Corelations are the prop for extraspecial commutative Frobenius monoids

Coya, Brandon; Fong, Brendan

doi:10.48550/arxiv.1601.02307

Cited by 4 publications

(7 citation statements)

References 13 publications

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“…For example in [FZ18], corelations are shown to be a certain pushout, leading to characterizations of equivalence relations, partial equivalence relations, linear subspaces and others. Corelations have also been shown to be the prop for certain Frobenius monoids [CF16]. The present development is potentially a starting point for double-categorical versions of these results.…”

Section: Corelationsmentioning

confidence: 62%

Double Categories of Relations

Lambert¹

2021

Preprint

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A double category of relations is essentially a cartesian equipment with strong, discrete and functorial tabulators and for which certain local products satisfy a Frobenius Law. A double category of relations is equivalent to a double category whose proarrows are relations on some ordinary category admitting a proper and stable factorization system. This characterization is based closely on the recent characterization of double categories of spans due to Aleiferi. The overall development can be viewed as a double-categorical version of that of the notion of a "tabular allegory" or that of a "functionally complete bicategory of relations.

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Section: Corelationsmentioning

confidence: 62%

Double Categories of Relations

Lambert¹

2021

Preprint

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“…The process of forcing a cospan to become jointly epic defines a unique functor [10] H ′ : FinCospan → FinCorel.…”

Section: Proposition 23mentioning

confidence: 99%

“…It can be shown that the object (1, m, i, d, e) in FinCorel is an extraspecial commutative Frobenius monoid. The precise connection between such objects and corelations was worked out by Fong and the author [10]. To summarize, strict monoidal functors from FinCorel to symmetric monoidal categories correspond to extraspecial commutative Frobenius monoids.…”

Section: Proposition 23mentioning

confidence: 99%

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A Compositional Framework for Bond Graphs

Coya¹

2017

Preprint

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Electrical circuits made only of perfectly conductive wires can be seen as partitions between finite sets: that is, isomorphism classes of jointly epic cospans. These are also known as "corelations" and are the morphisms in the category FinCorel. The two-element set has two different Frobenius monoid structures in FinCorel. These two Frobenius monoids are related to "series" and "parallel" junctions, which are used to connect pairs of wires and electrical components together. We show that these Frobenius monoids interact to form a "weak bimonoid" as defined by Pastro and Street. We conjecture a presentation for the subcategory of FinCorel generated by the morphisms associated to these two Frobenius monoids, which we call FinCorel • . We are interested in "bond graphs," which are studied in electrical engineering and are built from series and parallel junctions. Although the morphisms of FinCorel • resemble bond graphs, there is not a perfect correspondence. This motivates the search for a category whose morphisms more precisely model bond graphs. We approach this by considering a subcategory, LagRel • k , of the category of Lagrangian relations, LagRel k . This is because both bond graphs and circuits determine Lagrangian relations between symplectic vector spaces. The categories FinCorel • and LagRel • k have a correspondence between their sets of generating morphisms. Thus we define the category BondGraph by using generators and imposing equations that are found in both FinCorel • and LagRel • k . We study the functorial semantics of BondGraph by giving two different functors from it to the category LagRel k and a natural transformation between them. Given a bond graph, the first functor picks out a Lagrangian relation in terms of "effort" and "flow," while the second picks one out in terms of "potential" and "current." The natural transformation arises from the way that effort and flow relate to potential and current.

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“…A symmetric monoidal category containing the universal special commutative Frobenius algebra was exhibited by Lack [7], and independently by Rosebrugh, Sabadini and Walters [8] in their study of general processes: it is the category whose objects are finite sets and whose morphisms are cospans; the composition of cospans is given by pushout. Special commutative Frobenius objects and cospan categories have also been studied recently by Baez and Fong [2] in the context of electrical network theory, and Coya and Fong [3] have shown that the category of jointly surjective cospans contains the universal extraspecial commutative Frobenius algebra, meaning that also unit-followed-by-counit is required to be the identity. 1 In this note we observe that "specialness" is a result of the simple-minded nature of pushouts in the category of sets: if one replaces pushouts by homotopy pushouts, the resulting cospan category contains the universal commutative Frobenius object, rather than the special one.…”

Section: Introductionmentioning

confidence: 97%

Homotopy composition of cospans

Kock

Spivak

2016

Commun. Contemp. Math.

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It is well known that the category of finite sets and cospans, composed by pushout, contains the universal {\em special} commutative Frobenius algebra. In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.Comment: 4 pages. In this version: A more direct introduction and a additional few details on the tubular neighborhood argument. Final version, to appear in Commun. Contemp. Mat

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Corelations are the prop for extraspecial commutative Frobenius monoids

Cited by 4 publications

References 13 publications

Double Categories of Relations

Double Categories of Relations

A Compositional Framework for Bond Graphs

Homotopy composition of cospans

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