Simplicial descent categories

AbstractLet D be a category and E a class of morphisms in D. In this paper we study the question of how to transfer homotopic structure from the category of simplicial objects in D, Δ∘D, to D through a ‘good’ functor s:Δ∘D→D, which we call simple functor. For instance, the Bousfield–Kan homotopy colimit in a Quillen simplicial model category is a good simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on Δ∘D. We use these facts to produce cofiber sequences on the localized category of D by E, which give rise to a natural Verdier triangulated structure in the stable case