nLab properad




A properad in a symmetric monoidal category CC is a monoid in the monoidal category of bisymmetric sequences in CC (i.e., functors Σ×ΣC\Sigma\times\Sigma\to C) equipped with a version of substitution product modeled on connected directed graphs with 2 levels instead of corollas, which are used for operads. See §1.2 in Vallette for details. The idea is that operations in a properad can have multiple inputs and outputs, as opposed to a single output in an operad.

Relation to polycategories, dioperads and PROPs

The notion of coloured properads is more general than that of polycategories in that properads have composition-operations along more than one object. See at polycategory – Relation to properads for a more detailed explanation.

Properads are less general than PROPs, which also allow a “composition” of nonconnected graphs, by tensoring morphisms with \otimes.

The pluricategories of Kavanagh (Definition 2.1.11) are similar to properads, except that they have identity morphisms for each list of objects, rather than a unary morphism for each object.


Properads are called compact polycategories in:

  • Ross Duncan. “Types for quantum computing.” Phd thesis (2006).

The notion of pluricategory is defined in:

  • Ryan Kavanagh. Communication-Based Semantics for Recursive Session-Typed Processes. Thesis. Carnegie Mellon University, 2021 (pdf)

About the (∞,1)-category version

  • Shaul Barkan, Jan Steinebrunner, The equifibered approach to ∞-properads (arXiv:2211.02576).

Last revised on January 19, 2023 at 21:46:56. See the history of this page for a list of all contributions to it.