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.
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Emily Riehl: On the ∞topos semantics of homotopy type theory: All ∞toposes have...  Lecture 3

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Taste of topology: Open Sets

Show that (0, 1] is not compact  Topology  Compact sets
Transcription
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 property
References
 ^ Lurie 2009, Definition 6.1.0.4.
 ^ Lurie 2009, Theorem 6.1.0.6.
Further reading
 Spectral Algebraic Geometry  Charles Rezk (gives a downenoughtoearth introduction)
 Lurie, Jacob (2009). Higher Topos Theory (PDF). Princeton University Press. arXiv:math/0608040. ISBN 9780691140490.