Realizable homotopy colimits

TL;DR

This paper demonstrates that the Bousfield-Kan homotopy colimit is the absolute left derived functor of the colimit in any model category, establishing it as a realizable homotopy colimit via a 2-category approach.

Contribution

It proves the Bousfield-Kan homotopy colimit is a realizable homotopy colimit and characterizes such colimits with exact coproducts using a geometric realization pattern.

Findings

Bousfield-Kan homotopy colimit is the absolute left derived functor of colimit.

Realizable homotopy colimits can be characterized via a geometric realization formula.

The approach uses a 2-category of relative categories.

Abstract

In this paper we prove that for any model category, the Bousfield-Kan construction of the homotopy colimit is the absolute left derived functor of the colimit. This is achieved by showing that the Bousfield-Kan homotopy colimit is moreover a realizable homotopy colimit, defined by means of a suitable 2-category of relative categories. In addition, in the case of exact coproducts, we characterize the realizable homotopy colimits that satisfy a cofinality property as those given by a formula following the pattern of Bousfield-Kan construction: they are the composition of a "geometric realization" with the simplicial replacement.