Categories of Relations and Functional Relations

Jayewardene, Romaine; Wyler, Oswald

doi:10.1007/978-94-017-2529-3_16

Cited by 5 publications

(6 citation statements)

References 22 publications

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“…is controllable if gcd(B2, C1) = 1. Given the observability assumption, this implies gcd(C 2 B2, B 1 C1) = 1, and so the interconnected behaviour C • B represented by (8) is controllable.…”

Section: Comparison To Matrix Methodsmentioning

confidence: 99%

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A categorical approach to open and interconnected dynamical systems

Fong

Sobociński²,

Rapisarda³

2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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In his 1986 Automatica paper Willems introduced the influential behavioural approach to control theory with an investigation of linear time-invariant (LTI) discrete dynamical systems. The behavioural approach places open systems at its centre, modelling by tearing, zooming, and linking. In this paper, we show that these ideas are naturally expressed in the language of symmetric monoidal categories.Our main result implies an intuitive sound and fully complete string diagram algebra for reasoning about LTI systems. These string diagrams are closely related to the classical notion of signal flow graphs, but in contrast to previous work, they endowed with semantics as multi-input multi-output transducers that process streams with an infinite past as well as an infinite future. At the categorical level, the algebraic characterisation is that of the prop of corelations, instead of a pushout of the props of spans and cospans. Using this framework, we then derive a novel structural characterisation of controllability, and consequently provide a methodology for analysing controllability of networked and interconnected systems. We argue that this methodology has the potential of providing elegant, simple, and efficient solutions to problems arising in the analysis of systems over networks, a vibrant research area at the crossing of control theory and computer science.

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Section: Comparison To Matrix Methodsmentioning

confidence: 99%

“…Composition is given by pushout, as in the category of cospans, followed by factorising the copairing of the resulting cospan. This is a dualisation of a well-known construction of relations from spans [8]; the details can be found in [7]. mapping a cospan to its jointly-epic counterpart given by the factorisation system.…”

Section: Corelationsmentioning

confidence: 99%

A categorical approach to open and interconnected dynamical systems

Fong

Sobociński²,

Rapisarda³

2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Next, call a cospan X → N ← Y jointly epic if the induced morphism X + Y → N is an epimorphism. If monomorphisms in C are preserved under pushout, we may construct a symmetric monoidal category Corel(C) with objects again those of C, but this time morphisms isomorphism classes of jointly epic cospans in C, and composition taking the pushout of representative cospans, before corestricting to the jointly epic part [Mi00,JW96]. 2 The unitors, associator, and braiding are inherited from C.…”

Section: Examplementioning

confidence: 99%

“…More generally, a category of corelations may be constructed from any finitely cocomplete category equipped with a (E, M)-factorisation system such that M is preserved under pushout[JW96]. In related papers, we have shown that this construction can be used to model interconnection of 'black-boxed' systems; that is, to model systems in which only the internal structure is obscured, leaving only the external behaviour[BF, FRS, Fo].…”

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confidence: 99%

Corelations are the prop for extraspecial commutative Frobenius monoids

Coya,

Fong

2016

Preprint

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Just as binary relations between sets may be understood as jointly monic spans, so too may equivalence relations on the disjoint union of sets be understood as jointly epic cospans. With the ensuing notion of composition inherited from the pushout of cospans, we call these equivalence relations corelations. We define the category of corelations between finite sets and prove that it is equivalent to the prop for extraspecial commutative Frobenius monoids. Dually, we show that the category of relations is equivalent to the prop for special commutative bimonoids. Throughout, we emphasise how corelations model interconnection.

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“…This category inherits from the bicategory Span(C) the structure of a (strict) 2-category, with 2-cells given by order; actually, Rel(C) is the prototypical example of a (unitary and tabular) allegory: see [Freyd, Scedrov 1990], [Johnstone 2002]. More generally, as done in [Pavlović 1995] (with predecessors of this work presented under more restrictive conditions in [Klein 1970] and [Meisen 1974], and with a weakening of the pullback-stability constraint given in [Jayewardene, Wyler 1996]), without any epior mono restrictions one may consider an arbitrary stable factorization system (E, M) of a category C with binary products and pullbacks and form the category…”

Section: Introductionmentioning

confidence: 99%

Quotients of span categories that are allegories and the representation of regular categories

Hosseini¹,

Nasab²,

Tholen³

et al. 2021

Preprint

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We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)factorization system, we establish a quotient category Span E (C) that is isomorphic to the category Rel M (C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E • containing E such that Span E• (C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.

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Categories of Relations and Functional Relations

Cited by 5 publications

References 22 publications

A categorical approach to open and interconnected dynamical systems

A categorical approach to open and interconnected dynamical systems

Corelations are the prop for extraspecial commutative Frobenius monoids

Quotients of span categories that are allegories and the representation of regular categories

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