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Indexed Category
In category theory, a branch of mathematics, a ''C''-indexed category is a pseudofunctor from ''C''''op'' to Cat, where Cat is a Strict_2-category, 2-category of categories. Any indexed category has an associated Grothendieck construction, which gives rise to a fibred category. References {{cattheory-stub Category theory ... [...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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudofunctor
In mathematics, a pseudofunctor ''F'' is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that F(f \circ g) = F(f) \circ F(g) and F(1) = 1 do not hold as exact equalities but only up to ''coherent isomorphisms''. The Grothendieck construction associates to a pseudofunctor a fibered category. See also *Lax functor *Prestack (an example of pseudofunctor) *Fibered category Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of ... References *C. SorgerLectures on moduli of principal G-bundles over algebraic curves External links *http://ncatlab.org/nlab/show/pseudofunctor Functors {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strict 2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or ''weak'' 2-''category''), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77. Definition A 2-category C consists of: * A class of 0-''cells'' (or ''objects'') , , .... * For all objects and , a category \mathbf(A,B). The objects f,g: A \to B of this category are called 1-''cells'' and its morphisms \alpha: f \Rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Construction
The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory. Definition Let F\colon \mathcal \rightarrow \mathbf be a functor from any small category to the category of small categories. The Grothendieck construction for F is the category \Gamma(F) (also written \textstyle\int_ F, \textstyle\mathcal \int F or F \rtimes \mathcal), with * objects being pairs (c,x), where c\in \operatorname(\mathcal) and x\in \operatorname(F(c)); and * morphisms in \operatorname_((c_1,x_1),(c_2,x_2)) being pairs (f, g) such that f: c_1 \to c_2 in \mathcal, and g: F(f)(x_1) \to x_2 in F(c_2). Composition of morphisms is defined by (f,g) \circ (f',g') = (f \circ f', g \circ F(f)(g')). Slogan "The Grothendieck construction takes structured, tabulated data and flattens it by throwing it all into one big space. The projection functor is then tasked with remembering which box each datum originally came from." Example If G is a gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibred Category
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space ''X'' to another topological space ''Y'' is associated the pullback functor taking bundles on ''Y'' to bundles on ''X''. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
