On the Category of Props

Hackney, Philip; Robertson, Marcy

doi:10.1007/s10485-014-9369-4

Cited by 18 publications

(26 citation statements)

References 19 publications

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“…For name management we also have new equations for scope extrusion (Eqn. [11][12][13][14]. Finally, the connection between variables and the structural connectors, namely identity, is given in Eq.…”

Section: Figure 3: Equational Theory Of Uniflow Diagramsmentioning

confidence: 99%

A Structural and Nominal Syntax for Diagrams

Ghica

Lopez

2018

Electron. Proc. Theor. Comput. Sci.

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The correspondence between monoidal categories and graphical languages of diagrams has been studied extensively, leading to applications in quantum computing and communication, systems theory, circuit design and more. From the categorical perspective, diagrams can be specified using (name-free) combinators which enjoy elegant equational properties. However, conventional notations for diagrammatic structures, such as hardware description languages (VHDL, VERILOG) or graph languages (DOT), use a different style, which is flat, relational, and reliant on extensive use of names (labels). Such languages are not known to enjoy nice syntactic equational properties. However, since they make it relatively easy to specify (and modify) arbitrary diagrammatic structures they are more popular than the combinator style. In this paper we show how the two approaches to diagram syntax can be reconciled and unified in a way that does not change the semantics and the existing equational theory. Additionally, we give sound and complete equational theories for the combined syntax. Specifying graphsGraphs and their visual representations (diagrams) are an appealing way of describing many kinds of systems, in particular circuits. Work originated in the study of quantum computation [1] has exploited the connection between various classes of graphs and monoidal categories, going back to the seminal work of Joyal and Street [14], to add a layer of structure which makes reasoning about and with diagrams not just intuitive but also mathematically rigorous. Subsequently this connection was extended in many surprising and interesting directions, from computational linguistics [17], to modelling signal flow [4] and synchronous [9] or asynchronous [8] circuits. New automated reasoning tools based on diagrams, rather than the usual linear algebraic syntax, are a particularly exciting development (see http://globular.science/). This convenient and elegant interplay between the dual categorical and diagrammatic methods are by now a mature and rich area of research [18].Although these developments convincingly establish the usefulness of categorical diagrammatics in reasoning about many kinds of systems, we note that little has been suggested in terms of a workable syntax which is conceptually compatible with it, but also with conventional notations, which are unstructured and relational. Examples of the latter are hardware description languages such as VERILOG and VHDL, or graph languages such as DOT (see http://www.graphviz.org/). Indeed, in the literature the categorical combinators are usually taken as an implicit diagram syntax. Whereas such a syntax is expressive enough to describe the desired classes of graphs, it is often not as convenient as the alternatives. Although categorical combinators can be elegant and succinct in certain situation, point-free languages of combinators generally hold little appeal as concrete syntax (e.g. APL).Conventional diagram syntax lacks structure. It is a "flat" relational description of the graph. Whereas th...

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Section: Figure 3: Equational Theory Of Uniflow Diagramsmentioning

confidence: 99%

A Structural and Nominal Syntax for Diagrams

Ghica

Lopez

2018

Electron. Proc. Theor. Comput. Sci.

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“…If A ∈ D s is objectwise-free, then there exists an object s ∈ S and a bo morphism f : F(Disc(s)) ։ A in D s . By the F ⊣ U adjunction 8 8 Note that we continue to commit the abuse of notation writing U for Uι.…”

Section: A3 Objectwise-free Monoidal Traced and Compact Categoriesmentioning

confidence: 99%

String diagrams for traced and compact categories are oriented 1-cobordisms

Spivak

Schultz

Rupel

2017

Journal of Pure and Applied Algebra

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We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob /O , where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob /O in the sense that there is an equivalence between Cob /O -algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories.

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“…Towards hypergraph categories, it is instrumental to describe first the free symmetric monoidal category generated by a theory (Σ, C), which is called a C-coloured prop [18] (product and permutation category). This works in analogy with the single-sorted case C = {c}, in which monoidal theories act as presentations for ({c}-coloured) props [23].…”

Section: Props and Hypergraph Categoriesmentioning

confidence: 99%

“…Lemma 5.6. Consider a critical pair in Hyp Σ,C as in (18). If both K 1 and K 2 are discrete hypergraphs, so is the interface J.…”

Section: Proposition 52 ([3]mentioning

confidence: 99%

Rewriting in Free Hypergraph Categories

Zanasi

2017

Electron. Proc. Theor. Comput. Sci.

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We study rewriting for equational theories in the context of symmetric monoidal categories where there is a separable Frobenius monoid on each object. These categories, also called hypergraph categories, are increasingly relevant: Frobenius structures recently appeared in cross-disciplinary applications, including the study of quantum processes, dynamical systems and natural language processing. In this work we give a combinatorial characterisation of arrows of a free hypergraph category as cospans of labelled hypergraphs and establish a precise correspondence between rewriting modulo Frobenius structure on the one hand and double-pushout rewriting of hypergraphs on the other. This interpretation allows to use results on hypergraphs to ensure decidability of confluence for rewriting in a free hypergraph category. Our results generalise previous approaches where only categories generated by a single object (props) were considered.

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On the Category of Props

Cited by 18 publications

References 19 publications

A Structural and Nominal Syntax for Diagrams

A Structural and Nominal Syntax for Diagrams

String diagrams for traced and compact categories are oriented 1-cobordisms

Rewriting in Free Hypergraph Categories

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