| 词条 | ∞-topos |
| 释义 |
In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces 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 spaces on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some 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[1] as an ∞-category X such that there is a small ∞-category C and a left exact localization functor from the ∞-category of presheaves of spaces on C to X. 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 an ∞-topos is an ∞-category behaving like the category of sheaves of spaces. See also{{Portal|Category theory}}{{Portal|Topology}}
References1. ^{{harvnb|Lurie|loc=Definition 6.1.0.4.}} 2. ^{{harvnb|Lurie|loc=Theorem 6.1.0.6.}}
3 : Higher category theory|Sheaf theory|Topos theory |
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