Distributors on a tensor category

Abstract

Let 𝒜 be a tensor category and let 𝒱 denote the category of vector spaces. A distributor on 𝒜 is a functor 𝒜op × 𝒜 ⟶ 𝒱. We are concerned with distributors with two-sided 𝒜-action. Those distributors form a tensor category, which we denoted by 𝒜D(𝒜, 𝒜)𝒜. The functor category Hom(𝒜op, 𝒱) is also a tensor category and has the center Z(Hom(𝒜op, 𝒱)). We show that if A is rigid, then 𝒜D(𝒜, 𝒜)𝒜 and Z(Hom(𝒜op, 𝒱)) are equivalent as tensor categories. © 2006 by the University of Notre Dame. All rights reserved.