Enriched model categories in equivariant contexts

Abstract

We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V. For any V, a discrete group G gives a Hopf group, denoted I[G]. When V is cartesian monoidal, the Hopf groups are just the group objects in V. When V is the category of modules over a commutative ring R, I[G] is the group ring R[G] and the general Hopf groups are the cocommutative Hopf algebras over R. We show how all of the usual constructs of equivariant homotopy theory, both categorical and model theoretic, generalize to Hopf groups for any V. This opens up some quite elementary unexplored mathematical territory, while systematizing more familiar terrain.