Tensor envelopes of regular categories

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,δ) depending on a degree function δ. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows:

Let N be the maximal proper tensor ideal of T(A,δ). We show that T(A,δ)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories.
Using lattice theory, we give a simple numerical criterion for the vanishing of N.
We determine all degree functions for which T(A,δ) is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S_n, the hyperoctahedral groups S_n⋉ℤ₂ⁿ, or the general linear groups GL(n,F_q) over a fixed finite field.

This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S_n. It also extends (and provides proofs for) a previous paper on the special case of abelian categories.

Appeared in: Advances in Mathematics 214 (2007) 571-617

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Last updated: September 19, 2007