Monoidal Abelian Envelopes with a quotient property

TL;DR

This paper investigates conditions under which pseudo-tensor categories have abelian envelopes with a quotient property, expanding the understanding of tensor category extensions and their preservation under various constructions.

Contribution

It provides an intrinsic criterion for the existence of such abelian envelopes and interprets key tensor category operations as these envelopes, broadening the class of categories with known extension properties.

Findings

Established a criterion for abelian envelopes with the quotient property.

Interpreted scalar extension and Deligne tensor product as abelian envelopes.

Showed that subcategories of RepG have abelian envelopes with the quotient property.

Abstract

We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the existence of an abelian envelope satisfying this quotient property. This allows us to interpret the extension of scalars and Deligne tensor product of tensor categories as abelian envelopes, and to enlarge the class of tensor categories for which all extensions of scalars and tensor products are known to remain tensor categories. For an affine group scheme G, we show that pseudo-tensor subcategories of RepG have abelian envelopes with the quotient property, and we study many other such examples. This leads us to conjecture that all abelian envelopes satisfy the quotient property.