Colimits in Topoi

Paré, Robert

doi:10.1090/s0002-9904-1974-13497-x

Cited by 38 publications

(10 citation statements)

References 5 publications

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“…doi: 10.1006Âaima.2000.1947, available online at http:ÂÂwww.idealibrary.com on We also prove, under a certain hypothesis on E (satisfied for instance by all Grothendieck toposes or by all essential localizations of any topos of internal presheaves) that the opposite of the category of distributions on E is monadic over E by means of a double dualization monad. Our results relativize the Stone theory internal to a topos [20], as well as a related monadicity theorem [22].…”

Section: Introductionsupporting

confidence: 82%

Distribution Algebras and Duality

Bunge

Funk

Jibladze³

et al. 2000

Advances in Mathematics

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Since being introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [5 9, 12, 15, 19, 24]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) lattice-theoretic notion of distribution algebra. We characterize the distribution algebras in E relative to S as the S-bicomplete S-atomic Heyting algebras in E. As an illustration, we employ distribution algebras explicitly in order to give an alternative description of the display locale (complete spread) of a distribution [7,10,12].

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Section: Introductionsupporting

confidence: 82%

Distribution Algebras and Duality

Bunge

Funk

Jibladze³

et al. 2000

Advances in Mathematics

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“…The powerset P(X) is given by the exponential Ω X , where Ω is the lattice of truth values, which we shall discuss further in §7. Paré showed that the contravariant functor Ω (−) is monadic in the above sense [Par74].…”

Section: 2mentioning

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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“…We now show that the topos of V-definable sets in M is complete by constructing a product for an arbitrary indexed family xi (i E Z) of objects, where Z E V. (Cocompleteness can be proved similarly or deduced from completeness via [7].) By the preceding discussion (and the axiom of choice in the metatheory, i.e., in V), we can fix an indexed family of parameters p = (P~)~., E V such that, for each i, Xi is (with truth value 1 in M) the unique solution of 6(Xi, iii).…”

Section: Theorem 1 Let M Be Any Model Of Set Theory Its V-definablementioning

confidence: 99%

Complete topoi representing models of set theory

Blass

Scedrov

1992

Annals of Pure and Applied Logic

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Blass, A. and A. Scedrov, Complete topoi representing models of set theory, Annals of Pure and Applied Logic 57 (1992) l-26. By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued, pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos % of M is said to represent M if it is complete and its cumulative hierarchy, as defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given.

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Colimits in Topoi

Cited by 38 publications

References 5 publications

Distribution Algebras and Duality

Distribution Algebras and Duality

Foundations for Computable Topology

Complete topoi representing models of set theory

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