## 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 [1] 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

A coloured properad is more general than a polycategory in that it allows composition along more than one object. See Polycategory: Relation to properads for a more detailed explanation.

Unlike with polycategories and dioperads (and multicategories and operads), there doesn’t appear to be different terminology in the literature for one-coloured versus multicoloured properads.

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

## References

Last revised on March 25, 2020 at 18:34:54. See the history of this page for a list of all contributions to it.