Homotopy colimits in stable representation theory

TL;DR

This paper investigates the existence and uniqueness of homotopy colimits in stable representation theory, providing conditions for their existence and applications to K-theory and deformation theory.

Contribution

It offers new conditions for homotopy colimit existence in settings lacking model structures and generalizes classical theorems in relative homological algebra.

Findings

Homotopy cofibers may fail to exist with objects of positive finite projective dimension.

Established conditions ensure homotopy colimits are well-defined and unique.

Applications to Waldhausen K-theory and deformation-theoretic methods.

Abstract

We study the problem of existence and uniqueness of homotopy colimits in stable representation theory, where one typically does not have model category structures to guarantee that these homotopy colimits exist or have good properties. We get both negative results (homotopy cofibers fail to exist if there exist any objects of positive finite projective dimension!) and positive results (reasonable conditions under which homotopy colimits exist and are unique, even when model category structures fail to exist). Along the way, we obtain relative-homological-algebraic generalizations of classical theorems of Hilton-Rees and Oort. We describe some applications to Waldhausen $K$-theory and to deformation-theoretic methods in stable representation theory.