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Grothendieck toposes have Boolean points - a new proof. (English) Zbl 0358.18011

It is a new proof of the following theorem of M. Barr [J. Pure Appl. Algebra 5, 265–280 (1974; Zbl 0294.18009)]: Every Grothendieck topos has an exact cotripleable functor to a topos \(\$h_{can}(\mathbb B)\) of sheaves for some complete Boolean algebra. It is shorter and more in the spirit of elementary topos than that of Grothendieck topos. The theorem is proved, first, for a presheaf topos and then lifted to sheaves topos, by computing topologies in the sense of Lawvere-Tierney. In a manuscript paper, the title of which is: “La démonstration de Diaconescu du théorème de Barr, revue et corrigée” [Séminaire de théorie des catégories de Jean Benabou, Paris, novembre 1975], M. Coste points out an error and repairs it.

MSC:

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18A15 Foundations, relations to logic and deductive systems

Citations:

Zbl 0294.18009
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Full Text: DOI

References:

[1] Barr M., Journal of Pure and Applied Algebra 5 pp 265– (1975) · Zbl 0294.18009 · doi:10.1016/0022-4049(74)90037-1
[2] Lecture Notes in Mathematics 269 (1972)
[3] Tierney, M. 12-21 Sept 1971.Axiomatic sheaf theory:some constructions and applications, in Categories and Commutative Algebra, 12-21 Sept, 249–326. Edizioni Cremonese, Roma: C.I.M.E. Varenna. 1973
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