Deligne-Knop tensor categories and functoriality

Abstract

A general construction of Knop creates a symmetric monoidal category $\mathcal{T}(\mathcal{A},\hat{I}´)$ from any regular category $\mathcal{A}$ and a fixed degree function $\hat{I}´$. A special case of this construction are the Deligne categories $\underline{\mathrm{Rep}}(S_t)$ and $\underline{\mathrm{Rep}}(G{L}_t({\mathbb{F}}_q))$. We discuss when a functor $F:\mathcal{A}\to {\mathcal{A}}^{\prime }$ between regular categories induces a symmetric monoidal functor $\mathcal{T}(\mathcal{A},\hat{I}´)\to \mathcal{T}({\mathcal{A}}^{\prime },\hat{I}{´}^{\prime })$. We then give a criterion when a pair of adjoint functors between two regular categories $\mathcal{A},{\mathcal{A}}^{\prime }$ lifts to a pair of adjoint functors between $\mathcal{T}(\mathcal{A},\hat{I}´)$ and $\mathcal{T}({\mathcal{A}}^{\prime },\hat{I}{´}^{\prime })$.