Geodesic
Composing PROBs
Theory and applications of categories, Tome 38 (2022), pp. 1050-1061.

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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.
Publié le :
Classification : 18M15, 16T10
Keywords: PROB, bimonoid, bialgebra, braided monoidal category, crossed simplicial group, distributive law 
@article{TAC_2022_38_a25,
     author = {Daniel Graves},
     title = {Composing {PROBs}},
     journal = {Theory and applications of categories},
     pages = {1050--1061},
     publisher = {mathdoc},
     volume = {38},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2022_38_a25/}
}
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Daniel Graves. Composing PROBs. Theory and applications of categories, Tome 38 (2022), pp. 1050-1061. http://geodesic.mathdoc.fr/item/TAC_2022_38_a25/