1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines
3.
Banach space
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In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn, Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces, the vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. All norms on a vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space, if X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T, X → Y is denoted by B. In infinite-dimensional spaces, not all maps are continuous. For Y a Banach space, the space B is a Banach space with respect to this norm, if X is a Banach space, the space B = B forms a unital Banach algebra, the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T, X → Y such that T, if one of the two spaces X or Y is complete then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, the Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ. Every normed space X can be embedded in a Banach space. More precisely, there is a Banach space Y and an isometric mapping T, X → Y such that T is dense in Y. If Z is another Banach space such that there is an isomorphism from X onto a dense subset of Z. This Banach space Y is the completion of the normed space X, the underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y. The completion of X is often denoted by X ^, the cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ∥ ∥1 = ∥ x ∥ + ∥ y ∥, ∥ ∥ ∞ = max. In this sense, the product X × Y is complete if and only if the two factors are complete. If M is a linear subspace of a normed space X, there is a natural norm on the quotient space X / M
4.
Linear map
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
5.
Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
6.
Approximation property
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In mathematics, a Banach space is said to have the approximation property, if every compact operator is a limit of finite-rank operators. Every Hilbert space has this property, there are, however, Banach spaces which do not, Per Enflo published the first counterexample in a 1973 article. However, a lot of work in area was done by Grothendieck. Later many other counterexamples were found, the space of bounded operators on ℓ2 does not have the approximation property. The spaces ℓ p for p ≠2 and c 0 have closed subspaces that do not have the approximation property, some other flavours of the AP are studied, Let X be a Banach space and let 1 ≤ λ < ∞. A Banach space is said to have bounded approximation property, if it has the λ -AP for some λ, a Banach space is said to have metric approximation property, if it is 1-AP. A Banach space is said to have compact approximation property, if in the definition of AP an operator of rank is replaced with a compact operator. Every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, every subspace of an arbitrary product of Hilbert spaces possesses the approximation property. Every separable Frechet space that contains a Schauder basis possesses the approximation property, every space with a Schauder basis has the AP, thus a lot of spaces with the AP can be found. For example, the ℓ p spaces, or the symmetric Tsirelson space, a counterexample to the approximation property in Banach spaces. Grothendieck, A. Produits tensoriels topologiques et espaces nucleaires, paul R. Halmos, Has progress in mathematics slowed down. MR1066321 William B. Johnson Complementably universal separable Banach spaces in Robert G. Bartle,1980 Studies in functional analysis, on Enflos example of a Banach space without the approximation property. Séminaire Goulaouic–Schwartz 1972—1973, Équations aux dérivées partielles et analyse fonctionnelle, mR407569 Lindenstrauss, J. Tzafriri, L. Classical Banach Spaces I, Sequence spaces,1977. Nedevski, P. Trojanski, S. P. Enflo solved in the negative Banachs problem on the existence of a basis for every separable Banach space, history of Banach spaces and linear operators. Boston, MA, Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6, karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics,2002 Springer-Verlag, New York. Schaefer, Helmuth H. Wolff, M. P. Topological Vector Spaces, editura Academiei Republicii Socialiste România, Bucharest, Springer-Verlag, Berlin-New York,1981
7.
Kernel (category theory)
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Intuitively, the kernel of the morphism f, X → Y is the most general morphism k, K → X that yields zero when composed with f. Note that kernel pairs and difference kernels sometimes go by the kernel, while related. In order to define a kernel in the general category-theoretical sense, 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 = eq To be more explicit, the universal property can be used. Note that in many contexts, one would refer to the object K as the kernel. In any case, one can show that k is always a monomorphism, one may prefer to think of the kernel as the pair rather than as simply K or k alone. Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of modules over a fixed ring. Note that in the category of monoids, category-theoretic kernels exist just as for groups, therefore, the notion of kernel studied in monoid theory is slightly different. In the category of rings, there are no kernels in the sense, indeed. Nevertheless, there is still a notion of kernel studied in ring theory that corresponds to kernels in the category of pseudo-rings. In the category of pointed spaces, if f, X → Y is a continuous pointed map, then the preimage of the distinguished point. The inclusion map of K into X is the kernel of f. The dual concept to that of kernel is that of cokernel and that is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As mentioned above, a kernel is a type of binary equaliser, conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the f and g is the kernel of the difference g − f. It is because of this fact that binary equalisers are called difference kernels, every kernel, like any other equaliser, is a monomorphism. Conversely, a monomorphism is called if it is the kernel of some morphism. A category is called normal if every monomorphism is normal, abelian categories, in particular, are always normal
8.
