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 $Votimes N$, for any finite dimensional vector space $V$ and object $N$ to be the unique object representing the functor $Nmapsto Hom(V, Hom(N, N))$ (really internal hom).



In general let $mathcal{M}, mathcal{N}$ be right, left $mathcal{C}$-module categories for $mathcal{C}$ any tensor category. Then the $relative$ $tensor$ $product $ $mathcal{M}boxtimes_{mathcal{C}}mathcal{N}$ is defined as the unique (up to a unique equivalence) universal object for right exact in each variable $mathcal{C}$-balanced bifunctors from the cartesian product $mathcal{M}timesmathcal{N}$. As such it follows that $mathcal{M}boxtimes_{Vec}mathcal{N}=mathcal{M}boxtimesmathcal{N}$ where $mathcal{M}boxtimesmathcal{N}$ denotes the product of abelian categories defined in ``Categories Tannakiennes".

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