Composing PROBs

TL;DR

This paper introduces PROBs, a categorical framework for encoding algebraic structures like monoids and comonoids within braided monoidal categories, and demonstrates their composition to model bimonoids.

Contribution

It defines PROBs for monoids and comonoids, shows their composition to form bimonoids, and generalizes previous machinery for PROPs to PROBs.

Findings

PROBs encode algebraic structures in braided categories

Categories of algebras for PROBs are equivalent to monoids and comonoids

PROBs can be composed to model complex structures like bimonoids

Abstract

A PROB is a "product and braid" category. Such categories can be used to encode the structure borne by an object in a braided monoidal category. In this paper we provide PROBs whose categories of algebras in a braided monoidal category are equivalent to the categories of monoids and comonoids using the category associated to the braid crossed simplicial group of Fiedorowicz and Loday. We show that PROBs can be composed by generalizing the machinery introduced by Lack for PROPs. We use this to define a PROB for bimonoids in a braided monoidal category as a composite of the PROBs for monoids and comonoids.