Adjointable monoidal functors and quantum groupoids

TL;DR

This paper explores the structure of monoidal functors and their factorizations through bimodule categories, providing new characterizations of forgetful functors related to bialgebroids and weak bialgebras.

Contribution

It introduces a canonical factorization of monoidal functors via bimodule categories and characterizes forgetful functors of bialgebroids and weak bialgebras using bimonad theory.

Findings

Canonical factorization of monoidal functors through bimodule categories

Characterization of forgetful functors for bialgebroids

Application of bimonad description to weak bialgebras

Abstract

Every monoidal functor G: C --> M has a canonical factorization through the category of bimodules over some monoid R in M such that the factor U: C -->_R M_R is strongly unital. Using this result and the characterization of the forgetful functors M_A -->_R M_R of bialgebroids A over R given by Schauenburg together with their bimonad description given by the author recently here we characterize the "long" forgetful functors M_A -->_R M_R --> M of both bialgebroids and weak bialgebras.