∞-topos

Higher categorical generalization of a topos

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In mathematics, an ∞-topos (infinity-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 an (accessible) 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.

## Lurie characterization theorem

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It says:

Theorem—Let  X {\displaystyle X} be an ∞-category. Then the following are equivalent.

(a)  X {\displaystyle X} is an ∞-topos.
(b)  X {\displaystyle X} satisfies Giraud's axioms in the ∞-category setting: (1) it is a presentable ∞-category, (2) colimits in X are universal, (3) coproducts in X are disjoint and (4) every groupoid object in X is effective.

## See also

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* Mathematics portal

* Bousfield localization
* Homotopy hypothesis – Hypothesis that the ∞-groupoids are equivalent to the topological spaces
* ∞-groupoid – Abstract homotopical model for topological spaces
* Simplicial set
* Kan complex – Concept in the theory of simplicial sets.

## References

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1. ^ Lurie 2009, Definition 6.1.0.4.
2. ^ Lurie 2009, Theorem 6.1.0.6.

## Further reading

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* Spectral Algebraic Geometry \- Charles Rezk (gives a down-enough-to-earth introduction)
* Lurie, Jacob (2009). Higher Topos Theory (PDF). Princeton University Press. arXiv:math/0608040. ISBN 978-0-691-14049-0.

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