Cokernel
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In mathematics, the cokernel of a linear mapping of vector spaces f, X → Y is the quotient space Y/im of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f, cokernels are dual to the kernels of category theory, hence the name, the kernel is a subobject of the domain, while the cokernel is a quotient object of the codomain. This is elaborated in intuition, below, often the map q is understood, and Q itself is called the cokernel of f. In many situations in abstract algebra, such as for groups, vector spaces or modules. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient, one can define the cokernel in the general framework of category theory. In order for the definition to make sense the category in question must have zero morphisms, the cokernel of a morphism f, X → Y is defined as the coequalizer of f and the zero morphism 0XY, X → Y. The cokernel of f, X → Y is an object Q together with a q, Y → Q such that the diagram commutes. Moreover, the morphism q must be universal for this diagram, like all coequalizers, the cokernel q, Y → Q is necessarily an epimorphism. Conversely an epimorphism is called if it is the cokernel of some morphism. A category is called conormal if every epimorphism is normal, in the category of groups, the cokernel of a group homomorphism f, G → H is the quotient of H by the normal closure of the image of f. In the case of groups, since every subgroup is normal. In a preadditive category, it makes sense to add and subtract morphisms, in such a category, the coequalizer of two morphisms f and g is just the cokernel of their difference, coeq = coker . In an abelian category the image and coimage of a morphism f are given by im = ker , in particular, every abelian category is normal. That is, every monomorphism m can be written as the kernel of some morphism. Formally, one may connect the kernel and the cokernel of a map T, V → W by the exact sequence 0 → ker T → V → W → coker T →0. As a simple example, consider the map T, R2 → R2, then for an equation T = to have a solution, we must have a=0, and in that case the solution space is, or equivalently stated, +. Additionally, the cokernel can be thought of as something that detects surjections in the way that the kernel detects injections. A map is if and only if its kernel is trivial
9.
Monoidal category
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The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, monoidal categories can be seen as a generalization of these and other examples. In category theory, monoidal categories can be used to define the concept of a monoid object and they are also used in the definition of an enriched category. Monoidal categories have applications outside of category theory proper. They are used to define models for the fragment of intuitionistic linear logic. They also form the foundation for the topological order in condensed matter. Braided monoidal categories have applications in information, quantum field theory. A monoidal category is a category C equipped with a monoidal structure, every monoidal category is monoidally equivalent to a strict monoidal category. Any category with products can be regarded as monoidal with the product as the monoidal product. Such a category is called a cartesian monoidal category. For example, Set, the category of sets with the Cartesian product, Cat, the bicategory of small categories with the product category, where the category with one object and only its identity map is the unit. Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product, as special cases one has, K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit. Ab, the category of groups, with the group of integers Z serving as the unit. For any commutative ring R, the category of R-algebras is monoidal with the product of algebras as the product. The category of pointed spaces is monoidal with the smash product serving as the product, the category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit. In the case E=Cat, we get the endofunctors example above, bounded-above meet semilattices are strict symmetric monoidal categories, the product is meet and the identity is the top element. As a special case of monoidal categories, we consider monoidal preorders and this sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. We now present the general case and its well-known that a preorder can be considered as a category C, such that for every two objects c, c ′ ∈ O b, there exists at most one morphism c → c ′ in C
10.
