Monoidal adjunctions and abelian envelopes

TL;DR

This paper demonstrates how monoidal adjunctions can establish the existence of monoidal abelian envelopes for pseudo-tensor categories, providing concrete criteria and new examples in combinatorial and algebraic settings.

Contribution

It introduces a method using monoidal adjunctions to prove the existence of monoidal abelian envelopes and applies it to new categories like hyperoctahedral and modified symmetric groups.

Findings

Established criteria for monoidal abelian envelopes

Provided combinatorial proofs for new categories

Extended the theory to interpolation categories

Abstract

We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria, which we then demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.