nLab homotopy coinvariants functor

Contents

Contents

Idea

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

Triv:MComod C,XXη Triv: \mathbf{M}\to \mathbf{Comod}_C, \,\,\,\,X\mapsto X\otimes\eta
Coinv:Comod CM,(M,ρ)M coC:=M CICoinv:\mathbf{Comod}_C\to \mathbf{M},\,\,\,\,(M,\rho)\mapsto M^{co C}:=M\Box_C I

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

If (M,,I)(\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

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

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

References

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

Last revised on January 19, 2023 at 16:30:54. See the history of this page for a list of all contributions to it.