Abstract
We show that each rigid monoidal category {textbf{a}} over a field defines a family of universal tensor categories, which together classify all faithful monoidal functors from {textbf{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 {textbf{a}}. This yields a theory of ‘local abelian envelopes’ which completes the notion of monoidal abelian envelopes.
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