Categorifying equivariant monoids

TL;DR

This paper develops categorical frameworks combining PROPs, PROBs, and crossed simplicial groups to encode equivariant monoids, comonoids, bimonoids, and involutive structures with group actions.

Contribution

It constructs new categories that precisely encode equivariant monoid structures using extensions of crossed simplicial groups.

Findings

Categories of algebras correspond to equivariant monoids, comonoids, and bimonoids.

Extended the theory to balanced braided monoidal categories.

Encoded involutive monoids with compatible group actions.

Abstract

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of PROPs and PROBs with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action.