Composing PROBs

Daniel Graves

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arxiv 2105.13166 v2 pith:4TMSGFFV submitted 2021-05-27 math.CT math.AT

Composing PROBs

Daniel Graves This is my paper

classification math.CT math.AT

keywords categoryprobsbraidedcategoriesmonoidalbraidcomonoidsmonoids

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A PROB is a "product and braid" category. Such categories can be used to encode the structure borne by an object in a braided monoidal category. In this paper we provide PROBs whose categories of algebras in a braided monoidal category are equivalent to the categories of monoids and comonoids using the category associated to the braid crossed simplicial group of Fiedorowicz and Loday. We show that PROBs can be composed by generalizing the machinery introduced by Lack for PROPs. We use this to define a PROB for bimonoids in a braided monoidal category as a composite of the PROBs for monoids and comonoids.

Composing PROBs

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