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 Vec. Here we define VotimesN, for any finite dimensional vector space V and object N to be the unique object representing the functor NmapstoHom(V,Hom(N,N)) (really internal hom).



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

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