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Generator (category Theory)
In mathematics, specifically category theory, a family of generators (or family of separators) of a category \mathcal C is a collection \mathcal G \subseteq Ob(\mathcal C) of objects in \mathcal C, such that for any two ''distinct'' morphisms f, g: X \to Y in \mathcal, that is with f \neq g, there is some G in \mathcal G and some morphism h : G \to X such that f \circ h \neq g \circ h. If the collection consists of a single object G, we say it is a generator (or separator). Generators are central to the definition of Grothendieck categories. The dual concept is called a cogenerator (or coseparator). Examples * In the category of abelian groups, the group of integers \mathbb Z is a generator: If ''f'' and ''g'' are different, then there is an element x \in X, such that f(x) \neq g(x). Hence the map \mathbb Z \rightarrow X, n \mapsto n \cdot x suffices. * Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator. * In the categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar to function composition. Morphisms and objects are constituents of a category. Morphisms, also called ''maps'' or ''arrows'', relate two objects called the ''source'' and the ''target'' of the morphism. There is a partial operation, called ''composition'', on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition ( associativity of composition when it is defined, and existence of an identity morphism for every object). Morphisms and categories recur in much of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Category
In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's 1962 thesis. To every algebraic variety V one can associate a Grothendieck category \operatorname(V), consisting of the quasi-coherent sheaves on V. This category encodes all the relevant geometric information about V, and V can be recovered from \operatorname(V) (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories. Definition By definition, a Grothendieck category \mathcal is an AB5 category with a generator. Spelled out, this means that * \mathcal is an abelian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual (category Theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''op. (''C''op is composed by reversing every morphism of ''C''.) Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement ''S'' is true about ''C'', then its dual statement is true about ''C''op. Also, if a statement is false about ''C'', then its dual has to be false about ''C''op. (Compactly saying, ''S'' for ''C'' is true if and only if its dual for ''C''op is true.) Given a concrete category ''C'', it is often the case that the opposite category ''C''op per se is abstract. ''C''op need not b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Groups
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbol � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of morphisms is the composition of functions. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or both). Properties of the category of sets The axioms of a category are satisfied by Set because composition of functions is associative, and because every set ''X'' has an identity function id''X'' : ''X'' → ''X'' which serves as identity element for function composition. The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions as morph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator (ring Theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of . Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case of noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Definitions Let ''R'' be a ring, and let ''M'' be a left ''R''- module. Choose a non-empty subset ''S'' of ''M''. The ''annihilator'' of ''S'', denoted Ann''R''(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''s'' in ''S'', . In set notation, :\mathrm_R(S)=\ It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set). Subsets of right modules may be used as well, after the modifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categories For The Working Mathematician
''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on his lectures on the subject given at the University of Chicago, the Australian National University, Bowdoin College, and Tulane University. It is widely regarded as the premier introduction to the subject. Contents The book has twelve chapters, which are: :Chapter I. Categories, Functors, and Natural Transformations. :Chapter II. Constructions on Categories. :Chapter III. Universals and Limits. :Chapter IV. Adjoints. :Chapter V. Limits. :Chapter VI. Monads and Algebras. :Chapter VII. Monoids. :Chapter VIII. Abelian Categories. :Chapter IX. Special Limits. :Chapter X. Kan Extensions. :Chapter XI. Symmetry and Braiding in Monoidal Categories :Chapter XII. Structures in Categories. Chapters XI and XII were added in the 1998 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
