nLab
homotopy coinvariants functor

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 $\mathrm{Triv}⊣\mathrm{Coinv}$ of adjoint functors:

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

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

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

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

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

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