|
Gabriel–Popescu Theorem
In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by . It characterizes certain abelian categories (the Grothendieck categories) as 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 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 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 and faithful and has an exact left adjoint. This implies that ''A'' is equivalent to the Serre quotient category of Mod-''R'' by a certain localizing subcategory ''C''. (A localizing subcategory of Mod-''R'' is a full subcategor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Journal Of Pure And Applied Algebra
The ''Journal of Pure and Applied Algebra'' is a monthly peer-reviewed scientific journal covering that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications. Its founding editors-in-chief were Peter J. Freyd (University of Pennsylvania) and Alex Heller (City University of New York). The current managing editors are Eric Friedlander (University of Southern California), Charles Weibel (Rutgers University), and Srikanth Iyengar (University of Utah). Abstracting and indexing The journal is abstracted and indexed in Current Contents/Physics, Chemical, & Earth Sciences, Mathematical Reviews, PASCAL, Science Citation Index, Zentralblatt MATH, and Scopus. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left Exact Functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that ''fail'' to be exact, but in ways that can still be controlled. Definitions Let P and Q be abelian categories, and let be a covariant additive functor (so that, in particular, ''F''(0) = 0). We say that ''F'' is an exact functor if whenever :0 \to A\ \stackrel \ B\ \stackrel \ C \to 0 is a short exact sequence in P then :0 \to F(A) \ \stackrel \ F(B)\ \stackrel \ F(C) \to 0 is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→''A''→''B''→''C''→0 is exact, then 0→''F''(''A'')→''F''(''B'')→''F''(''C'')→0 is also exact".) Further, we say that ''F'' is *left-exact if whenever 0→''A''→''B''↠... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (category Theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism ''f'' : ''X'' → ''Y'' is the "most general" morphism ''k'' : ''K'' → ''X'' that yields zero when composed with (followed by) ''f''. Note that kernel pairs and difference kernels (also known as binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article. Definition Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary morphism in C, then a kernel of ''f'' is an equaliser of ''f'' and the zero morphism from ''X'' to ''Y''. In symbols: :ker(''f'') = eq(''f'', 0''XY'') To be more explicit, the following universal pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localizing Subcategory
In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category. Serre subcategories Let \mathcal be an abelian category. A non-empty full subcategory \mathcal is called a ''Serre subcategory'' (or also a ''dense subcategory''), if for every short exact sequence 0\rightarrow A' \rightarrow A\rightarrow A''\rightarrow 0 in \mathcal the object A is in \mathcal if and only if the objects A' and A'' belong to \mathcal. In words: \mathcal is closed under subobjects, quotient objects and extensions. Each Serre subcategory \mathcal of \mathcal is itself an abelian category, and the inclusion functor \mathcal\to\mathcal is exact. The importance of this notion stems from the fact that kernels of exact functors between abelian categories are Serre subcategories, and that one can build (for locally small \mathcal) the q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> American Journal Of Mathematics")
