Enriched model categories in equivariant contexts

TL;DR

This paper develops a broad framework for equivariant model category theory using Hopf groups in various symmetric monoidal categories, unifying and extending classical concepts in equivariant homotopy theory.

Contribution

It introduces Hopf groups as a generalization of groups in symmetric monoidal categories, enabling a unified approach to equivariant model categories across different contexts.

Findings

Generalizes equivariant homotopy constructs to Hopf groups

Unifies classical and new equivariant frameworks

Explores mathematical territory with broad categorical applicability

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.