### Abstract

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 language | English |
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Pages (from-to) | 147-163 |

Number of pages | 17 |

Journal | Theory and Applications of Categories |

Volume | 13 |

Issue number | 9 |

Publication status | Published - 2004 |

Externally published | Yes |

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### Keywords

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