Equivariant closure operators and trisp closure maps

A trisp closure map ϕ is a special map on the vertices of a trisp (triangulated space) T with the property that T collapses onto the subtrisp induced by the image of ϕ. We study the interaction between trisp closure maps and group operations on the trisp, and give conditions such that the quotient map is again a trisp closure map. Special attention is on the case that the trisp is the nerve of an acyclic category, and the relationship between trisp closure maps and closure operators on posets is studied.