∞-topos

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

• 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 – Map between simplicial sets with lifting propertyPages displaying short descriptions of redirect targets

References

Further reading

• 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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Category theory

Key concepts

Key concepts Category Abelian Additive Concrete Pre-abelian Preadditive Bicategory Adjoint functors CCC Commutative diagram End Exponential Functor Kan extension Morphism Natural transformation Universal property Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations Magmas Groups Abelian groups Rings (Fields) Modules (Vector spaces) Constructions on categories Free category Functor category Kleisli category Opposite category Quotient category Product category Comma category Subcategory

Higher category theory

Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories Weak n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category n-group n-monoid

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