Introduction to extensive and distributive categories

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

doi:10.1016/0022-4049(93)90035-r

Cited by 207 publications

(232 citation statements)

References 3 publications

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“…In fact, Theorem 7 holds in the more general setting of full cartesian Lawvere category whose coproducts satisfy the Beck-Chevalley condition and whose base categories satisfy extensivity [8]. Moreover, Theorem 6 extends to show that these properties are also preserved by the change-of-base construction provided all fibres of the original fibration satisfy extensivity.…”

Section: Lemma 2 If the Distributive Law λ Satisfies Non-introductiomentioning

confidence: 82%

When Is a Type Refinement an Inductive Type?

Atkey

Johann

Ghani

2011

Foundations of Software Science and Computational Structures

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Abstract. Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the Nindexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purposespecific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations often store redundant information about data and their refinements. This paper shows how to generically derive inductive characterisations of refinements of inductive types, and argues that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. These characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined.

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Section: Lemma 2 If the Distributive Law λ Satisfies Non-introductiomentioning

confidence: 82%

When Is a Type Refinement an Inductive Type?

Atkey

Johann

Ghani

2011

Foundations of Software Science and Computational Structures

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“…We assume that C is extensive [1]-that is, C has binary coproducts, which are disjoint and stable under pullback 5 .…”

Section: Assumptionmentioning

confidence: 99%

Transformation and Refinement of Rigid Structures

Danos

Heckel

Sobociński

2014

Graph Transformation

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Abstract. Stochastic rule-based models of networks and biological systems are hard to construct and analyse. Refinements help to produce systems at the right level of abstraction, enable analysis techniques and mappings to other formalisms. Rigidity is a property of graphs introduced in Kappa to support stochastic refinement, allowing to preserve the number of matches for rules in the refined system. In this paper: 1) we propose a notion of rigidity in an axiomatic setting based on adhesive categories; 2) we show how the rewriting of rigid structures can be defined systematically by requiring matches to be open maps reflecting structural features which ensure that rigidity is preserved; and 3) we obtain in our setting a notion of refinement which generalises that in Kappa, and allows a rule to be partitioned into a set of rules which are collectively equivalent to the original. We illustrate our approach with an example of a social network with dynamic topology.

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“…[Gir72]. This included stable disjoint coproducts, but they were later reformulated in a new way called extensivity [Coc93,CLW93]. This is very useful, because it is often technically easier to check that it holds in many categories of "spaces", understood in the very general sense of §4.2.…”

Section: Discrete Mathematicsmentioning

confidence: 99%

Foundations for Computable Topology

Taylor¹

2011

The Western Ontario Series in Philosophy of Science

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Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines.This method is applied to devising a new paradigm for general topology, called Abstract Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpiński space, which plays a key role as the classifier for both open and closed subspaces.In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory.The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0, 1] in ASD is compact."In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl.)"Point-set topology is a disease from which the human race will soon recover." (Henri Poincaré, quoted in Comic Sections by Des MacHale.)

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Introduction to extensive and distributive categories

Cited by 207 publications

References 3 publications

When Is a Type Refinement an Inductive Type?

When Is a Type Refinement an Inductive Type?

Transformation and Refinement of Rigid Structures

Foundations for Computable Topology

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