Cone complexes and group actions

We introduce the notion of cone complexes to conveniently handle the quotients of simplicial complexes by group actions. Cone complexes are a generalization of simplicial complexes described purely in terms of posets (partially ordered sets). Our main focus of this paper is to study group actions on cone complexes, and compare them with the case of small categories. We clarify how conditions of group actions related to quotients are changed by functors between cone complexes and small categories.