|
β-topos
In mathematics, an β-topos (infinity-topos) is, roughly, an β-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 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 (mathematics), 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 an (accessible β-category, accessible) left exact localization of an β-category, localization functor from the β-category of presheaf of spaces, presheaves of spaces on ''C'' to ''X''. A theorem of Lurie states that an β-category is an β ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this progra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
