## Definition

A properad in a symmetric monoidal category $C$ is a monoid in the monoidal category of bisymmetric sequences in $C$ (i.e., functors $\Sigma\times\Sigma\to C$) 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.

## References

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)