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Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,Mac Lane, p. 126 and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements. The dual concept to a subobject is a . This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc. Definitions In detail, let ''A'' be an object of some category. Given two monomorphisms :u: S \to A \ \text \ v: T\to A with codomain ''A'', we define an equivalence relation by u \equiv v if there exists an isomorphism \phi: S \to T with u = v \circ \phi. Equivalently, we write u \leq v if u factors through ''v''—that is, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Category Of Sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total 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 and/or restrict the arrows to functions of a particular kind. 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] |
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Quotient Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G\,/\,N, where G is the original group and N is the normal subgroup. (This is pronounced G\bmod N, where \mbox is short for modulo.) Much of the importan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the codomain or image of a function. A codomain is part of a function if is defined as a triple where is called the '' domain'' of , its ''codomain'', and its '' graph''. The set of all elements of the form , where ranges over the elements of the domain , is called the '' image'' of . The image of a function is a subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ... of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements in its codomain for which the equation does not have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Category Of Groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to sets. M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set→Mon→Grp, where F is the free functor; this functor assigns to every set ''S'' the free group on ''S.'' Categorical properties The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join (mathematics), join) and a unique infimum (also called a greatest lower bound or meet (mathematics), meet). An example is given by the power set of a set, partially ordered by Subset, inclusion, for which the supremum is the Union (set theory), union and the infimum is the Intersection (set theory), intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic Identity (mathematics), identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilatti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Equipotent
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', there is exactly one element ''x'' of ''A'' with ''f''(''x'') = ''y''. Equinumerous sets are said to have the same cardinality (number of elements). The study of cardinality is often called equinumerosity (''equalness-of-number''). The terms equipollence (''equalness-of-strength'') and equipotence (''equalness-of-power'') are sometimes used instead. Equinumerosity has the characteristic properties of an equivalence relation. The statement that two sets ''A'' and ''B'' are equinumerous is usually denoted :A \approx B \, or A \sim B, or , A, =, B, . The definition of equinumerosity using bijections can be applied to both finite and infinite sets, and allows one to state whether two sets have the same size even if they are infinite. Georg Cant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category ''C'' is a monomorphism in the dual category ''C''op). Many authors in abstract algebra and universal algebra define an epimorphism simply as an ''onto'' or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Locally Small Category
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains ''elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. 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. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work '' Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Proper Class
Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for algebraic varieties * Proper transfer function, a transfer function in control theory in which the degree of the numerator does not exceed the degree of the denominator * Proper equilibrium, in game theory, a refinement of the Nash equilibrium * Proper subset * Proper space * Proper complex random variable Other uses * Proper (liturgy), the part of a Christian liturgy that is specific to the date within the Liturgical Year * Proper frame, such system of reference in which object is stationary (non moving), sometimes also called a co-moving frame * Proper (heraldry), in heraldry, means depicted in natural colors * Proper Records, a UK record label * Proper (album), an album by Into It. Over It. released in 2011 * Proper (often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |