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The theory was rounded out by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning, since it involved a Grothendieck topology. The idea of a Grothendieck topology (also known as a site ) has been characterised by John Tate as a bold pun on the two senses of Riemann surface .
In mathematics, a topos (US: / ˈ t ɒ p ɒ s /, UK: / ˈ t oʊ p oʊ s, ˈ t oʊ p ɒ s /; plural topoi / ˈ t ɒ p ɔɪ / or / ˈ t oʊ p ɔɪ /, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).
A locale is a sort of a space but perhaps not with enough points. [3] The topos theory is sometimes said to be the theory of generalized locales. [4]Jean Giraud's gros topos, Peter Johnstone's topological topos, [5] or more recent incarnations such as condensed sets or pyknotic sets.
A theorem of Lurie [2] states that an ∞-category is an ∞-topos if and only if it satisfies an ∞-categorical version of Giraud's axioms in ordinary topos theory. A "topos" is a category behaving like the category of sheaves of sets on a topological space. In analogy, Lurie's definition and characterization theorem of an ∞-topos says that ...
Kleene, S. C. (1945). "On the interpretation of intuitionistic number theory". Journal of Symbolic Logic. 10 (4): 109– 124. doi:10.2307/2269016. JSTOR 2269016. S2CID 40471120. Phoa, Wesley (1992). An introduction to fibrations, topos theory, the effective topos and modest sets (Technical report). Laboratory for Foundations of Computer Science ...
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In mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products. [1] Deligne's completeness theorem says a coherent topos has enough points. [2] William Lawvere noticed that Deligne's theorem is a variant of the Gödel completeness theorem for first-order logic. [3]
For any morphism f in there is an associated "pullback functor" := which is key in the proof of the theorem. For any other morphism g in which shares the same codomain as f, their product is the diagonal of their pullback square, and the morphism which goes from the domain of to the domain of f is opposite to g in the pullback square, so it is the pullback of g along f, which can be denoted as .