A properad in a symmetric monoidal category is a monoid in the monoidal category of bisymmetric sequences in (i.e., functors ) 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.
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 .
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:
The notion of pluricategory is defined in:
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Last revised on January 19, 2023 at 21:46:56. See the history of this page for a list of all contributions to it.