A simplicial approach to multiplier bimonoids

TL;DR

This paper introduces a novel simplicial set framework to classify multiplier bimonoids, extending the concept of (co)monoids beyond traditional monoidal categories, providing new insights into their structure.

Contribution

It develops a simplicial approach to classify multiplier bimonoids, connecting them to (co)monoids in a more general setting than monoidal categories.

Findings

Classifies multiplier bimonoids via simplicial maps from the Catalan set.

Shows certain multiplier bimonoids can be regarded as comonoids in a simplicial context.

Provides a new perspective on the structure of multiplier bimonoids.

Abstract

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.