|
Cartesian Closed Category
In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal category, closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum computation, quantum and classical computation. Etymology Named after René Descartes (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product. Definition The category C is called Cartesian closed iff it satisfies the following three propert ... [...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] |
Empty Product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of addition, adding no numbers—is by convention 0, zero, or the additive identity. When numbers are implied, the empty product becomes 1, one. The term ''empty product'' is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing Set theory, set-theoretic intersections, categorical products, and products in computer programming. Nullary arithmetic product Definition Let ''a''1, ''a''2, ''a''3, ... be a sequence of numbers, and let :P_m = \prod_^m a_i = a_1 \cdots a_m be the product of the first ''m'' elements of the sequence. Then :P_m = P_ a_m for all ''m'' = 1, 2, ... provided that we use the c ... [...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] |
Section (fiber Bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi. In other words, if E is a fiber bundle over a base space, B: : \pi \colon E \to B then a section of that fiber bundle is a continuous map, : \sigma \colon B \to E such that : \pi(\sigma(x)) = x for all x \in B . A section is an abstract characterization of what it means to be a graph. The graph of a function g\colon B \to Y can be identified with a function taking its values in the Cartesian product E = B \times Y , of B and Y : :\sigma\colon B\to E, \quad \sigma(x) = (x,g(x)) \in E. Let \pi\colon E \to B be the projection onto the first factor: \pi(x,y) = x . Then a graph is any function \sigma for which \pi(\sigma(x)) = x . The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product. If \pi\colon E \to B is a fibre bundle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback (category Theory)
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit (category theory), limit of a diagram (category theory), diagram consisting of two morphisms and with a common codomain. The pullback is written :. Usually the morphisms and are omitted from the notation, and then the pullback is written :. The pullback comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative diagram, commutative square. The Dual (category theory), dual concept of the pullback is the ''Pushout ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Composition
In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \circ f) is pronounced "the composition of and ". Reverse composition, sometimes denoted f \mapsto g , applies the operation in the opposite order, applying f first and g second. Intuitively, reverse composition is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as #Properties, associativity. Examples * Composition of functions on a finite set (mathematics), set: If , and , then , as shown in the figure. * Composition of functions on an infinite set: If (where is the set of all real numbers) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Application
Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial dee" **Partial differential equation, a differential equation that contains unknown multivariable functions and their partial derivatives Other uses * Partial application, in computer science the process of fixing a number of arguments to a function, producing another function * Partial charge or net atomic charge, in chemistry a charge value that is not an integer or whole number * Partial fingerprint, impression of human fingers used in criminology or forensic science * Partial seizure or focal seizure, a seizure that initially affects only one hemisphere of the brain * Partial or Part score, in contract bridge a trick score less than 100, as well as other meanings * Partial or Partial wave, one sound wave of which a complex tone is comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comma Category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by William Lawvere, F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some Limit (category theory), limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13). Definition The most general comma ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (category Theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as product (category theory), products, pullback (category theory), pullbacks and inverse limits. The duality (category theory), dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushout (category theory), pushouts and direct limits. Limits and colimits, like the strongly related notions of universal property, universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category (mathematics), category C are defined by means of diagrams in C. Formally, a diagram (category theory), diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is tho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D (both from C to D), then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hom-set
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 contempo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
