Cartesian bicategories I

Carboni, A.; Walters, R. F. C.

doi:10.1016/0022-4049(87)90121-6

Cited by 216 publications

(223 citation statements)

References 4 publications

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“…We identify two classes of terminating rewriting systems for which confluence can be decided by means of critical pair analysis. The first one concerns SMTs containing a special Frobenius structure [14] (yielding categories alternatively called well-supported compact closed [13], p-and dgs-monoidal [25,9], or recently hypergraph categories [22,31]). For arbitrary SMTs, not necessarily equipped with a special Frobenius structure, we identify a second class of rewriting systems for which confluence can be decided.…”

Section: Introductionmentioning

confidence: 99%

Confluence of Graph Rewriting with Interfaces

Bonchi

Gadducci

Kissinger

et al. 2017

Programming Languages and Systems

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Abstract. For terminating double-pushout (DPO) graph rewriting systems confluence is, in general, undecidable. We show that confluence is decidable for an extension of DPO rewriting to graphs with interfaces. This variant is important due to it being closely related to rewriting of string diagrams. We show that our result extends, under mild conditions, to decidability of confluence for terminating rewriting systems of string diagrams in symmetric monoidal categories.

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Section: Introductionmentioning

confidence: 99%

Confluence of Graph Rewriting with Interfaces

Bonchi

Gadducci

Kissinger

et al. 2017

Programming Languages and Systems

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“…This leads us to introducing the notion of a special †-compact Frobenius algebra, which refines the usual topological quantum field theoretic notion of a normalized special Frobenius algebra [26]. The defining equality is due to Carboni and Walters [6]. 8 …”

Section: Classical Objectsmentioning

confidence: 99%

“…A symmetric Frobenius algebra is an internal commutative monoid (X, µ, ν) together with an internal commutative comonoid (X, δ, ǫ) which 6) that is, in a picture:…”

Section: Quantum Measurements Without Sumsmentioning

confidence: 99%

Quantum measurements without sums

Coecke¹,

Pavlović²

2007

Mathematics of Quantum Computation and Quantum Technology

107

183

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Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capability. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras.

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“…FRel does not naturally form an allegory, but does have features in common with them. Here we discuss the relationship of FRel to allegories, and make some brief comments directed toward the relationship between FRel and other categories generalizing Rel [3,1]. Definition 7.1 An allegory is a 2-category C where homsets form posets that are meet semilattices, equipped with an involution † that is the identity on objects, that satisfies the modular law…”

Section: Comparison With Abstract Categories Of Relationsmentioning

confidence: 99%

Categories with fuzzy sets and relations

Harding¹,

Walker²,

Walker³

2014

Fuzzy Sets and Systems

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We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these results to the setting of type-2 fuzzy sets.

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Cartesian bicategories I

Abstract: The notion of cartesian bicategory, introduced in [C&W] for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory.

Cited by 216 publications

References 4 publications

Confluence of Graph Rewriting with Interfaces

Confluence of Graph Rewriting with Interfaces

Quantum measurements without sums

Categories with fuzzy sets and relations

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