Coalgebraic Models for Combinatorial Model Categories

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant. Let A be a cofibrantly generated model category. Garner’s version of the small object argument, described in [7], shows that there is a comonad c on A for which the underlying endofunctor is a cofibrant replacement functor for the model structure on A. If X has a coalgebra structure for the comonad c, then X is a retract of cX, so in particular X is cofibrant. We therefore refer to a coalgebra for the comonad c as an algebraically cofibrant object of A, though this terminology hides the fact that we have chosen a specific comonad c and that there is potentially more than one coalgebra structure on a given cofibrant object. When c is defined via the Garner small object argument, the c-coalgebras include the ‘presented cell complexes’ defined as chosen composites of pushouts of coproducts of generating cofibrations, with morphisms between them given by maps that preserve the cellular structure (see [2]). We fix such a comonad c and denote the category of c-coalgebras by Ac. The forgetful functor u : Ac → A has a right adjoint given by taking the cofree coalgebra on an object. We abuse notation slightly and denote this right adjoint also by c : A→ Ac. Our goal in this paper is a study of the forgetful/cofree adjunction (0.1) u : Ac A : c. We show that when A is a combinatorial and simplicial model category, there is a (combinatorial and simplicial) model structure on Ac that is ‘left-induced’ by that on A. This means that a morphism g in Ac is a weak equivalence or cofibration if and only if u(g) is a weak equivalence, or respectively a cofibration, in A. Given this, it is easily follows that (0.1) is a Quillen equivalence. We conjecture that the result holds without the hypothesis that A be simplicial but we have not been able to prove it in this generality. Dugger shows in [5] that for any presentable model category C (that is, a model category that is Quillen equivalent to a combinatorial model category), there exists a Quillen equivalence A C in which A is combinatorial and simplicial. Composing with (0.1) we obtain a single Quillen equivalence Ac C such that every object in Ac is cofibrant. This improves on Dugger’s [5, Corollary 1.2] in that we have a single Quillen equivalence rather than a zigzag. 1 2 MICHAEL CHING AND EMILY RIEHL Our result is dual to a corresponding theorem of Nikolaus [12] which shows that the category of algebraically fibrant objects has a right-induced model structure, provided that every trivial cofibration is a monomorphism. Our approach is rather different and requires instead the conditions that the underlying model structure be combinatorial (not just cofibrantly generated) and simplicially-enriched. Our result is an application of work of Bayeh et al. [3] in which general conditions are established for the existence of left-induced model structures. There are two main parts: (1) we show that when A is locally presentable, the category Ac is also locally presentable; (2) we verify the conditions of [3, Theorem 2.21] by showing that any morphism in Ac with the right lifting property with respect to all cofibrations is a weak equivalence. The proof of part (1) depends crucially on the fact that accessible categories and accessible functors are closed under a certain class of 2-categorical limit constructions. This is closely related to the essential ingredient in Theorem 2.21 of [3], which is a result of Makkai and Rosicky [10] that states that locally presentable categories with a cofibrantly generated weak factorization system, and appropriate functors between such, are closed under the same class of 2-categorical limits. We defer this part of our proof to the appendix, allowing for a more leisurely treatment. Acknowledgments. This paper was written while both authors were visitors at the Mathematical Sciences Research Institute. The first author was supported by National Science Foundation grant DMS-1144149, and the second author by National Science Foundation Postdoctoral Research Fellowship DMS-1103790. The impetus for this paper was its application to the homotopic descent theory developed by the first author and Greg Arone, whom we thank for suggesting we look at it further. We also owe gratitude to Kathryn Hess and Brooke Shipley for their work on model categories of coalgebras over a comonad. 