PROPs, dioperads, and polycategories

A PROP , short for “products and permutations category,” is a strict symmetric monoidal category whose monoid of objects is freely generated by a single object, hence is isomorphic to \((\mathbb{N},+,0)\). More generally, a colored PROP is a strict symmetric monoidal category whose monoid of objects is freely generated, hence isomorphic to a list monoid. PROs (“products categories”) and colored PROs are defined similarly, but are not necessary symmetric as monoidal categories.

Although PROs and PROPs are “just” monoidal categories, they have a distinct flavor and are more closely associated with things like Lawvere theories, operads, dioperads, polycategories (aka colored dioperads, see MO ), and properads .

Literature

PROPs

  • Hackney & Robertson, 2015: On the category of props (doi, arxiv)

Unifying frameworks

Several authors have proposed frameworks to unify the sprawling family of definitions around PROPs, properads, dioperads, and so on.

  • Yau & Johnson, 2015: A foundation for PROPs, algebras, and modules (doi, MR )
  • Kaufmann & Ward, 2017: Feynman categories (online , arxiv)
    • Kaufmann, 2018: Lectures on Feynman categories (doi, arxiv)
    • Batanin, Kock, Weber, 2018: Regular patterns, substitudes, Feynman categories and operads (pdf, arxiv)

Combinatorial aspects

  • Hackney, 2021: Categories of graphs for operadic structures (arxiv)
  • Hackney, 2022: Segal conditions for generalized operads (arxiv)