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.

http://www.tac.mta.ca/tac/volumes/13/9/13-09.pdf

Revised 2021-09-21. Original version at

http://www.tac.mta.ca/tac/volumes/13/9/13-09a.pdf