Direct limit
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In mathematics, a direct limit is a colimit of a directed family of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition, in this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules, algebras, etc. With this in mind, homomorphisms are understood in the corresponding setting, let ⟨ I, ≤ ⟩ be a directed set. Then the pair ⟨ A i, f i j ⟩ is called a system over I. Here, if x i ∈ A i and x j ∈ A j, x i ∼ x j if there is some k ∈ I such that f i k = f j k. Heuristically, two elements in the disjoint union are equivalent if and only if they become equal in the direct system. An equivalent formulation that highlights the duality to the limit is that an element is equivalent to all its images under the maps of the directed system. One naturally obtains from this definition canonical functions ϕ i, A i → A sending each element to its equivalence class, the algebraic operations on A are defined such that these maps become morphisms. An important property is that taking direct limits in the category of modules is an exact functor, the direct limit can be defined in an arbitrary category C by means of a universal property. Let ⟨ X i, f i j ⟩ be a system of objects. A target is a pair ⟨ X, ϕ i ⟩ where X is an object in C and ϕ i, X i → X are morphisms such that ϕ i = ϕ j ∘ f i j. A direct limit is a universally repelling target in the sense that for each target ⟨ Y, ψ i ⟩, the direct limit of ⟨ X i, f i j ⟩ is often denoted lim → X i = X. Unlike for algebraic objects, the limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense and we note that a direct system in a category C admits an alternative description in terms of functors. Any directed poset ⟨ I, ≤ ⟩ can be considered as a small category I where the morphisms consist of arrows i → j if, a direct system is then just a covariant functor I → C. In this case a direct limit is a colimit, a collection of subsets M i of a set M can be partially ordered by inclusion. If the collection is directed, its limit is the union ⋃ M i. Let I be any directed set with a greatest element m, the direct limit of any corresponding direct system is isomorphic to Xm and the canonical morphism φm, Xm → X is an isomorphism
11.
Inverse limit
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In mathematics, the inverse limit is a construction that allows one to glue together several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category and we start with the definition of an inverse system of groups and homomorphisms. Then the pair is called a system of groups and morphisms over I. We define the limit of the inverse system as a particular subgroup of the direct product of the Ais. The inverse limit, A, comes equipped with natural projections πi, the inverse limit and the natural projections satisfy a universal property described in the next section. This same construction may be carried out if the Ais are sets, semigroups, topological spaces, rings, modules, algebras, etc. the inverse limit will also belong to that category. The inverse limit can be defined abstractly in a category by means of a universal property. Let be a system of objects and morphisms in a category C. The inverse limit of this system is an object X in C together with morphisms πi, the inverse limit is often denoted X = lim ← X i with the inverse system being understood. In some categories, the limit does not exist. If it does, however, it is unique in a strong sense and we note that an inverse system in a category C admits an alternative description in terms of functors. Any partially ordered set I can be considered as a category where the morphisms consist of arrows i → j if. An inverse system is then just a contravariant functor I → C, and the inverse limit functor lim ←, C I o p → C is a covariant functor. The ring of integers is the inverse limit of the rings Z/pnZ with the index set being the natural numbers with the usual order. That is, one considers sequences of such that each element of the sequence projects down to the previous ones, namely. The natural topology on the integers is the one implied here. Pro-finite groups are defined as limits of finite groups. Let the index set I of a system have a greatest element m
12.
