β-topos
In mathematics, an β-topos is, roughly, an β-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 β-topos is the β-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the β-category of Γ©tale sheaves on some scheme is not the β-category of sheaves on any topological space but it is still an β-topos.
Precisely, in Lurie's Higher Topos Theory, an β-topos is defined as an β-category X such that there is a small β-category C and a left exact localization functor from the β-category of presheaves of spaces on C to X. A theorem of Lurie states that an β-category is an β-topos if and only if it satisfies an β-categorical version of Giraudβs 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 β-topos says that an β-topos is an β-category behaving like the category of sheaves of spaces.
