View source for ∞-topos

In [[Mathematics|mathematics]], an '''∞-topos''' is, roughly, an ∞-category such that its objects behave like [[Sheaf (mathematics)|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 notion is due to [[Biography:Alexander Grothendieck|Alexander Grothendieck]], who considered ''n''-topos.--> 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 (mathematics)|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<ref>{{harvnb|Lurie|2009|loc=Definition 6.1.0.4.}}</ref> 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<ref>{{harvnb|Lurie|2009|loc=Theorem 6.1.0.6.}}</ref> 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|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 ==
*{{annotated link|Homotopy hypothesis}}
*{{annotated link|∞-groupoid}}
*[[Simplicial set]]

== References ==
{{reflist}}

== Further reading ==
*[https://web.archive.org/web/20200425203513/https://faculty.math.illinois.edu/~rezk/sag-chapter.pdf Spectral Algebraic Geometry] - Charles Rezk (gives a down-enough-to-earth introduction)
*{{cite book |last1=Lurie |first1=Jacob |year=2009 |title=Higher Topos Theory |url=https://www.math.ias.edu/~lurie/papers/HTT.pdf |publisher=Princeton University Press |arxiv=math/0608040 |isbn=978-0-691-14049-0 }}

{{Category theory}}

[[Category:Foundations of mathematics]]
[[Category:Higher category theory]]
[[Category:Sheaf theory]]
[[Category:Topos theory]]



{{Sourceattribution|∞-topos}}