Coproduct
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The coproduct of a family of objects is essentially the least specific object to which each object in the family admits a morphism. It is the dual notion to the categorical product, which means the definition is the same as the product. Despite this seemingly innocuous change in the name and notation, coproducts can be, let C be a category and let X1 and X2 be objects in that category. That is, the diagram commutes, The unique arrow f making this diagram commute may be denoted f1 ∐ f2 or f1 ⊕ f2 or f1 + f2 or. The morphisms i1 and i2 are called canonical injections, although they need not be injections nor even monic, the definition of a coproduct can be extended to an arbitrary family of objects indexed by a set J. That is, the diagrams commute, The coproduct of the family is often denoted X = ∐ j ∈ J X j or X = ⨁ j ∈ J X j. Sometimes the morphism f may be denoted f = ∐ j ∈ J f j, ∐ j ∈ J X j → Y to indicate its dependence on the individual fj. The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps, for example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups, in the case of topological spaces coproducts are disjoint unions with their disjoint union topologies. That is, it is a disjoint union of the sets. In the category of pointed spaces, fundamental in homotopy theory, the coproduct construction given above is actually a special case of a colimit in category theory. The coproduct in a category C can be defined as the colimit of any functor from a discrete category J into C, as with any universal property, the coproduct can be understood as a universal morphism. Let Δ, C → C×C be the functor which assigns to each object X the ordered pair. Then the coproduct X+Y in C is given by a morphism to the functor Δ from the object in C×C. The coproduct indexed by the empty set is the same as an object in C. If J is a set such that all coproducts for families indexed with J exist, the coproduct of the family is then often denoted by ∐j Xj, and the maps ij are known as the natural injections. That this map is a surjection follows from the commutativity of the diagram and that it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram, thus the contravariant hom-functor changes coproducts into products
13.
Product (category theory)
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Essentially, the product of a family of objects is the most general object which admits a morphism to each of the given objects. Let C be a category with some objects X1 and X2, the morphisms π1 and π2 are called the canonical projections or projection morphisms. Above we defined the binary product, instead of two objects we can take an arbitrary family of objects indexed by some set I. Then we obtain the definition of a product, × Xn and the product of morphisms is denoted < f1. Alternatively, the product may be defined through equations, so, for example, for the binary product, Existence of f is guaranteed by existence of the operation < -, - >. Commutativity of the diagrams above is guaranteed by the equality ∀f1, ∀f2, ∀i∈, πi o < f1, uniqueness of f is guaranteed by the equality ∀g, Y → X, < π1 o g, π2 o g > = g. The product is a case of a limit. This may be seen by using a category as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components, if we regard this diagram as a functor, it is a functor from the index set I considered as a discrete category. The definition of the product then coincides with the definition of the limit, i being a cone, just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the property of limits. The diagonal functor Δ, C → C × C assigns to each object X the ordered pair, the product X1 × X2 in C is given by a universal morphism from the functor Δ to the object in C × C. This universal morphism consists of an object X of C and a morphism → which contains projections, in the category of sets, the product is the cartesian product. Given a family of sets Xi the product is defined as Πi∈I Xi, = with the canonical projections πj, Πi∈I Xi → Xj, πj, = xj. Given any set Y with a family of functions fi, Y → Xi, other examples, In the category of topological spaces, the product is the space whose underlying set is the cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous, in the category of modules over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication. In the category of groups, the product is the product of groups given by the cartesian product with multiplication defined componentwise. In the category of relations, the product is given by the disjoint union, in the category of algebraic varieties, the categorical product is given by the Segre embedding
14.
Monoid (category theory)
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In the above notations, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop, suppose that the monoidal category C has a symmetry γ. A monoid M in C is commutative when μ o γ = μ, a monoid object in Set, the category of sets, is a monoid in the usual sense. A monoid object in Top, the category of spaces, is a topological monoid. A monoid object in the category of monoids is just a commutative monoid and this follows easily from the Eckmann–Hilton theorem. A monoid object in the category of complete join-semilattices Sup is a unital quantale, a monoid object in, the category of abelian groups, is a ring. For a commutative ring R, an object in, the category of modules over R, is an R-algebra. The category of graded modules is a graded R-algebra, the category of chain complexes is a dg-algebra. A monoid object in K-Vect, the category of spaces, is a K-algebra. For any category C, the category of its endofunctors has a monoidal structure induced by the composition, a monoid object in is a monad on C. Given two monoids and in a monoidal category C, a morphism f, M → M is a morphism of monoids when f o μ = μ o, f o η = η, in other words, the following diagrams, commute. The category of monoids in C and their morphisms is written MonC. Act-S, the category of monoids acting on sets Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories, Walter de Gruyter, Berlin ISBN 3-11-015248-7
15.
