(Co)Simplicial Descent Categories

TL;DR

This paper explores how to transfer homotopic structures from simplicial objects to their base categories using simple functors, establishing equivalences and triangulated structures in various contexts.

Contribution

It introduces a framework for transferring homotopic structures via simple functors, including examples beyond Quillen models, and establishes equivalences and triangulated structures.

Findings

Simple functor induces equivalence on localized categories.

Natural Brown category structure on sD established.

Produces cofiber sequences leading to Verdier triangulated structures.

Abstract

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.