Homotopy Limits and Homotopy Colimits of Chain Complexes

TL;DR

This paper presents a formula for computing homotopy limits and colimits of chain complex diagrams using bar and cobar constructions, extending to framed model categories.

Contribution

It introduces a Bousfield--Kan formula for homotopy limits and colimits of chain complexes and demonstrates its applicability in framed model categories.

Findings

The Bousfield--Kan formula computes homotopy colimits in framed model categories.

Provides explicit formulas for homotopy limits and colimits of chain complexes.

Extends known formulas to a broader categorical context.

Abstract

We give a formula for homotopy limits and homotopy colimits of diagrams of chain complexes using the cobar and bar constructions, also known as the Bousfield--Kan formula. Along the way, we show that the Bousfield--Kan formula computes homotopy colimits in any framed model category.