Homological kernels of monoidal functors

TL;DR

This paper introduces a framework where rigid monoidal categories over a field generate universal tensor categories that classify faithful monoidal functors, extending the concept of monoidal abelian envelopes through homological kernels.

Contribution

It develops a theory of homological kernels for monoidal functors, providing universal tensor categories that classify these functors and generalize monoidal abelian envelopes.

Findings

Universal tensor categories classify faithful monoidal functors.

Each universal tensor category corresponds to a homological kernel.

The theory extends monoidal abelian envelopes via sheaf category realizations.

Abstract

We show that each rigid monoidal category A over a field defines a family of universal tensor categories, which together classify all faithful monoidal functors from A to tensor categories. Each of the universal tensor categories classifies monoidal functors of a given 'homological kernel' and can be realised as a sheaf category, not necessarily on A. This yields a theory of 'local abelian envelope' which completes the notion of monoidal abelian envelopes.