Composing PROPs

Stephen Lack*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

117 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)147-163
Number of pages17
JournalTheory and Applications of Categories
Issue number9
Publication statusPublished - 2004
Externally publishedYes


  • Algebra
  • Bialgebra
  • Distributive law
  • Monad
  • PROP
  • Symmetric monoidal category


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