  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.
