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Keywords:
Combinatorics; Species; Properad; Protoperad
Combinatorics; Species; Properad; Protoperad
Summary:
This paper is the first of two articles which develop the notion of protoperads. In this one, we construct a new monoidal product on the category of reduced $\frak {S}$-modules. We study the associated monoids, called {\it protoperads}, which are a type of generalised operad. As operads encode algebraic operations with several inputs and one output, protoperads encode algebraic operations with the same number of inputs and outputs. We describe the underlying combinatorics of protoperads, and show that there exists a notion of free protoperad. We also show that the monoidal product introduced here is related to Vallette’s one on the category of $\frak {S}$-bimodules, via the induction functor.
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