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Injective Cogenerator
In category theory, a branch of mathematics, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation. More precisely: * A generator of a category with a zero object is an object ''G'' such that for every nonzero object H there exists a nonzero morphism f:''G'' → ''H''. * A cogenerator is an object ''C'' such that for every nonzero object ''H'' there exists a nonzero morphism f:''H'' → ''C''. (Note the reversed order). The abelian group case 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 gro ... [...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] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tietze Extension Theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary. Formal statement If X is a normal space and f : A \to \R is a continuous map from a closed subset A of X into the real numbers \R carrying the standard topology, then there exists a of f to X; that is, there exists a map F : X \to \R continuous on all of X with F(a) = f(a) for all a \in A. Moreover, F may be chosen such that \sup \ ~=~ \sup \, that is, if f is bounded then F may be chosen to be bounded (with the same bound as f). History L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when X is a finite-dimensional real vector space. Heinrich Tietze extended it to all metric spaces, and Pavel Urysohn proved the theorem as stated here, for normal topological spaces. Equivalent statements This the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. The circle group forms a subgroup of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure \theta: \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Injective Module
In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding. Definitions Let be a ring, and a left -module. Consider a system of infinitely many linear equations :\sum_ r_x_j = m_i, where both sets and may be infinite, m_i\in M, and for each the number of nonzero r_\in R is finite. The goal is to decide whether such a system has a ''solution'', that is whether there exist elements of such that all the equations of the system are simultaneously satisfied. (It is not required th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faithful Functor
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor. Formal definitions Explicitly, let ''C'' and ''D'' be (locally small) categories and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function :F_\colon\mathrm_(X,Y)\rightarrow\mathrm_(F(X),F(Y)) for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be *faithful if ''F''''X'',''Y'' is injectiveJacobson (2009), p. 22 *full if ''F''''X'',''Y'' is surjectiveMac Lane (1971), p. 14 *fully faithful (= full and faithful) if ''F''''X'',''Y'' is bijective for each ''X'' and ''Y'' in ''C''. A mnemonic for remembering the term "full" is that the image of the function fills the codomain; a mnemonic for remembering the term "faithful" is that you can trust (have faith) that F(X)=F(Y) implies X=Y. Properties A faithful functor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Module
In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective. Definition The group (\mathbb/\mathbb, +), the group of rational numbers modulo 1, can be considered as a \mathbb-module in the natural way. Let M be an additive group which is also considered as a \mathbb-module. Then the group M^* = \operatorname_\mathbb (M, \mathbb / \mathbb) of \mathbb-homomorphisms from M to \mathbb / \mathbb is called the ''character group associated to M''. The elements in this group are called ''characters''. If M is a left R-module over a ring R, then the character group M^* is a right R-module and called the ''character module associated to'' M. The module action in the character module for f \in \opera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Category
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 desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a -adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
