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# 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

#### Authors

#### 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 $\mathscr{V}$. 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.

#### Keywords

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