatthe above Quillen pair is not a Quillen equivalence, at least when R is a field. It wasshown that for a particular choice of fibrant object C in CoAlg R ( Ch ≥ R ), the counit N (Γ CoAlg ( C )) −→ C is not a weak equivalence (i.e. not a quasi-isomorphism). Therefore, on the Dwyer-Kan localizations, the normalization functor does notinduce an equivalence of ∞ -categories: N : N (cid:16) CoAlg ( sMod R ) (cid:17) h W − , CoAlg i N (cid:16) CoAlg ( Ch ≥ R ) (cid:17) h W − dg , CoAlg i . Here W ∆ , CoAlg ⊆ W ∆ and W dg , CoAlg ⊆ W dg denote the subclasses of their respectiveweak equivalences between coalgebra objects. However, Corollary 4.2 shows thatthe normalization functor induces an equivalence between the A ∞ -coalgebras: N : CoAlg A ∞ (cid:0) N ( sMod R ) (cid:2) W − (cid:3)(cid:1) CoAlg A ∞ (cid:16) N (cid:16) Ch ≥ R (cid:17) h W − dg i(cid:17) . ≃ By [Lur17, 7.1.3.10], the ∞ -categories N ( sMod R )[ W − ] and N ( Ch ≥ R )[ W − dg ] rep-resent the symmetric monoidal ∞ -category Mod ≥ HR of HR -modules in connectivespectra. From our above discussion, we obtain the following result which says thatwe cannot rigidify coassociative coalgebras in Mod ≥ HR in Ch ≥ R or sMod R . Corollary 4.3.
Let R be a commutative ring. Consider sMod R and Ch ≥ R withtheir standard monoidal model structures. Then there exists M ∈ { sMod R , Ch ≥ R } that does not satisfy the coassociative rigidification, i.e.: N ( CoAlg ( M ) c ) (cid:2) W − (cid:3) CoAlg A ∞ (cid:16) Mod ≥ HR (cid:17) , where W is the class of weak equivalences between coalgebra cofibrant objects in M . The Stable Dold-Kan Correspondance.
Our approach also gives a newproof of the stable Dold-Kan correspondance . This well-known result was formal-ized with ∞ -categories in [Lur17, 7.1.2.13] as follows. Let R be a commutativediscrete ring. Then the ∞ -category of HR -modules in spectra Mod HR is equivalentto ∞ -category of derived R -modules D ( R ) as symmetric monoidal ∞ -categories: Mod HR ≃ D ( R ). However, the equivalence was not described explicitly in Lurie. In[Shi07, 2.10], Shipley provided an explicit zig-zag of (weak monoidal) Quillen equiv-alences between the standard model category Mod HR of HR -modules in symmetricspectra and the projective model category of chain complexes over R : Mod HR Sp Σ ( sMod R ) Sp Σ (cid:16) Ch ≥ R (cid:17) Ch R . ⊥ ⊣ ⊥ Notice that the Dwyer-Kan localizations of
Mod HR and Ch R are precisely the ∞ -categories Mod HR and D ( R ) respectively. If we derive and combine the Quillenfunctors above, we obtain an explicit functor of ∞ -categories Θ : Mod HR → D ( R ).Recall that both sMod R and Ch ≥ R are left proper cellular symmetric monoidalmodel categories. Recall from [ABG18, B.3] that there is a compatibility withstabilization and the Dwyer-Kan localization of a left proper cellular simplicialsymmetric monoidal model category. Combining the above with Corollary 4.1 andTheorem 2.13 yields the following. Corollary 4.4 (The Stable Dold-Kan Correspondance) . Let R be a commutativediscrete ring. Then the ∞ -category of HR -modules is equivalent to ∞ -category of OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 13 derived R -modules as symmetric monoidal ∞ -categories via the functor: Θ :
Mod HR ≃ −→ D ( R ) . Moreoever, the functor induces an equivalence for any ∞ -operad O : CoAlg O ( Mod HR ) ≃ −→ CoAlg O ( D ( R )) . Coalgebras in Spectra
Based on the main result of [PS19], we prove here, in Corollary 5.6, that themonoidal model categories of symmetric spectra (see [HSS00]), orthogonal spec-tra (see [MMSS01] [MM02]), Γ-spaces (see [Seg74] [BF78]), and W -spaces (see[And74]), do not respect the coassociative rigidification, in the sense of Definition3.1. In other words, the strictly coassociative counital coalgebras in these monoidalcategories of spectra do not have the correct homotopy type.We work with the symmetric monoidal model category of symmetric spectra,denoted Sp Σ (see [HSS00]), and claim that similar results can be obtained withthe other categories mentioned above, following [PS19]. Notice that we have theequivalence of ∞ -categories: N ( Sp Σ )[ W − ] ≃ Sp , where W is the class of stable equivalences of symmetric spectra, and Sp is the ∞ -category of spectra as in [Lur17, 1.4.3.1].5.1. Model Structures for Coalgebras.
