---
abstract: 'Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.'
address:
- 'Wigner Research Centre for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary'
- 'Department of Mathematics, Macquarie University NSW 2109, Australia'
author:
- Gabriella Böhm
- Stephen Lack
title: Multiplier bialgebras in braided monoidal categories
---

[^1]

Introduction
============

A [*bimonoid*]{} (or [*bialgebra*]{}) $A$ in a braided monoidal category can equivalently be described without referring separately to the multiplication $m:A.A \to A$ and comultiplication $d:A \to A.A$ but only the unit, the counit, and the so-called [*fusion morphism*]{} [@Street:fusion] $$t_1:=\big(
\xymatrix{
A.A\ar[r]^-{d.1}&
A.A.A\ar[r]^-{1.m}&
A.A}\big).$$ In terms of these data, the axioms are given by the [*fusion equation*]{}, expressed by the commutativity of $$\label{eq:intr_fusion}
\xymatrix @C=35pt{
A^3 \ar[r]^-{1.t_1}\ar[d]_-{t_1.1}&
A^3 \ar[r]^-{b.1}&
A^3 \ar[r]^-{1.t_1}&
A^3 \ar[r]^-{b^{-1}.1}&
A^3\ar[d]^-{t_1.1}\\
A^3 \ar[rrrr]_-{1.t_1}&&&&
A^3}$$ (where $b$ stands for the braiding) and some compatibilities of $t_1$ with the unit and the counit. This is analyzed in [@LackStreet:skew Section 2] in terms of [*augmented lax tricocycloids*]{}: a morphism $t_1$ satisfies the fusion equation if and only if $t'=b^{-1}t_1$ satisfies the cocycle condition 