  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.
