COALGEBRAIC MODELS FOR COMBINATORIAL MODEL CATEGORIES

Abstract

Abstract. 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

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