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Codensity Monad
In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad (category theory), monad to a wide class of functors. Definition The codensity monad of a functor G: D \to C is defined to be the Kan extension, right Kan extension of G along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor T^G : C \to C. The monad structure on T^G stems from the universal property of the right Kan extension. The codensity monad exists whenever D is a small category (has only a set, as opposed to a proper class, of morphisms) and C possesses all (small, i.e., set-indexed) limits. It also exists whenever G has a left adjoint. By the general formula computing right Kan extensions in terms of end (category theory), ends, the codensity monad is given by the following formula: T^G(c) = \int_ G(d)^, where C(c, G(d)) denotes the set of morphisms in C between the indicated objects and the integral denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaf
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isbell Duality
Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell John Rolfe Isbell (October 27, 1930 – August 6, 2005) was an American mathematician, for many years a professor of mathematics at the University at Buffalo (SUNY). Biography Isbell was born in Portland, Oregon, the son of an army officer from I ...) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality between covariant and contravariant representable Presheaf (category theory), presheaves associated with an objects of categories under the Yoneda embedding. Also, says that; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics". Definition Yoneda embedding The (covariant) Yoneda embedding is a covariant functor from a small category \mathcal into the category of Presheaf (category theory), presheaves \left[\mathcal^, \mathcal \right] on \mathcal, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the tuple (X, \mathcal A) is called a measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma-algebra would be: \mathcal A_1 = \. Then \left(X, \mathcal A_1\right) is a measurable space. Another possible \sigma-algebra would be the power set on X: \mathcal A_2 = \mathcal P(X). With this, a second measurable space on the set X is given by \left(X, \mathcal A_2\right). Common measurable spaces If X is finite or countably infinite, the \sigma-algebra is most often the power set on X, so \mathcal A = \mathcal P(X). This leads to the measurable space (X, \mathcal P(X)). If X is a topological space In mathematics, a topological space is, rou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Space
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giry Monad , from The Phantom of the Opera
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Giry may refer to: People * Arthur Giry (1848–1899), French historian * Louis Giry (1596–1665), French lawyer, translator and writer * Odet-Joseph Giry (1699–1761), French clergyman * Sylvie Giry-Rousset (born 1965), French cross-country skier Places * Giry, Nièvre, France Fictional characters * Madame Giry, from The Phantom of the Opera * Meg Giry Meg is a feminine given name, often a short form of Megatron, Megan, Megumi (Japanese), etc. It may refer to: People *Meg (singer), a Japanese singer *Meg Cabot (born 1967), American author of romantic and paranormal fiction *Meg Burton Cahill ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone Space
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras. Equivalent conditions The following conditions on the topological space X are equivalent: * X is a Stone space; * X is homeomorphic to the projective limit (in the category of topological spaces) of an inverse system of finite discrete spaces; * X is compact and totally separated; * X is compact, T0 , and zero-dimensional (in the sense of the small inductive dimension); * X is coherent and Hausdorff. Examples Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space \Z_p of p-adic integers, where p is any prime number. Generalizing these examples, any product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Topological Space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Monad
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes and in functional programming languages, allowing languages with non-mutable states to do things such as simulate for-loops; see Monad (functional programming). Introduction and definition A monad is a certain type of endofunctor. For example, if F and G are a pair of adjoint functors, with F left adjoint to G, then the composition G \circ F is a monad. If F and G are inverse functors, the corresponding monad is the identity functor. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Cogenerator
In category theory, a branch of mathematics, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation. More precisely: * A generator of a category with a zero object is an object ''G'' such that for every nonzero object H there exists a nonzero morphism f:''G'' → ''H''. * A cogenerator is an object ''C'' such that for every nonzero object ''H'' there exists a nonzero morphism f:''H'' → ''C''. (Note the reversed order). The abelian group case Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of ''G'' until the morphism :''f'': Sum(''G'') →''H'' is surjective; and one can form direct products of ''C'' until the morphism :''f'':''H''→ Prod(''C'') is injective. For example, the integers are a generator of the category of abelian gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Object (mathematics)
In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition. Definition An object ''X'' in a category ''C'' which admits all filtered colimits (also known as direct limits) is called ''compact'' if the functor :\operatorname_C(X, \cdot) : C \to \mathrm, Y \mapsto \operatorname_C(X, Y) commutes with filtered colimits, i.e., if the natural map :\operatorname \operatorname_C(X, Y_i) \to \operatorname_C(X, \operatorname_i Y_i) is a bijection for any filtered system of objects Y_i in ''C''. Since elements in the filtered colimit at the left are represented by maps X \to Y_i, for some ''i'', the surjectivity of the above map amounts to requiring that a map X \to \operatorname_i Y_i factors over some Y_i. The terminology is motivated by an example arising from topology mentioned below. Several authors also use a terminology which is more closely related to algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
