Homological kernels of monoidal functors.

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 envelopes' which completes the notion of monoidal abelian envelopes. [ABSTRACT FROM AUTHOR]