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
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)
Combinatorial aspects