nLab homotopy coinvariants functor

Idea

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

$$Triv: \mathbf{M}\to \mathbf{Comod}_C, \,\,\,\,X\mapsto X\otimes\eta$$

$$Coinv:\mathbf{Comod}_C\to \mathbf{M},\,\,\,\,(M,\rho)\mapsto M^{co C}:=M\Box_C I$$

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 $(\mathbf{M},\otimes,I)$ 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

$$\mathbb{R}Coinv: Ho(\mathbf{Comod}_C)\to Ho(\mathbf{M}).$$

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

Related concepts

• coinvariant

References

• K. Hess, Homotopic Hopf-Galois extensions: foundations and examples, arxiv/0902.3393