  For $\Cc$ a $G$-category, we give a condition on a diagram of simplicial sets
indexed on $\Cc$ that allows us to define a natural $G$-action on its homotopy
colimit, and in some other simplicial sets and categories defined in terms of
the diagram. Well-known theorems on homeomorphisms and homotopy equivalences
are generalized to an equivariant version.
