An outline of functorial semantics

Linton, F. E. J.

doi:10.1007/bfb0083080

Cited by 47 publications

(34 citation statements)

References 9 publications

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“…So we are saying that we want S op to be a category of algebras over S. For this, we need a way of formulating (potentially infinitary) algebraic theories that works over an arbitrary category S, and not just over the category of sets. Such an account is provided by the categorical notion of monad [Lin69].…”

Section: Always Topologizementioning

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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Section: Always Topologizementioning

confidence: 99%

Foundations for Computable Topology

Taylor¹

2011

The Western Ontario Series in Philosophy of Science

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“…In contrast to coalgebras for a comonad, the coalgebra map ξ : X → T X encodes what the transition system (X, ξ) can perform in one step. From this point of view, the format [20] and Rosický [26], one can show that Alg(L) can always be described by operations and equations over A. But in that approach the arities describing Alg(L) are objects in A and not cardinalities in Set.…”

Section: Definition 7 (Presented Functor)mentioning

confidence: 99%

Presenting Functors by Operations and Equations

Bonsangue

Kurz

2006

Lecture Notes in Computer Science

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Abstract. We take the point of view that, if transition systems are coalgebras for a functor T, then an adequate logic for these transition systems should arise from the 'Stone dual' L of T. We show that such a functor always gives rise to an 'abstract' adequate logic for T-coalgebras and investigate under which circumstances it gives rise to a 'concrete' such logic, that is, a logic with an inductively defined syntax and proof system. We obtain a result that allows us to prove adequateness of logics uniformly for a large number of different types of transition systems and give some examples of its usefulness.

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“…Any functor preserves split coequalizer systems [Linton (1969a)]. In particular every triple does, and so Proposition 3 guarantees that coequalizers of U T -split pairs of A T -morphisms can be computed in A , as was stated in greater generality in [Linton (1969), Section 6].…”

Section: Catmentioning

confidence: 99%

“…This implies that Π * * Π has Π −1 * Π as its set of components and has no higher homotopy. (This is equivalent to the curiosity that Z( ) as an endofunctor on sets satisfies the hypotheses of the "precise" tripleableness theorem ([Beck (1967), Theorem 1] or [Linton (1969a)]. )…”

Section: Acyclicity and Coproductsmentioning

confidence: 99%