nLab
homotopy coinvariants functor

Contents

Idea

Given a monoidal category (M,,I)(M,\otimes,I) and a comonoid CC in MM 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: M\to Comod_C, \,\,\,\,X\mapsto X\otimes\eta
Coinv:Comod CM,(M,ρ)M coC:=M CICoinv:Comod_C\to 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)(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:HoComod CHoM. \mathbb{R}Coinv: Ho Comod_C\to Ho 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
Revised on November 14, 2013 23:07:45 by Urs Schreiber (82.169.114.243)