isomorphism

A category C is given by two pieces of data: a class of objects and a class of morphisms. There are two operations defined on every morphism, the domain (or source) and the codomain (or target). Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. The set of all morphisms from X to Y is denoted homC(X ...

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Morphism, Morphism - Definition, Morphism - Types of morphisms, Morphism - Examples

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Before giving a precise definition of the fundamental group, we try to describe the general idea in non-mathematical terms. Take some space, and some point in it, and consider all the loops at this point -- paths which start at this point, wander around as much they like and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second. The set of all the loops with this method of combining them is the fundamental group, except that for technical reasons it is necessary to consider two loops to be th ...

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Fundamental group, Fundamental group - Intuition and definition, Fundamental group - Examples, Fundamental group - Functoriality, Fundamental group - Relationship to first homology group, Fundamental group - Related concepts, Fundamental group - Fundamental groupoid

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The Catholic Church for some time now, especially with Pope Benedict XVI, head of the Congregation of the Doctrine of the Faith when he was a cardinal, has identified relativism as one of the problems of today. [1] According to the Church and some philosophers, relativism, as a denial of absolute truth, leads to moral license and a denial of the possibility of sin and of God. Relativism, they say, is a denial of the capacity of our mind and reason to arrive at truth. Truth, according to Catholic theologians and philosophers, fo ...

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Relativism, Relativism - Advocates of relativism, Relativism - Arguments against relativism, Relativism - Counter-arguments, Relativism - The Catholic Church and relativism, Relativism - John Paul II, Relativism - Benedict XVI, Relativism - See Also

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In the category of sets the pullback of f and g is the set X ×Z Y = {(x, y) ∈ X × Y | f(x) = g(y)}, together with the restrictions of the projection maps π1 and π2 to X ×Z Y . This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f o ...

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Pullback category theory, Pullback category theory - Universal property, Pullback category theory - Examples

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In the eight queens puzzle, if the eight queens are considered to be distinct, there are 3 709 440 distinct solutions. Normally however, the queens are considered to be identical, and one says "there are 92 (= 3709440/8!) unique solutions up to permutations of the queens," signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, but the same squares on th ...

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Up to, Up to - Examples

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The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive iff it is isomorphic to a lattice of sets (closed under set union and intersection). It is easy to check that set union and intersection are indeed distributive in the above sense. The other direction is less trivial -- it will follow from the representation theorems mentioned below. The important insight from this characterization is that the identities (equations) that hold in all distributive lattices are exactly ...

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Distributive lattice, Distributive lattice - Formal definition, Distributive lattice - Morphisms, Distributive lattice - Examples, Distributive lattice - Characteristic properties, Distributive lattice - Representation theory, Distributive lattice - Free distributive lattices, Distributive lattice - Literature

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Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case: The characteristic constant can be expressed in terms of its logarithm: When expressed in this way, the real number ρ becomes an expansion factor. It indicates how repulsive the fixed point γ1 is, and how attractive γ2 i ...

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Möbius transformation, Möbius transformation - Overview, Möbius transformation - Definition, Möbius transformation - Projective matrix representations, Möbius transformation - Properties, Möbius transformation - Classification, Möbius transformation - Fixed points, Möbius transformation - Normal form, Möbius transformation - Geometric interpretation of the characteristic constant, Möbius transformation - Elliptic transformations, Möbius transformation - Hyperbolic transformations, Möbius transformation - Loxodromic transformations, Möbius transformation - Stereographic projection, Möbius transformation - Iterating a transformation, Möbius transformation - Poles of the transformation, Möbius transformation - Specifying a transformation by three points

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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, 0XY) To be more explicit, the following universal property can be used. A kernel of f is any morphism k : See also:

Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

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Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects. The automorphism group of an initial (or terminal) object I is trivial. Aut(I) = Hom(I,I) = { idI }. ...

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Initial object, Initial object - Properties, Initial object - Examples

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Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as homomorphisms. For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). Note that f(x) = 3x is a homomorphism, since f(a + b< ...

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Homomorphism, Homomorphism - Informal discussion, Homomorphism - Formal definition, Homomorphism - Types of homomorphisms, Homomorphism - Kernel of a homomorphism

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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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Inverse limit - Algebraic objects. We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (Ai)i∈I be a family of groups and suppose we have a family of homomorphisms fij : Aj → Ai for all i ≤ j (note the order) with the following properties:< ...

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Inverse limit, Inverse limit - Formal definition, Inverse limit - Algebraic objects, Inverse limit - General definition, Inverse limit - Examples, Inverse limit - Related concepts and generalizations

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As is true for any notion of isomorphism, we have the following general properties formally similar to an equivalence relation: any category C is isomorphic to itself if C is isomorphic to D, then D is isomorphic to C if C is isomorphic to D and D is isomorphic to E, then C is isomorphic to E. A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets. This criterion can be convenient as it av ...