1. Combinatorial simplicial model categories In this section, we explore the interaction between the locally presentable and simplicially enriched structures on a model category that is both simplicial and combinatorial. We also prove that there exists a cofibrant replacement comonad satisfying the conditions needed for our main theorem. We adopt the convention that all categories appearing in this paper are locally small. First, recall the following definitions. Definition 1.1. Let A be a cocomplete category. For a regular cardinal λ, an object X in A is λpresentable if the functor A(X,−) : A→ Set preserves colimits of λ-filtered diagrams. A category A is locally λ-presentable if it is cocomplete and has a (small) set P of λ-presentable objects such that every object in A is isomorphic to a λ-filtered colimit of objects in P. A category is locally presentable if it is locally λ-presentable for some regular cardinal λ. Locally presentable categories are abundant, particularly in ‘algebraic’ or ‘simplicial’ contexts. We refer the reader to [1] for more details. Jeff Smith introduced the notion of a combinatorial model category: one that is cofibrantly generated and for which the underlying category is locally presentable. In this paper we are interested in model categories A that are both combinatorial and simplicial. The following result gives us control over how these two structures interact. 1A category C is λ-filtered if every subcategory of C with fewer than λ morphisms has a cocone in C. A diagram is λ-filtered when its indexing category is λ-filtered. COALGEBRAIC MODELS FOR COMBINATORIAL MODEL CATEGORIES 3 Lemma 1.2. Let A be a combinatorial and simplicial model category. Then there are arbitrarily large cardinals λ such that: (1) A is locally λ-presentable; (2) A is cofibrantly generated with a set of generating cofibrations for which the domains and codomains are λ-presentable objects; (3) an object X ∈ A is λ-presentable if and only if the functor Hom(X,−) : A→ sSet, given by the simplicial enrichment of A, preserves λ-filtered colimits. Proof. Since A is combinatorial, it is locally μ-presentable for some cardinal μ. There is also then some cardinal μ′ > μ such that the domains and codomains of the generating cofibrations are μ′-presentable. Now consider the functors − ⊗ ∆n : A → A, for n ≥ 0, given by the simplicial tensoring on A. These functors preserve all colimits and so by [1, 2.19] there are arbitrarily large cardinals λ such that all of these functors take λ-presentable objects to λ-presentable objects. Note that for any such λ > μ′, conditions (1) and (2) of the lemma are satisfied. Suppose that X ∈ A is λ-presentable and (Y )j∈J is a λ-filtered diagram in A. By construction of λ, X ⊗∆n is also λ-presentable and so Hom(X, colim J Y )n ∼= sSet(∆,Hom(X, colim J Y )) ∼= A(X ⊗∆, colim J Y ) ∼= colim J A(X ⊗∆, Y ) ∼= colim J sSet(∆,Hom(X,Y )) ∼= colim J Hom(X,Y )n which implies that Hom(X,−) preserves λ-filtered colimits, because colimits in sSet are defined pointwise. Conversely, if Hom(X,−) preserves λ-filtered colimits, then by a similar sequence of bijections, so does A(X ⊗∆n,−) for all n ≥ 0. Hence, taking n = 0, X is λ-presentable. Thus λ satisfies condition (3) also. As described in the introduction, any cofibrantly generated model category A has a cofibrant replacement comonad c : A→ A constructed using Garner’s small object argument. Blumberg and Riehl [4] have noted that when A is, in addition, a simplicial model category, the comonad c can be taken to be simplicially enriched. In this case algebraically cofibrant objects also include ‘enriched cell complexes’ built from tensors of generating cofibrations with arbitrary simplicial sets. For our main result we require a cofibrant replacement comonad that is both simplicial and preserves λ-filtered colimits. Lemma 1.3. Let A be a combinatorial and simplicial model category with λ as in Lemma 1.2. Then there is a simplicially-enriched cofibrant replacement