A PROP is a way of encoding structure borne by an object of a symmetric monoidal category. We describe a notion of distributive law for PROPs, based on Beck's distributive laws for monads. A distributive law between PROPs allows them to be composed, and an algebra for the composite PROP consists of a single object with an algebra structure for each of the original PROPs, subject to compatibility conditions encoded by the distributive law. An example is the PROP for bialgebras, which is a composite of the PROP for coalgebras and that for algebras.
Keywords: symmetric monoidal category, PROP, monad, distributive law, algebra, bialgebra
2000 MSC: 18D10, 18C10, 18D35
Theory and Applications of Categories,
Vol. 13, 2004,
No. 9, pp 147-163.