Weak multiplier bimonoids

TL;DR

This paper introduces a new algebraic structure called regular weak multiplier bimonoids in braided monoidal categories, generalizing existing concepts and establishing their categorical properties.

Contribution

It defines regular weak multiplier bimonoids via weakly counital fusion morphisms and explores their properties and module categories within braided monoidal categories.

Findings

Base object carries a coseparable comonoid structure

Modules form a monoidal category with a strict functor to bicomodules

Applicable to categories like modules, graded modules, bornological spaces, and Hilbert spaces

Abstract

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided monoidal categories. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.