comonad c : A → A that preserves λ-filtered colimits. Proof. The functorial factorization produced by the simplicially enriched version of the Garner small object argument is defined by an iterated colimit process described in Theorem 13.2.1 of [15]. We apply this construction to the usual set of generating cofibrations to obtain a simplicially enriched functorial factorization of any map as a cofibration followed by a trivial fibration in the simplicial model structure on A. The simplicially-enriched cofibrant replacement comonad c is extracted by restricting the factorization to maps whose domain is initial; see [15, Corollary 13.2.4]. It remains to argue that c preserves λ-filtered colimits. The functor c is built from various colimit constructions, which commute with all colimits, and from functors Sq(i,−) : A2 → sSet defined for 4 MICHAEL CHING AND EMILY RIEHL each generating cofibration i. For a morphism f in A, the simplicial set Sq(i, f) is the ‘space of commutative squares from i to f ’ defined via the pullback Sq(i, f) // y Hom(dom i,domf) Hom(cod i, codf) // Hom(dom i, codf) The domain and codomain functors preserve all colimits, because they are defined pointwise. The enriched representables Hom(dom i,−) and Hom(cod i,−) preserve λ-filtered colimits by 1.2.(2). Finally, the pullback, a finite limit, commutes with all filtered colimits. The main result we need from this section is the following. Theorem 1.4. Let A be a combinatorial and simplicial model category and let c be as in Lemma 1.3. Then the category Ac of c-coalgebras is locally presentable, simplicially enriched, and the forgetful/cofree adjunction u : Ac A : c is simplicial. Moreover, Ac is tensored and cotensored over simplicial sets. Proof. We defer the proof that Ac is locally presentable to Proposition A.1 in the appendix. It is well-known that the comonadic adjunction for a simplicially enriched comonad on a simplicial category is simplicially enriched; see, e.g., [8, §3.2]. The tensor X ⊗K of a c-coalgebra X with a simplicial set K is defined as in [8, Lemma 3.11]. Observe that u(X ⊗K) ∼= uX ⊗K. In this way, each simplicial set defines a functor − ⊗K : Ac → Ac, which preserves all colimits because these colimits are created by the forgetful functor u : Ac → A, and tensoring with K preserves colimits in A. By the special adjoint functor theorem, any cocontinuous functor between locally presentable categories has a right adjoint, so Ac is also cotensored over simplicial sets. It turns out that the λ-presentable objects in Ac can be easily described. Lemma 1.5. Let λ and c be as in Lemma 1.3. Then a c-coalgebra X ∈ Ac is λ-presentable if and only if the underlying object uX is λ-presentable in A. Proof. First note that because the comonad c preserves λ-filtered colimits and the left adjoint u : Ac → A creates them, the cofree functor c : A→ Ac also preserves λ-filtered colimits. Now suppose that uX is λ-presentable and consider a λ-filtered diagram (Y )j∈J in Ac. Then we have Ac(X, colimY ) = lim(A(uX, u colimY )⇒ A(uX, ucu colimY )). Since c, u, A(uX,−), and finite limits of sets commute with λ-filtered colimits, so too does Ac(X,−) and thus X is λ-presentable. Conversely, suppose that X is λ-presentable and consider a λ-filtered diagram (A)j∈J in A. Then, by the u a c adjunction, we have A(uX, colimA) ∼= Ac(X, c colimA). Since c and Ac(X,−) commute with λ-filtered colimits, so too does A(uX,−) and so uX is λpresentable. 2See [1, 1.58] for a proof that locally presentable categories are co-wellpowered. COALGEBRAIC MODELS FOR COMBINATORIAL MODEL CATEGORIES 5 We also need the following result from [13, 3.6] which slightly strengthens that of Dugger [5, 7.3]. Lemma 1.6 (Raptis-Rosický). Let A be a cofibrantly generated model category with λ as in Lemma 1.2.(2). Then taking λ-filtered colimits preserves weak equivalences. Proof. We wish to show that the colimit functor colim: A → A preserves weak equivalences, when J is λ-filtered. Endowing the domain with the projective model structure, colim is left Quillen. A given pointwise weak equivalence between J-indexed diagrams may be factored as a projective trivial cofibration followed by a pointwise trivial fibration. As colim sends projective trivial cofibrations to trivial cofibrations, it suffices to show that colim carries pointwise trivial fibrations to trivial fibrations. This follows because the domains and codomains of the generating cofibrations are λ-presentable and J is λ-filtered. 