morphism

The first origins of sheaf theory are hard to pin down — they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology. 1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering. 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochainsSee 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

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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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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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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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The success in the early nineteenth century of the theory of elliptic functions in giving a basis for the theory of elliptic integrals left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions of two complex variables, having four independent periods (i.e. p ...

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Abelian variety, Abelian variety - History and motivation, Abelian variety - Analytic theory, Abelian variety - Definition, Abelian variety - Riemann conditions, Abelian variety - The Jacobian of an algebraic curve, Abelian variety - Abelian functions, Abelian variety - Algebraic definition, Abelian variety - Structure of the group of points, Abelian variety - Polarization and dual abelian variety, Abelian variety - Dual abelian variety, Abelian variety - Polarizations, Abelian variety - Polarizations over the complex numbers, Abelian variety - Abelian scheme

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Two equivalent definitions of abelian variety over a general field are commonly in use: a connected and complete algebraic group over k a connected and projective algebraic group over k When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, elliptic curves are abelian varieties of dimension 1. In the early 40s, Weil used the former definition but could not at first prove that it implied the second. Only in 1948 did he prove that com ...

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Abelian variety, Abelian variety - History and motivation, Abelian variety - Analytic theory, Abelian variety - Definition, Abelian variety - Riemann conditions, Abelian variety - The Jacobian of an algebraic curve, Abelian variety - Abelian functions, Abelian variety - Algebraic definition, Abelian variety - Structure of the group of points, Abelian variety - Polarization and dual abelian variety, Abelian variety - Dual abelian variety, Abelian variety - Polarizations, Abelian variety - Polarizations over the complex numbers, Abelian variety - Abelian scheme

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A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms: Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) Identity element: there exists an element e in M, such that for all a in M, a*e = e*a = a. One often sees the additional axiom Closure: for all a, b in M, a*b is in M though, strictly speaking, this isn't necessary as it is implied by the ...

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Monoid, Monoid - Definition, Monoid - Examples, Monoid - Properties, Monoid - Monoid homomorphisms, Monoid - Relation to category theory

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A homomorphism between two monoids (M, *) and (M′, @) is a function f : M → M′ such that f(x*y) = f(x)@f(y) for all x, y in M f(e) = e′ where e and e′ are the identities on M and M′ respectively. Not every magma homomorphism is a monoid homomorphism since it may not preserve the identity. Contrast this with the case of group homomo ...

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Monoid, Monoid - Definition, Monoid - Examples, Monoid - Properties, Monoid - Monoid homomorphisms, Monoid - Relation to category theory

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Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Stanislaw Ulam, that comparable ideas were current in the later 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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A morphism f : a → b is called a monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : x → a. an epimorphism (or epic) if g1f = g2f implies g1 = g2 for all morphisms g1, g2 : b → x.See also:

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

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Alexander Grothendieck - Childhood and studies. Born to a Russian Jewish father and German Protestant mother in Berlin, he was a displaced person during much of his childhood due to the upheavals of World War II. Alexander lived with his father, Alexander Shapiro, and his mother, Hanka Grothendieck, both of whom were socialist revolutionaries. Until 1933 they lived together in Berlin. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Alexander with a family in Hambur ...

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Alexander Grothendieck, Alexander Grothendieck - Mathematical achievements, Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles, Alexander Grothendieck - Life, Alexander Grothendieck - Childhood and studies, Alexander Grothendieck - Politics and retreat from scientific community, Alexander Grothendieck - Manuscripts written in the 1980s, Alexander Grothendieck - Disappearance

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By the definitions, an abelian variety is a group variety. Its group of points can be proven to be commutative. By the Lefschetz principle, for every algebraically closed field of characteristic zero (and in particular, for C), the torsion group of an abelian variety of dimension g is isomorphic to (Q/Z)2g. Hence, its n-torsion part is isomorphic to (Z/nZ)2g, i.e. the product of 2g copies of the cyclic group of order n. Its free part is uncountable, since eve ...