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Isomorphism of categories, Isomorphism of categories - Properties, Isomorphism of categories - Examples

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It is natural to ask, for which classes of functions and for which interpolation nodes the sequence of interpolating polynomials converges to the interpolated function? Convergence may be understood in different ways, e.g. pointwise, uniform or in some integral norm. The aspects of uniform convergence are discussed below. The following theorem seems to be a rather encouraging answer: For any function f(x) continuous on an interval [a,b] there exists a table of nodes for which the sequenc ...

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Polynomial interpolation, Polynomial interpolation - Applications, Polynomial interpolation - Definition, Polynomial interpolation - Constructing the interpolation polynomial, Polynomial interpolation - Non-Vandermonde solutions, Polynomial interpolation - Interpolation error, Polynomial interpolation - Lebesgue constants, Polynomial interpolation - Convergence properties, Polynomial interpolation - Related concepts

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Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx: The defining properties of a group guarantee that the set of orbits of X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ~ y iff there exists a g in G with g·x< ...

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Group action, Group action - Definition, Group action - Examples, Group action - Types of actions, Group action - Orbits and stabilizers, Group action - Morphisms and isomorphisms between G-sets, Group action - Continuous group actions, Group action - Strongly continuous group action and smooth vector, Group action - Generalizations

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Complex number - The complex number field. Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations: So defined, the complex numbers form a field, the complex number field, denoted by C. We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginar ...

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Complex number, Complex number - Definition, Complex number - The complex number field, Complex number - The complex plane, Complex number - Absolute value conjugation and distance, Complex number - Complex number division, Complex number - Matrix representation of complex numbers, Complex number - Geometric interpretation of the operations on complex numbers, Complex number - Some properties, Complex number - Real vector space, Complex number - Solutions of polynomial equations, Complex number - Algebraic characterization, Complex number - Characterization as a topological field, Complex number - Complex analysis, Complex number - Applications, Complex number - Control theory, Complex number - Signal analysis, Complex number - Improper integrals, Complex number - Quantum mechanics, Complex number - Relativity, Complex number - Applied mathematics, Complex number - Fluid dynamics, Complex number - Fractals, Complex number - History

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There are two matroids associated with a signed graph, called the signed-graphic matroid (or the frame matroid or bias matroid) and the lift matroid, both of which generalize the cycle matroid of a graph. They are special cases of the same matroids of a biased graph. The signed-graphic matroid M(G) (Zaslavsky, 1982) has for its ground set the edge set E. An edge set is independent if each component contains either no circles or just one circle, which is negative. (In matroid theo ...

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Signed graph, Signed graph - Examples, Signed graph - Adjacency matrix, Signed graph - Orientation, Signed graph - Incidence matrix, Signed graph - Switching, Signed graph - Fundamental theorem, Signed graph - Matroid theory, Signed graph - Other kinds of signed graph, Signed graph - Generalizations

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Specifically, let (M1, ω1) and (M2, ω2) by symplectic manifolds. A map f : M1 → M2 is a symplectomorphism if it is a diffeomorphism and the pullback of ω2 under f is equal to ω1: Examples of symplectomorphisms are the canonical transformations of classical mechanics and theoretical physics. A Hamiltonian symplectomorphism is a symplectomorphism that arises as the flow of a Hamiltonian ...

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Symplectomorphism, Symplectomorphism - Formal definition, Symplectomorphism - Flows, Symplectomorphism - Comparison with Riemannian geometry, Symplectomorphism - Quantizations, Symplectomorphism - The group of Hamiltonian symplectomorphisms, Symplectomorphism - Arnold conjecture

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The original form of the theorem, contained in a third-century book by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao, is a statement about simultaneous congruences (see modular arithmetic). Suppose n1, ..., nk are positive integers which are pairwise coprime (meaning gcd (ni, nj) = 1 whenever i ≠ j). Then, for any given integers a1, ..., ak, there exists an in ...

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Chinese remainder theorem, Chinese remainder theorem - Simultaneous congruences of integers, Chinese remainder theorem - Statement for principal ideal domains, Chinese remainder theorem - Statement for general rings, Chinese remainder theorem - Applications of the Chinese remainder theorem

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The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is: Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I is contained in some prime ideal of B that is disjoint from F. We refer to this statement as BPI. This situation can be expressed ...

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Boolean prime ideal theorem, Boolean prime ideal theorem - Prime ideal theorems, Boolean prime ideal theorem - Boolean prime ideal theorem, Boolean prime ideal theorem - Further prime ideal theorems, Boolean prime ideal theorem - Applications, Boolean prime ideal theorem - Literature

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Model example: if U and V are two open subsets of , a differentiable map f from U to V is a diffeomorphism if it is a bijection, its differential df is invertible (as the matrix of all , ), which means the ...

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Diffeomorphism, Diffeomorphism - Local description, Diffeomorphism - Diffeomorphism group, Diffeomorphism - Homeomorphism and diffeomorphism

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