2. The model structure on algebraically cofibrant objects We now turn to the proof of our main theorem, that the adjunction u a c ‘left-induces’ a model structure on Ac. We therefore make the following definitions. Definition 2.1. A morphism f in Ac is a weak equivalence if the underlying morphism u(f) is a weak equivalence in A, and f is a cofibration if u(f) is a cofibration in A. According to [3, Theorem 2.21] it is now sufficient to show that every morphism g in Ac, which has the right lifting property with respect to all cofibrations, is a weak equivalence. Our basic strategy is to show that weak equivalences are detected by the simplicial enrichment. For this we use the following bar resolutions of the objects in Ac. Definition 2.2. By Theorem 1.4, Ac is locally λ-presentable for some regular cardinal λ, and, by increasing λ if necessary, we may assume that λ also satisfies the conditions of Lemma 1.2 (and hence also Lemmas 1.5 and 1.6). Choose a (small) set P of λ-presentable objects in Ac such that every object of Ac is a λ-filtered colimit of objects of P. Let Homc(−,−) denote the simplicial enrichment of Ac. For any X ∈ Ac we then define a simplicial object B•(X) in Ac with Br(X) := ∨ P0,...,Pr∈P X ⊗ [Homc(P0, P1)× · · · ×Homc(Pr, X)] where the face maps are given by composition in the simplicial category Ac and degeneracy maps are given by the unit maps ∗ → Homc(Pi, Pi). This simplicial object is naturally augmented by a map B0(X)→ X which therefore induces a natural transformation with components ηX : |B•(X)| → X. Lemma 2.3. For each X ∈ Ac, the map ηX is a weak equivalence in Ac. Proof. For P ∈ P, the augmented simplicial object B•(P ) → P has extra degeneracies given by the unit map ∗ → Homc(P, P ) and so the map ηP is a simplicial homotopy equivalence in Ac. Therefore u(ηP ) is a simplicial homotopy equivalence, and hence a weak equivalence, in A. So ηP is a weak equivalence in Ac. 6 MICHAEL CHING AND EMILY RIEHL For any X ∈ Ac, we can write X ∼= colim j∈J P j for some λ-filtered diagram (P )j∈J in Ac where P j ∈ P. We now claim that for any P ∈ P, the natural map of simplicial sets (2.4) colim J Homc(P, P )→ Homc(P, colim J P ) ∼= Homc(P,X) is an isomorphism. Since colimits of simplicial sets are formed levelwise, it is sufficient to show that, for each integer n ≥ 0, the map colim J Ac(P ⊗∆, P )→ Ac(P ⊗∆, colim J P ) is an isomorphism of sets, i.e. that P ⊗∆n is a λ-presentable object of Ac. Now u(P ) is λ-presentable in A by Lemma 1.5 and so u(P ⊗ ∆n) = u(P ) ⊗ ∆n is λ-presentable by condition (3) of Lemma 1.2. So by Lemma 1.5 again, P ⊗ ∆n is λ-presentable in Ac. This establishes the isomorphisms (2.4). Since finite products, tensors, coproducts, and geometric realization all commute with λ-filtered colimits, it follows that the natural map colim J |B•(P )| → |B•(X)| is an isomorphism. Thus the map u(ηX) can be written as the λ-filtered colimit of the maps u(ηP j ), each of which is a weak equivalence in A. By Lemma 1.6, u(ηX) is a weak equivalence in A, and hence ηX is a weak equivalence in Ac. We now use the above bar resolutions to complete the proof of our main theorem. Theorem 2.5. Let A be a combinatorial and simplicial model category and c as in Lemma 1.3. Then the weak equivalences and cofibrations of Definition 2.1 make Ac into a combinatorial and simplicial model category such that the adjunction u a c is a simplicial Quillen equivalence. Proof. To get the model structure on Ac we apply [3, 2.21]. We have already shown in Theorem 1.4 that u a c is an adjunction between locally presentable categories. Now let g : X → Y be a morphism in Ac that has the right lifting property with respect to all cofibrations. We claim first that for any P ∈ Ac, g induces a weak equivalence of simplicial sets g∗ : Homc(P,X)→ Homc(P, Y ). In fact, we show that g∗ is a trivial fibration of simplicial sets. To see this, consider a lifting problem