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In mathematics, a topos (USˈtɒpɒs, UKˈtoʊpoʊs,_ˈtoʊpɒs; plural topoi ˈtɒpɔɪ or ˈtoʊpɔɪ, or toposes) is a that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the and possess a notion of localization; they are a direct generalization of point-set topology. 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. Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme. Another illustration of the capability of Grothendieck toposes to incarnate the “essence” of different mathematical situations is given by their use as bridges for connecting theories which, albeit written in possibly very different languages, share a common mathematical content. A Grothendieck topos is a C which satisfies any one of the following three properties. (A theorem of Jean Giraud states that the properties below are all equivalent.) There is a D and an inclusion C ↪ Presh(D) that admits a finite-limit-preserving left adjoint. C is the category of sheaves on a Grothendieck site. C satisfies Giraud's axioms, below. Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a . Giraud's axioms for a C are: C has a small set of s, and admits all small colimits. Furthermore, fiber products distribute over coproducts.

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