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Cauchy Completion (category Theory)
In category theory, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way somehow analogous to a completion in topology. These are: (ignoring the set-theoretic matters for simplicity), *free cocompletion, free completion. These are obtained by freely adding colimits or limits. Explicitly, the free cocompletion of a category ''C'' is the Yoneda embedding of ''C'' into the category of presheaves on ''C''. The free completion of ''C'' is the free cocompletion of the opposite of ''C''. **ind-completion. This is obtained by freely adding filtered colimits. *Cauchy completion of a category ''C'' is roughly the closure of ''C'' in some ambient category so that all functors preserve limits. For example, if a metric space is viewed as an enriched category (see generalized metric space), then the Cauchy completion of it coincides with the usual completion of the space. *Isbell completion (also called reflexive completion), introduced by Isbell in 1 ... [...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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion (metric Space)
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots in a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d\left(x_m, x_n\right) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: :#Every |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Embedding
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studying the locally small category \mathcal , one should study the category of all functors of \mathcal into \mathbf (the category of sets wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaf (category Theory)
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on C into a category, and is an example of a functor category. It is often written as \widehat = \mathbf^. A functor into \widehat is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, ''A'') for some object ''A'' of C is called a representable presheaf. Some authors refer to a functor F\colon C^\mathrm\to\mathbf as a \mathbf-valued presheaf. Examples * A simplicial set is a Set-valued presheaf on the simplex category C=\Delta. Properties * When C is a small category, the functor category \widehat=\mathbf^ is cartes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ind-completion
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from a small filtered category ''I'' to ''C''. The dual concept is the pro-completion, Pro(''C''). Definitions Filtered categories Direct systems depend on the notion of ''filtered categories''. For example, the category N, whose objects are natural numbers, and with exactly one morphism from ''n'' to ''m'' whenever n \le m, is a filtered category. Direct systems A ''direct system'' or an ''ind-object'' in a category ''C'' is defined to be a functor :F : I \to C from a small filtered category ''I'' to ''C''. For example, if ''I'' is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in ''C'' together with morphisms as displayed. The ind-completion Ind-objects in ''C'' form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtered Colimit
In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category, which will be recalled below. Filtered categories A category J is filtered when * it is not empty, * for every two objects j and j' in J there exists an object k and two arrows f:j\to k and f':j'\to k in J, * for every two parallel arrows u,v:i\to j in J, there exists an object k and an arrow w:j\to k such that wu=wv. A filtered colimit is a colimit of a functor F:J\to C where J is a filtered category. Cofiltered categories A category J is cofiltered if the opposite category J^ is filtered. In detail, a category is cofiltered when * it is not empty, * for every two objects j and j' in J there exists an object k and two arrows f:k\to j and f':k \to j' in J, * for every two parallel arrows u,v:j\to i in J, there exists an obj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Metric Space
In mathematics, specifically in category theory, a generalized metric space is a metric space but without the symmetry property and some other properties.namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite. Precisely, it is a category enriched over , \infty/math>, the one-point compactification of \mathbb. The notion was introduced in 1973 by Lawvere who noticed that a metric space can be viewed as a particular kind of a category. The categorical point of view is useful since by Yoneda's lemma, a generalized metric space can be embedded into a much larger category in which, for instance, one can construct the Cauchy completion In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ... of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isbell Conjugacy
Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality between covariant and contravariant representable 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 presheaves \left mathcal^, \mathcal \right/math> on \mathcal, taking X \in \mathcal to the contravariant representable functor: Y \; (h^) :\mathcal \rightarrow \left mathcal^, \mathcal \right/math> X \mapsto \mathrm (-,X). and the co-Yoneda embedding (a.k.a. contravariant Yoneda embedding or the dual Yoneda embedding) is a contravariant functor (a covar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isbell Envelope
Isbell may refer to: People * Alvertis Isbell (born 1940), American musician, songwriter, producer *Arnold J. Isbell (1899–1945), American naval officer *Cecil Isbell (1915–1985), American football player * Clayton Isbell (born 2000), American football player *Frank Isbell (1875–1941), American baseball player *Harris Isbell (1910-1994), American physician *Jane Isbell (1927-1981), American actress *Jason Isbell (born 1979), American musician * Jeffrey Dean Isbell (Izzy Stradlin) (born 1962), American musician *Joe Isbell (born 1940), American football player *John R. Isbell (1930–2005) American mathematician *Larry Isbell (1930–1978), American football player *Lynne Isbell (born 1955), American anthropologist * Marion William Isbell (1904-1988), American founder of Ramada Motels *Ruwellyn Isbell (born 1993), South African rugby union player * Virginia Isbell, American politician * Charles Lee Isbell, Jr., American computer scientist Places *Isbell, unincorporated town i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Completion
In category theory, a branch of mathematics, the exact completion constructs a Barr-exact category from any finitely complete category. It is used to form the effective topos and other realizability toposes. Construction Let ''C'' be a category with finite limits. Then the ''exact completion'' of ''C'' (denoted ''C''''ex'') has for its objects pseudo-equivalence relations in ''C''. A pseudo-equivalence relation is like an equivalence relation except that it need not be jointly monic. An object in ''C''''ex'' thus consists of two objects ''X''0 and ''X''1 and two parallel morphisms ''x''0 and ''x''1 from ''X''1 to ''X''0 such that there exist a reflexivity morphism ''r'' from ''X''0 to ''X''1 such that ''x''0''r'' = ''x''1''r'' = 1''X''0; a symmetry morphism ''s'' from ''X''1 to itself such that ''x''0''s'' = ''x''1 and ''x''1''s'' = ''x''0; and a transitivity morphism ''t'' from ''X''1 × ''x''1, ''X''0, ''x''0 ''X''1 to ''X''1 such that ''x''0''t'' = ''x''0''p'' and ''x''1''t'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
