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Density Theorem (category Theory)
In category theory, a branch of mathematics, the density theorem states that every presheaf of sets is a colimit of representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form \Delta^n = \operatorname(-, (called the standard ''n''-simplex) so the theorem says: for each simplicial set ''X'', :X \simeq \varinjlim \Delta^n where the colim runs over an index category determined by ''X''. Statement Let ''F'' be a presheaf on a category ''C''; i.e., an object of the functor category \widehat = \mathbf(C^\text, \mathbf). For an index category over which a colimit will run, let ''I'' be the category of elements of ''F'': it is the category where # an object is a pair (U, x) consisting of an object ''U'' in ''C'' and an element x \in F(U), # a morphism (U, x) \to (V, y) consists of a morphism u: U \to V in ''C'' such that (Fu)(y) = x. It comes with the forg ... [...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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Presheaf Of Sets
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 cartesian closed. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of 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 C are defined by means of diagrams in C. Formally, a 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 thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Representable Functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. Definition Let C be a locally small category and let Set be the category of sets. For each object ''A'' of C let Hom(''A'',–) be the hom functor that maps object ''X'' to the set Hom(''A'',''X''). A functor ''F'' : C → Set is said to be representable if it is naturally isomorphic to Hom(''A'',–) for some object ''A'' of C. A representation of ''F'' is a pair (''A'', Φ) where :Φ : Hom(''A'',& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Simplicial Set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Functor Category
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in the category). Functor categories are of interest for two main reasons: * many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; * every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting. Definition Suppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D), ,D/math>, or D ^C, has as objects the covariant functors from C to D, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Category Of Elements
In category theory, if is a category and is a set-valued functor, the category of elements of (also denoted ) is the following category: * Objects are pairs (A,a) where A \in \mathop(C) and a \in FA. * Morphisms (A,a) \to (B,b) are arrows f: A \to B of C such that (Ff)a = b. A more concise way to state this is that the category of elements of is the comma category , where is a singleton (a set with one element). The category of elements of comes with a natural projection that sends an object to , and an arrow to its underlying arrow in . The category of elements of a presheaf In some texts (e.g. Mac Lane, Moerdijk) the category of elements is used for presheaves. We state it explicitly for completeness. If is a presheaf, the category of elements of (again denoted by , or, to make the distinction to the above definition clear, ) is the following category: * Objects are pairs (A,a) where A \in \mathop(C) and a \in P(A). * Morphisms (A,a) \to (B,b) are arrows f:A \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Diagram (category Theory)
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a ''function'' from a fixed index ''set'' to the class of ''sets''. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a ''functor'' from a fixed index ''category'' to some ''category''. The universal functor of a diagram is the diagonal functor; its right adjoint is the limit of the diagram and its left adjoint is the colimit. The natural transformation from the diagonal functor to some arbitrary diagram is called a cone. Definition Formally, a diagram of type ''J'' in a category ''C'' is a ( covariant) functor The category ''J'' is called the index category or the scheme of the diagram ''D''; the functor is sometimes called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Constant Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) ( ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |