nLab
bicomodule

Let C,DC,D be comonoids in a monoidal category A=(A,,1)A = (A,\otimes,1). A CC-DD bicomodule is an object MM in AA, with left CC-coaction λ C:MCM\lambda_C : M\to C\otimes M and right DD-coaction ρ D:MMD\rho_D: M\to M\otimes D which commute in the sense that

(λ Cid D)ρ D=(id Cρ D)λ C. (\lambda_C\otimes id_D)\circ\rho_D = (id_C\otimes \rho_D)\circ \lambda_C.

Typical cases are when AA is the category of kk-modules where kk is a commutative unital ring (the comonoids are then kk-coalgebras), and the more general case of bicomodules over corings, where AA is the category of kk-bimodules where kk is a possibly noncommutative ring.

There is an operation of cotensor product for bicomodules over coalgebras/corings; however it is not associative in general, unlike the tensor product of bimodules over rings!

Revised on October 11, 2011 01:26:20 by Todd Trimble (69.118.58.208)