Fiber functors, monoidal sites and Tannaka duality for bialgebroids

TL;DR

This paper develops a new Tannaka duality framework for bialgebroids by characterizing fiber functors on small additive monoidal categories, revealing a sheaf-theoretic perspective and extending classical results.

Contribution

It introduces a novel Tannaka duality theorem for bialgebroids and provides a sheaf-theoretic interpretation, generalizing previous work and establishing existence results for fiber functors.

Findings

Reconstructed bialgebroid H has a comodule category equivalent to T-sheaves.

Established a sheaf-theoretic interpretation of the duality.

Proved an existence theorem for fiber functors with bounded fusion and weak kernels.

Abstract

What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phung Ho Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstructed bialgebroid H has comodule category equivalent to the category of T-sheaves w.r.t. a monoidal Grothendieck topology on C. We also prove an existence theorem for fiber functors on small additive monoidal categories with bounded fusion and weak kernels. For certain autonomous categories a generalized Ulbrich Theorem can be formulated which relates fiber functors to Hopf algebroid Galois extensions.