If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x + ... + x (n summands) and (−n)x = −(nx). In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups. Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classifi ...

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Abelian group, 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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A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b where a and b are in ob(C). We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or homC(a, b)) to denote the hom-class of all morphisms from a to b. (Some a ...

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Category mathematics, Category mathematics - Definition, Category mathematics - Examples, Category mathematics - Types of morphisms, Category mathematics - Types of categories

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Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (that is, depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as op ...

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Sheaf mathematics, 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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It is possible to define a cohomology theory for sheaves of abelian groups (sheaf cohomology) that can give much useful, more concrete information. The main issue is the existence of the long exact sequence coming from an exact sequence of sheaves. In applications emphasis was placed on sheaves on spaces that were less well-behaved than finite complexes. For example, in algebraic geometry spaces carry ...

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Sheaf mathematics, 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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To apply the Grothendieck group to purely algebraic settings, it is useful to generalize it to the case of an essentially small abelian category. To do this, let be an essentially small abelian category. Let F be the free abelian group generated by isomorphism classes of objects of the category. (This is where the hypothesis of essential smallness is necessary; without it, F would not be a set.) We will impose some relations on F. Call R the subgroup of F generated as follows: For each exact sequence 0→A→B→C→0 in , the element ...

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

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In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E together with a continuous map from E to X. More precisely: to every sheaf F of sets on X there exists a local homeomorphism π: E → X such that F is isomorphic to the sheaf o ...

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Sheaf mathematics, 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 cosmology, the concept of "curvature of space" is considered. A space without curvature is called a "flat space" or Euclidean space. A question often asked is "is the Universe flat"? According to Albert Einstein's theory of relativity, it probably is curved and warped due to gravity. See also. Curvature Flatness criterion Shape of the universe Flatness - External link. http://archive.ncsa.u ...

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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 soft sheaf F over X is one such that any section over any closed subset of X can be extended to a global section. Soft sheaves are acyclic over paracompact Hausdorff spaces. ...

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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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A flasque sheaf (also called a flabby sheaf) is a sheaf with the following property: if X is the base topological space on which the sheaf is defined and are open subsets, then the restriction map is surjective, as a map of groups (rings, modules, etc.). Flasque sheaves are useful because (by definition) sections of them extend. This means that they are some of the simplest sheaves to handle in terms of homological algebra. Any s ...

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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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The flatness of a surface is the degree to which it approximates a mathematical plane. The term is generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. See curvature. Flatness in homological algebra and algebraic geometry means, of an object A in an abelian category, that is an exact functor. See ...

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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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The relationship between K0 of a commutative monoid and K0 of an abelian category comes from the splitting principle. According to the splitting principle, we can always take an exact sequence 0→A→B→C→0 and find a closely related exact sequence in which the middle term splits, that is, it is the direct sum of the other two terms. Because of this, the Grothendieck group of the commutative monoid of vector bundles on a smooth manifold is the same as the Grothendieck group of the abelia ...

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

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If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; in this case, the groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes. If h: G -> G is a group homomorphism, we call it an endomorphism of G. If furthermore it is bijective and hence an isomorphism, it is called an automorphis ...

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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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To define sheaves we will proceed in two steps. The first step is to introduce the concept of a presheaf, which captures the idea of associating local information to a topological space. The second step is to introduce an additional axiom, called the gluing axiom or the sheaf axiom, which captures the idea of gluing local information to get global information. See also:

Sheaf mathematics, 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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If L is abelian (that is, the bracket is always 0), then U(L) is commutative; if a basis of the vector space L has been chosen, then U(L) can be identified with the polynomial algebra over K, with one variable per basis element. If L is the Lie algebra corresponding to the Lie group G, U(L) can be identified with the algebra of left-invariant differential operators (of all orders) on G; with L lying inside ...

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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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The Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (the category of sets with functions as morphisms). Set is the category we understand best, and a functor of C into Set can be seen as a "representation" of C in terms of known structures. The original category C is contained in this functor category, but new objects appear in the functor category which were absent and "hidden" in C. Treating these new objects just like the old ...

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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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Let L be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism h: L → UL, (notation as above) we say that U is the universal enveloping algebra of L if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: L → AL there exists a unique unital algebra homomorphism g: U → ASee 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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In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sections are very important examples. Suppose E and X are topological spaces and π : E → X is a continuous map. For every open set U in X, let F(U) be the set all continuous maps f : U → E such that π(f(x)) = x for all x in U. Such a function f is called a section of π. ...

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Sheaf mathematics, 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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A fine sheaf over X is one with "partitions of unity"; more precisely for any open cover of the space X we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover. Fine sheaves are usually only used over paracompact Hausdorff spaces X. Typical examples are the sheaf of continuous real functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings. Fine sheaves over pa ...

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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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Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of G until the morphism f:Sum(G) -> H is surjective; and one can form direct products of C until the morphism f:H -> Prod(C) is injective. For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group). This is the origin of the term generator. The approximatio ...

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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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Let F and G be two sheaves on X both with values in the category C. We define a morphism from G to F to be a family of morphisms φU : G(U) → F(U) in the category C for all opens U in X which commute with the restriction maps. That is, the following diagram must commute for each pair of open sets U ⊆ V in X. If F and G are considered as contravariant functors from TopXSee also:

Sheaf mathematics, 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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Fix a point x of X. We would like to study the behavior of F near the point x. In analytical terms, we would like to somehow take the limit as we get nearer and nearer to the point x. The corresponding concept is to take the direct limit of F(N) as N runs over the open neighbourhoods of x ordered by inclusion (in categorical terminology, this is an example of a colimit). We denote this limit by Fx and call it the stalk of F at x. If F ...

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Sheaf mathematics, 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

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - Stalks of a sheaf at a point and germs of functions