Enriched category
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In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. In an enriched category, the set of associated with every pair of objects is replaced by an opaque object in some fixed monoidal category of hom-objects. Enriched category theory thus encompasses within the same framework a wide variety of structures including ordinary categories where the hom-set carries additional structure beyond being a set. An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or a category in M. Due to Mac Lanes preference for the letter V in referring to the monoidal category, the first diagram expresses the associativity of composition, That is, the associativity requirement is now taken over by the associator of the hom-category M. In this case, each leading to C in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms from a → b → c → d from C, C and C. Commutativity of the diagram is then merely the statement that both orders of composition give the result, exactly as required for ordinary categories. The identity morphism 1C, C → C that M has for each of its objects by virtue of it being an ordinary category, ordinary categories are categories enriched over, the category of sets with Cartesian product as the monoidal operation, as noted above. 2-Categories are categories enriched over Cat, the category of small categories, in this case the 2-cells between morphisms a → b and the vertical-composition rule that relates them correspond to the morphisms of the ordinary category C and its own composition rule. Locally small categories are categories enriched over, the category of sets with Cartesian product as the monoidal operation. Locally finite categories, by analogy, are categories enriched over, the hom-objects 2 then simply deny or affirm a particular binary relation on the given pair of objects, for the sake of having more familiar notation we can write this relation as a ≤ b. And since all diagrams in 2 commute, this is the content of the enriched category axioms for categories enriched over 2. Preadditive categories are categories enriched over, the category of groups with tensor product as the monoidal operation. If there is a functor from a monoidal category M to a monoidal category N. Every monoidal category M has a monoidal functor M to the category of sets, in many examples this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties. An enriched functor is the generalization of the notion of a functor to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure, because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism. There is no longer any notion of an identity morphism, nor of a composition of two morphisms
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Hopf algebra
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Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. They have diverse applications ranging from Condensed-matter physics and quantum theory to string theory. Theorem Let A be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0, then A is a free exterior algebra with generators of odd degree. In the sumless Sweedler notation, this property can also be expressed as S c = c S = ϵ1 for all c ∈ H, as for algebras, one can replace the underlying field K with a commutative ring R in the above definition. The definition of Hopf algebra is self-dual, so if one can define a dual of H, in general, S is an antihomomorphism, so S2 is a homomorphism, which is therefore an automorphism if S was invertible. If S2 = idH, then the Hopf algebra is said to be involutive, if H is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive. If a bialgebra B admits an antipode S, then S is unique, the antipode is an analog to the inversion map on a group that sends g to g−1. A subalgebra A of a Hopf algebra H is a Hopf subalgebra if it is a subcoalgebra of H, in other words, a Hopf subalgebra A is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of H is restricted to A. The Nichols–Zoeller freeness theorem established that the natural A-module H is free of finite rank if H is finite-dimensional, as a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically semisimple. Similarly, a Hopf subalgebra A is left normal in H if it is stable under the left adjoint mapping defined by adl = haS. The two conditions of normality are equivalent if the antipode S is bijective, in which case A is said to be a normal Hopf subalgebra, a normal Hopf subalgebra A in H satisfies the condition, HA+ = A+H where A+ denotes the kernel of the counit on K. This normality condition implies that HA+ is a Hopf ideal of H, as a consequence one has a quotient Hopf algebra H/HA+ and epimorphism H → H/A+H, a theory analogous to that of normal subgroups and quotient groups in group theory. A group-like element is an element x such that Δ = x⊗x. The group-like elements form a group with inverse given by the antipode, a primitive element x satisfies Δ = x⊗1 + 1⊗x. Let A be a Hopf algebra, and let M and N be A-modules, then, M ⊗ N is also an A-module, with a, = Δ = = for m ∈ M, n ∈ N and Δ =. The relationship between Δ, ε, and S ensure that certain natural homomorphisms of vector spaces are indeed homomorphisms of A-modules, for instance, the natural isomorphisms of vector spaces M → M ⊗ K and M → K ⊗ M are also isomorphisms of A-modules. Also, the map of vector spaces M* ⊗ M → K with f ⊗ m → f is also a homomorphism of A-modules, however, the map M ⊗ M* → K is not necessarily a homomorphism of A-modules. This observation was actually a source of the notion of Hopf algebra, using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups
