Tannaka Reconstruction of Weak Hopf Algebras in Arbitrary Monoidal Categories

TL;DR

This paper develops a graphical calculus approach to Tannaka reconstruction, showing how weak Hopf algebras can be reconstructed from monoidal functors into braided autonomous categories, extending known results.

Contribution

It introduces a new graphical calculus variant and generalizes Tannaka reconstruction to weak Hopf algebras in arbitrary monoidal categories.

Findings

Reconstruction yields weak bialgebras in braided categories.

Autonomous categories lead to weak Hopf algebras.

Extension of Tannaka theory to broader categorical contexts.

Abstract

We introduce a variant on the graphical calculus of Cockett and Seely for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F: A --> B from a monoidal category A to a suitably complete or cocomplete braided autonomous category B, the usual formula for Tannaka reconstruction gives a weak bialgebra in B; if, moreover, A is autonomous, this weak bialgebra is in fact a weak Hopf algebra.