Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

Enriched model categories in equivariant contexts

Pages: 213 – 246

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10


Bertrand Guillou (Department of Mathematics, University of Kentucky, Lexington, Ky., U.S.A.)

J.P. May (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Jonathan Rubin (Department of Mathematics, University of Chicago, Illinois, U.S.A.)


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 $\mathscr{V}$. For any $\mathscr{V}$, a discrete group $G$ gives a Hopf group, denoted $I[G]$. When $\mathscr{V}$ is cartesian monoidal, the Hopf groups are just the group objects in $\mathscr{V}$. When $\mathscr{V}$ 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 $\mathscr{V}$. This opens up some quite elementary unexplored mathematical territory, while systematizing more familiar terrain.


enriched model category, equivariant model category, Hopf group

2010 Mathematics Subject Classification

55P91, 55U35

This work was partially supported by Simons Collaboration Grant No. 282316 and NSF Grant DMS-171037 held by the first author.

Copyright © 2018, Bertrand Guillou, J.P. May and Jonathan Rubin. Permission to copy for private use granted.

Received 22 August 2017

Received revised 1 July 2018

Published 10 October 2018