The subobject decomposition in enveloping tensor categories

TL;DR

This paper studies the structure of tensor categories derived from regular categories with degree functions, revealing a canonical decomposition of generating objects and explicitly computing morphisms and tensor products, generalizing Deligne's categories.

Contribution

It introduces a canonical subobject decomposition in enveloping tensor categories and provides explicit calculations of morphisms and tensor products within this framework.

Findings

Objects decompose canonically as a direct sum

Explicit formulas for morphisms and compositions

Generalization of Deligne's category construction

Abstract

To every regular category $\mathcal{A}$ equipped with a degree function $\delta$ one can attach a pseudo-abelian tensor category $\mathcal{T}(\mathcal{A},\delta)$. We show that the generating objects of $\mathcal{T}$ decompose canonically as a direct sum. In this paper we calculate morphisms, compositions of morphisms and tensor products of the summands. As a special case we recover the original construction of Deligne's category $\operatorname{Rep} S_t$.