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Duality (mathematics)
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Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues theorem is self-dual in this sense under the standard duality in projective geometry, many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. From a category theory viewpoint, duality can also be seen as a functor and this functor assigns to each space its dual space, and the pullback construction assigns to each arrow f, V → W its dual f∗, W∗ → V∗. In the words of Michael Atiyah, Duality in mathematics is not a theorem, the following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. A simple, maybe the most simple, duality arises from considering subsets of a fixed set S, to any subset A ⊆ S, the complement Ac consists of all those elements in S which are not contained in A. It is again a subset of S, taking the complement has the following properties, Applying it twice gives back the original set, i. e. c = A. This is referred to by saying that the operation of taking the complement is an involution, an inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. Given two subsets A and B of S, A is contained in Bc if and only if B is contained in Ac. This duality appears in topology as a duality between open and closed subsets of some fixed topological space X, a subset U of X is closed if, because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of sets is open, so dually. The interior of a set is the largest open set contained in it, because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. A duality in geometry is provided by the cone construction. Given a set C of points in the plane R2, unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set C. Instead, C ∗ ∗ is the smallest cone containing C which may be bigger than C. Therefore this duality is weaker than the one above, in that Applying the operation twice gives back a possibly bigger set, the other two properties carry over without change, It is still true that an inclusion C ⊆ D is turned into an inclusion in the opposite direction. Given two subsets C and D of the plane, C is contained in D ∗ if, a very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V*. Its elements are the k-linear maps φ, V → k, the three properties of the dual cone carry over to this type of duality by replacing subsets of R2 by vector space and inclusions of such subsets by linear maps. That is, Applying the operation of taking the dual vector space twice gives another vector space V**, there is always a map V → V**
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Pontryagin duality
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The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual. The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of compact abelian groups. Pontryagins treatment relied on the group being second-countable and either compact or discrete and this was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940. Moreover, any function on a group can be recovered from its discrete Fourier transform. Similarly, a group G and its dual group G ^ are not in general isomorphic, more categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see categorical considerations. A topological group is called compact if the underlying topological space is locally compact. Its called abelian if the group is abelian. Examples of locally compact abelian groups are, R n for n a positive integer, the positive real numbers R + with multiplication as operation. This group is isomorphic to by the exponential map, any finite abelian group, with the discrete topology. By the structure theorem for finite groups, all such groups are products of cyclic groups. The integers Z under addition, again with the discrete topology, the circle group, denoted T for torus. This is the group of numbers of modulus 1. T is isomorphic as a group to the quotient group R / Z. The field Q p of p-adic numbers under addition, with the usual p-adic topology, if G is a locally compact abelian group, a character of G is a continuous group homomorphism from G with values in the circle group T. The set of all characters on G can be made into a compact abelian group, called the dual group of G. This topology in general is not metrizable, however, if the group G is a separable locally compact abelian group, then the dual group is metrizable. This is analogous to the space in linear algebra, just as for a vector space V over a field K. More abstractly, these are examples of representable functors, being represented respectively by K and T