PROPs for involutive monoids and involutive bimonoids

TL;DR

This paper establishes an equivalence between involutive bimonoids in a symmetric monoidal category and algebras over a specific PROP derived from involutive non-commutative sets, linking algebraic structures with categorical frameworks.

Contribution

It introduces a PROP framework for involutive bimonoids, providing a new categorical perspective on involutive algebraic structures.

Findings

Equivalence between involutive bimonoids and PROP algebras

Construction of a PROP from involutive non-commutative sets

Categorical encoding of involution compatible structures

Abstract

The category of involutive non-commutative sets encodes the structure of an involution compatible with a (co)associative (co)multiplication. We prove that the category of involutive bimonoids in a symmetric monoidal category is equivalent to the category of algebras over a PROP constructed from the category of involutive non-commutative sets.