Distributors on a tensor category

Abstract

Let $\cA$ be a tensor category and let $\cV$ denote the category of vector spaces. A distributor on $\cA$ is a functor $\cA^{\op}\times \cA\to \cV$. We are concerned with distributors with two-sided $\cA$-action. Those distributors form a tensor category, which we denoted by ${}_{\cA}\bD(\cA,\cA)_{\cA}$. The functor category $\Hom(\cA^{\op},\cV)$ is also a tensor category and has the center $\bZ(\Hom(\cA^{\op},\cV))$. We show that if $\cA$ is rigid, then ${}_{\cA}\bD(\cA,\cA)_{\cA}$ and $\bZ(\Hom(\cA^{\op},\cV))$ are equivalent as tensor categories.