Saturday, 28 July 2007

reference request - Tensor product of abelian categories

A generalization of Deligne's construction may be found here: http://arxiv.org/abs/0911.4979. To recoved Deligne's construction one simply takes the perspective that any abelian category is a module category over VecVec. Here we define VotimesNVotimesN, for any finite dimensional vector space VV and object NN to be the unique object representing the functor NmapstoHom(V,Hom(N,N))NmapstoHom(V,Hom(N,N)) (really internal hom).



In general let mathcalM,mathcalNmathcalM,mathcalN be right, left mathcalCmathcalC-module categories for mathcalCmathcalC any tensor category. Then the relativerelative tensortensor productproduct mathcalMboxtimesmathcalCmathcalNmathcalMboxtimesmathcalCmathcalN is defined as the unique (up to a unique equivalence) universal object for right exact in each variable mathcalCmathcalC-balanced bifunctors from the cartesian product mathcalMtimesmathcalNmathcalMtimesmathcalN. As such it follows that mathcalMboxtimesVecmathcalN=mathcalMboxtimesmathcalNmathcalMboxtimesVecmathcalN=mathcalMboxtimesmathcalN where mathcalMboxtimesmathcalNmathcalMboxtimesmathcalN denotes the product of abelian categories defined in ``Categories Tannakiennes".

No comments:

Post a Comment