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Representable Functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category (mathematics), category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. Set (mathematics), sets and function (mathematics), functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. Definition Let C be a locally small category and let Set be the category of sets. For each object ''A'' of C let Hom(''A'',–) be the hom functor that maps object ''X'' to the set Hom(''A'',''X''). A functor ''F'' : C → Set is said to be representable if it is naturally isomorphic to Hom(''A'',–) for some object ''A'' of C. A representat ...
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Mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ... and number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...), formulas and related structures (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...), shapes and spaces in which they are contained (geometry Geometry (from the grc, ...
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Universal Morphism
In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism (see Universal property#Formal definition, Formal Definition). Universal morphisms can also be thought of more abstractly as Initial and terminal objects, initial or terminal objects of a comma category (see Universal property#Connection with comma categories, Connection with comma categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated. Universal properties may be used in other areas of mathematics implicitly, but the abstract and more precise definition of it can be studied in category theory. This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all free objects, ...
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Integer
An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ... ''integer'' meaning "whole") is colloquially defined as a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ... that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of integers consists of zero (), the positive natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' larges ...
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Variable (mathematics)
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a variable (from Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ... '' variabilis'', "changeable") is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. All (and ) is achieved th ... and placeholder for (historically) a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity an ...
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Polynomial Ring
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., especially in the field of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ..., a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...) formed from the set of polynomial In mathematics Mathematics (from Ancient Greek, ...
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Category Of Rings
In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is Category (mathematics)#Small and large categories, large, meaning that the class (set theory), class of all rings is proper class, proper. As a concrete category The category Ring is a concrete category meaning that the objects are set (mathematics), sets with additional structure (addition and multiplication) and the morphisms are function (mathematics), functions that preserve this structure. There is a natural forgetful functor :''U'' : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :''F'' : Set → Ring which assigns to each set ''X'' ...
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Category Of Groups
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ... Grp (or Gp) has the class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ... of all groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
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Singleton Set
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a singleton, also known as a unit set, is a set with exactly one element. For example, the set is a singleton containing the element ''null''. The term is also used for a 1-tuple In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... (a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... with one member). Prop ...
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Free Object
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the idea of a free object is one of the basic concepts of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema .... It is a part of universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ..., in the sense that it relates to all types of algebraic structure (with finitary In mathematics and logic, a ...
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Forgetful Functor
In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature (logic), signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case. Overview As an example, there are several forgetful functors from the category of commutative rings. A (unital algebra, unital) Ring (mathematics), ring, described in the language of universal algebra, is an ordered tuple (''R'', +, ×, ''a'', 0, 1) satisfying certain axioms, where "+" and "×" are binary functions on the set ''R'', ''a'' i ...
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Indicator Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an indicator function or a characteristic function of a subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... of a set is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... that maps elements of the subset to one, and all other elements of the set to zero. The indicator function of a subset of a set ...
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Inverse Image
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the image of a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... is the set of all output values it may produce. More generally, evaluating a given function f at each element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ... of a given subset A of its domain Domain may refer to: Mathematics *Domain of a ...
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