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Abelian variety, Abelian variety - History and motivation, Abelian variety - Analytic theory, Abelian variety - Definition, Abelian variety - Riemann conditions, Abelian variety - The Jacobian of an algebraic curve, Abelian variety - Abelian functions, Abelian variety - Algebraic definition, Abelian variety - Structure of the group of points, Abelian variety - Polarization and dual abelian variety, Abelian variety - Dual abelian variety, Abelian variety - Polarizations, Abelian variety - Polarizations over the complex numbers, Abelian variety - Abelian scheme

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Abelian variety - Dual abelian variety. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field). This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms fv: Bv → Av in a ...

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Abelian variety, Abelian variety - History and motivation, Abelian variety - Analytic theory, Abelian variety - Definition, Abelian variety - Riemann conditions, Abelian variety - The Jacobian of an algebraic curve, Abelian variety - Abelian functions, Abelian variety - Algebraic definition, Abelian variety - Structure of the group of points, Abelian variety - Polarization and dual abelian variety, Abelian variety - Dual abelian variety, Abelian variety - Polarizations, Abelian variety - Polarizations over the complex numbers, Abelian variety - Abelian scheme

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The first needs of sheaf theory were for sheaves of abelian groups; so taking the category C as the category of abelian groups was only natural. In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central. This, however, is not quite the same thing; one speaks instead of a locally ringed space, because it is not true, except in trite cases, that such a sheaf is a functor into a category of local rings. It is the stalks of the sheaf that are local rings, ...

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Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

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How to turn a given presheaf P into a sheaf F? This is called sheafification, and there is a rough intuition of what one should do, at least for a presheaf of sets. One should introduce an equivalence relation, which makes equivalent data given by different covers which 'become' equivalent by refining the covers. One approach is therefore to go to the stalks and recover the sheaf space of the best poss ...

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Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

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In any metric space M we can define the open balls as the sets of the form B(x; r) = {y in M : d(x,y) < r}, where x is in M and r is a positive real number, called the radius of the ball. A subset of M which is a union of (finitely or infinitely many) open balls is called an open set. The complement of an open set is called closed. Every metric space is automatically a topolo ...

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Metric space, Metric space - History, Metric space - Definition, Metric space - Examples, Metric space - Metric spaces as topological spaces, Metric space - Boundedness and compactness, Metric space - Separation properties and extension of continuous functions, Metric space - Distance between points and sets, Metric space - Equivalence of metric spaces, Metric space - Quotient metric space

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A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. The smallest possible such r is called the diameter of M. The space M is called precompact or totally bounded if for every r > 0 there exist finitely many open balls of radius r whose union equals M. Since the set of the centres of these balls is finite, it has finite diameter, from which it follows ...

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Metric space, Metric space - History, Metric space - Definition, Metric space - Examples, Metric space - Metric spaces as topological spaces, Metric space - Boundedness and compactness, Metric space - Separation properties and extension of continuous functions, Metric space - Distance between points and sets, Metric space - Equivalence of metric spaces, Metric space - Quotient metric space

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Comparing two metric spaces one can distinguish various degrees of equivalence. To preserve at least the topological structure induced by the metric, these require at least the existence of a continuous function between them (morphism preserving the topology of the metric spaces). Given two metric spaces (M1, d1) and (M2, d2): They are called topologically isomorphic (or homeomorphic) if there exists a homeomorphism between them. See also:

Metric space, Metric space - History, Metric space - Definition, Metric space - Examples, Metric space - Metric spaces as topological spaces, Metric space - Boundedness and compactness, Metric space - Separation properties and extension of continuous functions, Metric space - Distance between points and sets, Metric space - Equivalence of metric spaces, Metric space - Quotient metric space

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In the formal definition, a function represents a relationship between its domain and its codomain, rather than just a rule for taking an input to an output. A generalization of the notion of function is morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows: (f + g)(x) = f(x) + g(x) (f × g)(x) = f(x) × < ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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Functions can be odd or even continuous or discontinuous real or complex scalar or vectorial. Function mathematics - Ambiguous functions. An ambiguous function is a mathematical equation that can have more than one correct answer. For example, the square root of 4 can be either -2 or 2 as both answers squared would give 4. Strictly speaking, an ambiguous function is not truly a function because a mathematical function is defined as having "a unique output to each given input ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Properties of functions