In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an ∞-topos (infinity-
topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...
) is, roughly, an
∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ...
such that its objects behave like
sheaves of spaces with some choice of
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
; 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
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. 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 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
states that an ∞-category is an ∞-topos if and only if it satisfies an ∞-categorical version of Giraud's axioms in
ordinary topos
Ordinary or The Ordinary often refer to:
Music
* ''Ordinary'' (EP) (2015), by South Korean group Beast
* ''Ordinary'' (album) (2011), by Every Little Thing
* "Ordinary" (Alex Warren song) (2025)
* "Ordinary" (Two Door Cinema Club song) (2016 ...
theory. A "
topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...
" 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
It says:
See also
*
Bousfield localization In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences.
Bousfield localization is named a ...
*
*
*
Simplicial set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs.
Every simplicial set gives rise to a "n ...
*
References
Further reading
Spectral Algebraic Geometry- Charles Rezk (gives a down-enough-to-earth introduction)
*
{{Foundations-footer
Foundations of mathematics
Higher category theory
Sheaf theory
Topos theory