In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Abelian category - Definitions. A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Pe ...

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Abelian category - Definitions Abelian category - Examples Abelian category - Elementary properties Abelian category - Related concepts Abelian category - History

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In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. The zero object of Ab is the trivial group {0} which consists only of its neutral element. Note that Ab is a full subcategory of Grp, the category of all ...

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

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Sheaf mathematics - Introduction Sheaf mathematics - The formal definition
Sheaf mathematics - Definition of a presheaf Sheaf mathematics - The gluing axiom
Sheaf mathematics - Examples Sheaf mathematics - Morphisms of sheaves Sheaf mathematics - Stalks of a sheaf at a point and germs of functions Sheaf mathematics - The étale space of a sheaf Sheaf mathematics - Generalizations Sheaf mathematics - History

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In mathematics, the term abelian is used in many different definitions: Abelian - In group theory. An abelian group is a group in which the binary operation is commutative. The category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms. A metabelian group is a group where the commutator subgroup is contained in the center. Any group is "made abelian" by its abelianisation. Abelian - In Galois theoryIncluding:

Abelian - In group theory Abelian - In Galois theory Abelian - In real analysis Abelian - In functional analysis Abelian - In topology and number theory Abelian - In category theory

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In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. In other words, the order of elements in a product doesn't matter. Such groups are generally easier to understand. Abelian groups are named after Niels Henrik Abel. Groups that are not commutative are called non-abelian (rather than non-commutative). Abelian group - NotationIncluding:

Abelian group - Notation Abelian group - Examples Abelian group - Multiplication table Abelian group - Properties Abelian group - Finite abelian groups Abelian group - List of small abelian groups Abelian group - Relation to other mathematical topics Abelian group - A note on the typography

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There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. An early source occurs in the problem of forming an mth-degree equation having as its roots m of the roots of a given nth ...

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Group theory, Group theory - History, Group theory - Elementary introduction, Group theory - Some useful theorems, Group theory - Generalizations, Group theory - Miscellany

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Suppose C is a small category (i.e. the objects form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Funct(C,D) or DC, has as objects the covariant functors from C to D, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if μ(X) : F(X) → G(X) is a natural transformation from the functor ...

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Functor category, Functor category - Definition, Functor category - Examples, Functor category - Facts

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A morphism of ringed spaces is simply a morphism of sheaves. Explicitly, a morphism from (X, OX) to (Y, OY) is given by the following data: a continuous map f : X → Y a family of ring homomorphisms φV : OY(V) → OX(f -1(V)) for every open set V of Y which commute with the restriction maps. That is, if V1 ⊂ VSee also:

Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - OX modules

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Joseph Whitworth popularized the first practical method of making accurate flat surfaces during the 1830s, using engineer's blue and scraping techniques on three trial surfaces. By testing all three pairs against each other, it is ensured that the surfaces become flat. Using two surfaces would result in a concave surface and a convex surface. Eventually a point is reached when many points of contact are visible within each square inch, at which time the ...

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Flatness, Flatness - Flatness in mathematics, Flatness - Flatness in cosmology, Flatness - External link, Flatness - Flatness in mechanical engineering, Flatness - Reference, Flatness - External link, Flatness - Flatness in liquids

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A category C must have zero morphisms for the concept of normality to make complete sense. In that case, we say that a monomorphism is normal if it is the kernel of some morphism, and an epimorphism is normal (or conormal) if it is the cokernel of some morphism. C itself is normal if every monomorphism is normal. C is conormal if every epimorphism is normal. Finally, C is binormal if it's both normal and conormal. But note that some authors will use only the word "normal" t ...

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Normal morphism, Normal morphism - Definition, Normal morphism - Examples

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For general reasons having to do with universal properties, we can say that if a Lie algebra has a universal enveloping algebra, then this enveloping algebra is uniquely determined by L (up to a unique algebra isomorphism). By the following construction, which suggests itself on general grounds (for instance, as part of a pair of adjoint functors), we establish that indeed every Lie algebra does have a universal enveloping algebra. Starting with the tensor algebra T(L) on the vector space underlying LSee also:

Universal enveloping algebra, Universal enveloping algebra - Universal property, Universal enveloping algebra - Direct construction, Universal enveloping algebra - Examples in particular cases, Universal enveloping algebra - Further description of structure

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Yoneda lemma - General version. Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set. If C is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object A of C induces a natural functor to Set called a hom-functor. This functor is denoted: The hom-functor hA sends XSee also:

Yoneda lemma, Yoneda lemma - Generalities, Yoneda lemma - Formal statement, Yoneda lemma - General version, Yoneda lemma - Proof, Yoneda lemma - The Yoneda embedding, Yoneda lemma - Preadditive categories rings and modules

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In topological language, we try to find covers of unfamiliar objects. Finding a generator of an abelian category allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogene ...

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Injective cogenerator, Injective cogenerator - The abelian group case, Injective cogenerator - General theory, Injective cogenerator - In general topology

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We define the kernel of h to be ker(h) = { u in G : h(u) = eH } and the image of h to be im(h) = { h(u) : u in G }. The kernel is a normal subgroup of G (in fact, h(g-1 u g) = h(g)-1 h(u) h(g) = h(g)-1 eH h(g) = h(g ...

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Group homomorphism, Group homomorphism - Image and kernel, Group homomorphism - Examples, Group homomorphism - The category of groups, Group homomorphism - Isomorphisms endomorphisms and automorphisms, Group homomorphism - Homomorphisms of abelian groups

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An injective sheaf F is just a sheaf that is an injective element of the category of abelian sheaves; in other words, homomorphisms from A to F can always be lifted to any sheaf B containing A. The category of abelian sheaves has enough injective elements: this means that any sheaf is a subsheaf of an injective sheaf. This result of Grothendieck follows from the existence of a generator of the category (it can be written down explicitly, and is related to the subobject classifier). This is enough to show that right derived functors of any left exact func ...

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Injective sheaf, Injective sheaf - Injective sheaves, Injective sheaf - Fine sheaves, Injective sheaf - Soft sheaves, Injective sheaf - Flasque or flabby sheaves, Injective sheaf - Acyclic sheaves

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One can show that a functor F : C -> D yields an equivalence of categories if and only if has all of the following three properties: full, i.e. for any two objects c1 and c2 of C, the map MorC(c1,c2) -> MorD(Fc1,Fc2) induced by F is surjective; faithful, i.e. for any two objects c1 and c_{See also:}

Equivalence of categories, Equivalence of categories - Definition, Equivalence of categories - Equivalent characterizations, Equivalence of categories - Examples, Equivalence of categories - Properties

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To construct the Grothendieck group of a commutative monoid M, one forms the Cartesian product M×M. The two coordinates are meant to represent a positive part and a negative part: (m, n) is meant to correspond to m − n. Addition is defined coordinate-wise: (m1, m2) + (n1, n2) = (m1 + n1, m ...

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Grothendieck group, Grothendieck group - Explicit construction, Grothendieck group - Generalization, Grothendieck group - Splitting principle, Grothendieck group - Example

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The first version of sheaf cohomology to be defined was that based on Čech cohomology, in which the relatively small change was made of attributing to an open set U of a topological space X an element in F(U), an abelian group that 'varies' with U, rather than an abelian group A that is fixed ahead of time. This means that cochains are easy to write down rather concretely; in fact the model applications, such as the Cousin problems on meromorphic functions, stay within fairly familiar mathematical t ...

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Sheaf cohomology, Sheaf cohomology - Definitions, Sheaf cohomology - Applications, Sheaf cohomology - Euler characteristics

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In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicate ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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