(Co)Simplicial Descent Categories

In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD –> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of 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 sD. 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.

💡 Research Summary

The paper investigates how to transfer homotopical information from the category of simplicial objects sD, built over a fixed base category D, back to D itself. The central tool is a “simple functor” s : sD → D, which plays the role of a homotopy colimit but is defined in a way that works beyond the usual Quillen model setting. Two guiding examples are presented: the classical Bousfield‑Kan homotopy colimit in a simplicial model category, and Deligne’s simple functor for mixed Hodge complexes, which lives in a non‑model context.

The first major theorem establishes that s induces an equivalence after localizing both sides with respect to their respective weak equivalences. In other words, the derived functor of s yields an equivalence
 s : sD