  We introduce the concept of an infinite cochain sequence and initiate a
theory of homological algebra for them. We show how these sequences simplify
and improve the construction of infinite coclass families (as introduced by
Eick and Leedham-Green) and how they apply in proving that almost all groups in
such a family have equivalent Quillen categories. We also include some examples
of infinite families of p-groups from different coclass families that have
equivalent Quillen categories.
