∞-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

References

• Lurie, J., Higher Topos Theory.