Deligne-Knop tensor categories and functoriality

TL;DR

This paper explores the construction of symmetric monoidal categories from regular categories using Knop's method, focusing on functoriality and adjoint functors, with applications to Deligne categories.

Contribution

It provides criteria for when functors and adjoint pairs between regular categories induce corresponding functors and adjoints between their associated tensor categories.

Findings

Criteria for functor-induced symmetric monoidal functors

Conditions for lifting adjoint functors between tensor categories

Application to Deligne categories and their functorial properties

Abstract

A general construction of Knop creates a symmetric monoidal category $\mathcal{T}(\mathcal{A},\delta)$ from any regular category $\mathcal{A}$ and a fixed degree function $\delta$. A special case of this construction are the Deligne categories $\underline{\operatorname{Rep}}(S_t)$ and $\underline{\operatorname{Rep}}(GL_t(\mathbb{F}_q))$. We discuss when a functor $F:\mathcal{A} \to \mathcal{A}'$ between regular categories induces a symmetric monoidal functor $\mathcal{T}(\mathcal{A},\delta) \to \mathcal{T}(\mathcal{A}',\delta')$. We then give a criterion when a pair of adjoint functors between two regular categories $\mathcal{A}, \ \mathcal{A}'$ lifts to a pair of adjoint functors between $\mathcal{T}(\mathcal{A},\delta)$ and $\mathcal{T}(\mathcal{A}',\delta')$.