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 [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. On the other hand, a properad is less general than a prop, because a prop also allows nonconnected graphs in the definition of the substitution product.


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