Composing PROPs

Stephen Lack

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.

Revised 2021-09-21. Original version at

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