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Coherent Topos
In mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products.Jacob LurieCategorical Logic (278x) Lecture 11. Definition 6. See also *spectral space In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Definition Let ''X'' be a topological ... References External links *https://ncatlab.org/nlab/show/coherent+topos {{math-stub Topos theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the 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 formaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Space
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Definition Let ''X'' be a topological space and let ''K''\circ(''X'') be the set of all compact open subsets of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions: *''X'' is compact and T0. * ''K''\circ(''X'') is a basis of open subsets of ''X''. * ''K''\circ(''X'') is closed under finite intersections. * ''X'' is sober, i.e., every nonempty irreducible closed subset of ''X'' has a (necessarily unique) generic point. Equivalent descriptions Let ''X'' be a topological space. Each of the following properties are equivalent to the property of ''X'' being spectral: #''X'' is homeomorphic to a projective limit of finite T0-spaces. #''X'' is homeomorphic to the spectrum of a bounded distributive lattice ''L''. In this case, ''L'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
