  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.
