Topos theory

In mathematics, a topos (UK: /ˈtɒpɒs/, US: /ˈtoʊpoʊs, ˈtoʊpɒs/; plural topoi /ˈtoʊpɔɪ/ or /ˈtɒpɔɪ/, 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. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. In mathematics, a topos (UK: /ˈtɒpɒs/, US: /ˈtoʊpoʊs, ˈtoʊpɒs/; plural topoi /ˈtoʊpɔɪ/ or /ˈtɒpɔɪ/, 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. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. 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. Let C be a category. A theorem of Giraud states that the following are equivalent: A category with these properties is called a '(Grothendieck) topos'. Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's axioms for a category C are: The last axiom needs the most explanation. If X is an object of C, an 'equivalence relation' R on X is a map R→X×X in Csuch that for any object Y in C, the induced map Hom(Y,R)→Hom(Y,X)×Hom(Y,X) gives an ordinary equivalence relation on the set Hom(Y,X). Since C has colimits we may form the coequalizer of the two maps R→X; call this X/R. The equivalence relation is 'effective' if the canonical map