A category of multiplier bimonoids

TL;DR

This paper introduces a new categorical framework for multiplier bimonoids in braided monoidal categories, describing their structure as comonoids in a specially constructed monoidal category.

Contribution

It develops a novel categorical approach to multiplier bimonoids, defining a monoidal category of semigroups and characterizing bimonoids as comonoids within it.

Findings

Defines a category of semigroups with multiplicative morphisms

Equips this category with a monoidal structure

Characterizes multiplier bimonoids as comonoids in this category

Abstract

The central object studied in this paper is a multiplier bimonoid in a braided monoidal category C. Adapting the philosophy of Janssen and Vercruysse, and making some mild assumptions on the category C, we consider a category M whose objects are certain semigroups in C and whose morphisms from A to B can be regarded as suitable multiplicative morphisms from A to the multiplier monoid of B. We equip this category M with a monoidal structure and describe multiplier bimonoids in C (whose structure morphisms belong to a distinguished class of regular epimorphisms) as certain comonoids in M. This provides us with one possible notion of morphism between such multiplier bimonoids.