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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 .
When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise: A topos is a category that has the following two properties:
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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 ...
In mathematics, The fundamental theorem of topos theory states that the slice / of a topos over any one of its objects is itself a topos. Moreover, if there is a morphism f : A → B {\displaystyle f:A\rightarrow B} in E {\displaystyle \mathbf {E} } then there is a functor f ∗ : E / B → E / A {\displaystyle f^{*}:\mathbf {E} /B\rightarrow ...
The classifying topos for linear orders with distinct largest and smallest elements of a topos is the topos of simplicial sets. If G is a discrete group, the classifying topos for G-torsors over a topos is the topos BG of G-sets. The classifying space of topological groups in homotopy theory.