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Gabriel–Popescu Theorem
In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian category, abelian categories, introduced by . It characterizes certain abelian categories (the Grothendieck category, Grothendieck categories) as Quotient of an abelian category, quotients of module categories. There are several generalizations and variations of the Gabriel–Popescu theorem, given by (for an AB5 category with a set of Generator (category theory), generators), , (for triangulated categories). Theorem Let ''A'' be a Grothendieck category (an AB5 category with a generator), ''G'' a generator of ''A'' and ''R'' be the endomorphism ring, ring of endomorphisms of ''G''; also, let ''S'' be the functor from ''A'' to Mod-''R'' (the category of right ''R''-modules) defined by ''S''(''X'') = Hom(''G'',''X''). Then the Gabriel–Popescu theorem states that ''S'' is full functor, full and faithful functor, faithful and has an exact functor, exact adjoint functor, left adjoint. This impl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