Although not necessary to show thenon-rigidification, as seen in Remark 3.2, we provide here a model category forcoalgebras and cocommutative coalgebra in symmetric spectra. We shall use theleft-induced methods from [HKRS17]. In [HSS00, Section 5] there is a simplicial,combinatorial model structure on Sp Σ with all objects cofibrant called the (absolute)injective stable model stucture , see also [Sch, Remark III.4.13]. Proposition 5.1 ([HKRS17, 5.0.1, 5.0.2]) . For any S -algebra A in Sp Σ , thereexists an injective model structure on Mod A ( Sp Σ ) left-induced from the injectivestable model structure on Sp Σ : Mod A ( Sp Σ ) Sp Σ , U Hom Sp Σ ( A, − ) ⊥ with cofibrations the monomorphisms and weak equivalences the stable equivalences.This model structure on Mod A ( Sp Σ ) is simplicial and combinatorial. Let A be a commutative ring spectrum (i.e. a commutative S -algebra). Thesymmetric monoidal category ( Mod A ( Sp Σ ) , ∧ A , A ) is presentable and the smashproduct ∧ A preserves colimits in both variables. Thus we can apply [Por08, 2.7] andwe obtain the (forgetful-cofree)-adjunction between A -coalgebras and A -modules in Sp Σ : CoAlg A ( Sp Σ ) Mod A ( Sp Σ ) . U T ∨ ⊥ Proposition 5.2.
Let A be any commutative S -algebra in symmetric spectra Sp Σ .There exists a combinatorial model structure on A -coalgebras CoAlg A ( Sp Σ ) left-induced by the (forgetful-cofree) adjunction from the injective stable model structure on Mod A ( Sp Σ ) . In particular, the weak equivalences in CoAlg A ( Sp Σ ) are the under-lying stable equivalences, and the cofibrations are the underlying monomorphisms.Proof. We mimic the proof of [HKRS17, Theorem 5.0.3]. We apply [HKRS17,2.2.1]. Tensoring with a simplicial set lifts to A -coalgebras. Indeed, let K be asimplicial set and ( C, ∆ C , ε C ) be an A -coalgebra. Then the free S -module Σ ∞ + K is endowed with a unique (cocommutative) S -coalgebra structure (Σ ∞ + K, ∆ K , ε K ),see [PS19, Lemma 2.4], where the comultiplication ∆ K is induced by the diagonal K + → K + ∧ K + and the counit ε K is induced by the non-trivial map K + → S .Then the tensor K ⊗ C := Σ ∞ + K ∧ S C is an A -coalgebra with comultiplication:Σ ∞ + K ∧ S C ∆ K ∧ ∆ C −→ (Σ ∞ + K ∧ S Σ ∞ + K ) ∧ S ( C ∧ A C ) ∼ = (Σ ∞ + K ∧ S C ) ∧ A (Σ ∞ + K ∧ S C ) , and counit: Σ ∞ + K ∧ S C S ∧ S A ∼ = A. ε K ∧ ε C There is a good cylinder object in sSet given by the factorization: S a S ∆[1] + = I S . ∼ Since
Mod A ( Sp Σ ) is simplicial, all objects are cofibrant, and that the smash productof an A -coalgebra with this factorization in sSet lifts to CoAlg A ( Sp Σ ), this definesa good cylinder object in CoAlg A ( Sp Σ ) for any A -coalgebra C : C a C C ⊗ I C, ∼ as C ⊗ S ∼ = C , and colimits in CoAlg A ( Sp Σ ) are computed in Mod A ( Sp Σ ). Thecombinatorial statement follows from [BHK +
15, 2.23] combined with [HKRS17,3.3.4]. (cid:3)
We can also easily extends the results to cocommutative A -coalgebras. Proposition 5.3.
