Quasitopoi, logic and heyting-valued models

Monro, G. P.

doi:10.1016/0022-4049(86)90077-0

Cited by 13 publications

(13 citation statements)

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“…In part to provide such a foundation, since the early 1970's a variety of categorical structures have been studied which allow for mathematics to be done internally: elementary topoi (see [Kock and Wraith 1972, Lawvere 1972, Freyd 1972, Johnstone 1977, Goldblatt 1979, Bell 1988). quasitopoi ( [Penon 1977, Wyler 1991 ), various approaches using monoidal structures ( [Stout 1992, Höhle 1991, Monro 1986 ) and fibered categories [Jacobs 1999].…”

Section: How a Categorical Foundation Might Helpmentioning

confidence: 99%

Upsetting the Foundations for Mathematics

Stout¹

2005

philosophiascientiae

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Commençant par une revue sommaire des types de questions qu'une fondation des mathématiques devrait poser, cet article présente premièrement une critique des fondements basés sur la théorie des ensembles, puis propose l'idée que plusieurs fondements catégoriques, reliés les uns aux autres, seraient plus avantageux, et finalement indique une méthode pour retrouver la théorie des ensembles à travers une approche catégorique.

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Section: How a Categorical Foundation Might Helpmentioning

confidence: 99%

Upsetting the Foundations for Mathematics

Stout¹

2005

philosophiascientiae

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“…For this example, IndC is a quasitopos, introduced independently by G.P. MONRO [13] and D. PONASSE (see [14] of if-sets, iff SA(X) < SBUX^QX), for all x in |^4|.…”

Section: » I'mentioning

confidence: 99%

“…In the theory of fuzzy sets, there is strong evidence that functional relations rather than "crisp" maps are the appropriate morphisms; see e.g. [14] and [8], and from a categorical viewpoint [7] and [13]. These functional relations are obtained from factorization structures defined in Section 7 of this paper.…”

Section: Introductionmentioning

confidence: 99%

Categories of Relations and Functional Relations

Jayewardene

Wyler

2000

Papers in Honour of Bernhard Banaschewski

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“…Any quasitopos with the factorization system (Epis, Strong Monos) is an EM-category. This example is discussed in detail in [8]. EXAMPLE 3.…”

Section: Amentioning

confidence: 99%

Logic, sheaves, and factorization systems

Monro

1993

J. symb. log.

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In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.

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Quasitopoi, logic and heyting-valued models

Cited by 13 publications

References 2 publications

Upsetting the Foundations for Mathematics

Upsetting the Foundations for Mathematics

Categories of Relations and Functional Relations

Logic, sheaves, and factorization systems

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