# Contents

## Idea

Given a monoidal category $(M,\otimes,I)$ and a comonoid $C$ in $M$ with coaugmentation $\eta:I\to C$, one can define the following pair $Triv \dashv Coinv$ of adjoint functors:

$Triv: M\to Comod_C, \,\,\,\,X\mapsto X\otimes\eta$
$Coinv:Comod_C\to M,\,\,\,\,(M,\rho)\mapsto M^{co C}:=M\Box_C I$

where $\Box_C$ denotes the cotensor product bifunctor and $\rho:M\to M\otimes C$ is a right $C$-coaction. $Triv$ is called the (co)free or trivial comodule functor and $Coinv$ the functor of coinvariants.

If $(M,\otimes,I)$ is in fact a monoidal model category, then we can ask whether this pair of functors is a Quillen pair. If so then the the homotopy coinvariants functor is the total right derived functor

$\mathbb{R}Coinv: Ho Comod_C\to Ho M.$

Given a $C$-comodule $(M,\rho)$, any representative of $\mathbb{R}Coinv(M,\rho)$ is called a model of the homotopy coinvariants of $M$.

## References

• K. Hess, Homotopic Hopf-Galois extensions: foundations and examples, arxiv/0902.3393
Revised on November 14, 2013 23:07:45 by Urs Schreiber (82.169.114.243)