For a group with homotopical finiteness ${\cal F}_n$, there is a $K(\pi,1)$ whose universal cover admits a group action with finite fundamental domain for the action on the $n$-skeleton. Squier's property of finite derivation type corresponds to an action up to dimension $n=3$ in which the fundamental domain consists of ``directed'' cells. This property can then be applied to monoids as well as groups, and for groups it is equivalent to ${\cal F}_3$. In this talk I will discuss a homological analog of the definition of finite derivation type, an extension of this to higher dimensions, and connections to other finiteness conditions.