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.
Cite
@article{arxiv.1307.4488,
title = {Enriched model categories in equivariant contexts},
author = {Bertrand Guillou and J. P. May and Jonathan Rubin},
journal= {arXiv preprint arXiv:1307.4488},
year = {2017}
}
Comments
32 pages. v2. A third author, Jonathan Rubin, has been added. This is a significant departure from the previous version. Section 3 has moved to arXiv:1110.3571. The primary emphasis is now on equivariant homotopy theory with respect to a Hopf group object. A new section 4 applies this point of view to the study of modules over a cocommutative Hopf algebra