Let A be any commutative S -algebra in symmetric spectra Sp Σ .There exists a combinatorial model structure on the category of cocommutative A -coalgebras CoCAlg A ( Sp Σ ) left-induced by the (forgetful-cofree) adjunction from theinjective stable model structure on Mod A ( Sp Σ ) . In particular, the weak equivalencesin CoCAlg A ( Sp Σ ) are the underlying stable equivalences, and the cofibrations are theunderlying monomorphisms. The Failure of Rigidification.
We show here the failure of rigidification.Let A and B be commutative S -coalgebras. A map A → B is defined to be a positive flat cofibration of commutative S -algebras if it is a cofibration in the modelcategory of commutative S -algebras defined in [Shi04, 3.2] (or the positive flat stablemodel structure defined in [Sch, III.6.1]). As noted in [PS19, 2.4], every comonoidin ( sSet ∗ , ∧ , S ) is of the form Y + and the comultiplication is given by the diagonal Y + → ( Y × Y ) + ∼ = Y + ∧ Y + . Theorem 5.4 ([PS19, 3.4, 3.6]) . Let A be a positive flat cofibrant commutative S -algebra in Sp Σ . Then, given any counital coassociative A -coalgebra C in Sp Σ ,the comultiplication is cocommutative and induced by the following epimorphism of A -coalgebras: A ∧ C −→ C, OALGEBRAS IN THE DWYER-KAN LOCALIZATION OF A MODEL CATEGORY 15 where A ∧ C is given an A -coalgebra structure via the diagonal on the pointed space C → C ∧ C . Remark 5.5.
As noted in [PS19, 3.6], any E ∞ -ring spectrum is equivalent (as an E ∞ -ring spectrum) to a positive flat cofibrant commutative S -algebra in Sp Σ .Let A be any commutative S -algebra. Let CoAlg A ( Sp Σ c ) denote the comonoidin the cofibrant objects of A -modules in Sp Σ endowed with the absolute projectivestable model structure (as in [Sch, IV.6.1]). There is a natural map of ∞ -categories: α : N ( CoAlg A ( Sp Σ c )) (cid:2) W − (cid:3) CoAlg A ∞ ( Mod A ( Sp )) , where W is the class of stable equivalences between A -coalgebras. Corollary 5.6.
Let A be a positive flat cofibrant commutative S -algebra in Sp Σ .Then the ∞ -category of A -modules Mod A ( Sp ) does not satisfy the coassociativerigidification. In particular, for A = S we have: CoAlg A ∞ ( Sp ) N ( CoAlg S ( Sp Σ )) (cid:2) W − (cid:3) . Proof.
Let ( C, ∆ , ε ) be an A -coalgebra in Sp Σ that is cofibrant as an A -modules inthe (absolute) projective stable model structure. Suppose the functor: α : N ( CoAlg A ( Sp Σ c )) (cid:2) W − (cid:3) CoAlg A ∞ ( Mod A ( Sp )) , is an equivalence of ∞ -category. By Theorem 5.4, we see that α ( C ) is automaticallyan E ∞ -coalgebra. But there exist A ∞ -coalgebras in Sp that are not E ∞ -coalgebras.Indeed, take any compact topological group that is not Abelian (say O (2)), then A ∧ O (2) + is an A ∞ -algebra in Mod A ( Sp ) that is not commutative and is a compactspectrum. By Spanier-Whitehead duality, we obtain an A ∞ -coalgebra that is not E ∞ in spectra. (cid:3) Remark 5.7.
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