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.