Topos

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In mathematics, a topos (US: /ˈtɒpɒs/, UK: /ˈtoʊpoʊs, ˈtoʊpɒs/; plural topoi /ˈtɒpɔɪ/ or /ˈtoʊpɔɪ/, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology.^[1] The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.

The mathematical field that studies topoi is called topos theory.