Crossed

The term crossed is used as a modifier that describes how a situation gets complicated in the presence of groups acting on themselves and on each other. The action of a group on itself, if involved in the definition, is taken to be the action by conjugation. Note that the action on itself is trivial if and only if the group is a non-abelian group.

Crossed stuff is useful in understanding low-dimensional homology and cohomology and related ideas.