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∞-topos
In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects are sheaves with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of, say, abelian groups on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some affine scheme is not the ∞-category of sheaves on any topological space but it is still an ∞-topos.
Precisely, in Lurie's ''Higher Topos Theory'', an ∞-topos is defined as an ∞-category ''X'' such that there is an ∞-category ''C'' and a left exact localization functor from the ∞-category of presheaves of spaces on ''C'' to ''X''. A theorem of Lurie states that an ∞-category is an ∞-topos if and only if it satisfies an ∞-categorical version of Giraud’s axioms in ordinary topos theory. Authors including Wikipedia describes a "topos" as 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 an ∞-topos is an ∞-category behaving like the category of sheaves of spaces.
== See also ==
*Homotopy hypothesis
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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