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Fibered Categories
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] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphisms
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 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Category
In mathematics, an overcategory (also called a slice category) is a construction from category theory used in multiple contexts, such as with covering spaces (espace étalé). They were introduced as a mechanism for keeping track of data surrounding a fixed object X in some category \mathcal. The dual notion is that of an undercategory (also called a coslice category). Definition Let \mathcal be a category and X a fixed object of \mathcalpg 59. The overcategory (also called a slice category) \mathcal/X is an associated category whose objects are pairs (A, \pi) where \pi:A \to X is a morphism in \mathcal. Then, a morphism between objects f:(A, \pi) \to (A', \pi') is given by a morphism f:A \to A' in the category \mathcal such that the following diagram commutes\begin A & \xrightarrow & A' \\ \pi\downarrow \text & \text &\text \downarrow \pi' \\ X & = & X \endThere is a dual notion called the undercategory (also called a coslice category) X/\mathcal whose objects are pairs (B, \p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this progra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformations
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 mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Functor
In mathematics, especially homotopy theory, a cartesian fibration is, roughly, a map so that every lift exists that is a final object among all lifts. For example, the forgetful functor :\textrm \to \textrm from the category of pairs (X, F) of schemes and quasi-coherent sheaves on them is a cartesian fibration (see ). In fact, the Grothendieck construction says all cartesian fibrations are of this type; i.e., they simply forget extra data. See also: fibred category, prestack. The dual of a cartesian fibration is called an op-fibration; in particular, not a cocartesian fibration. A right fibration between simplicial sets is an example of a cartesian fibration. Definition Given a functor \pi : C \to S, a morphism f : x \to y in C is called \pi-cartesian or simply cartesian if the natural map :(f_*, \pi) : \operatorname(z, x) \to \operatorname(z, y) \times_ \operatorname(\pi(z), \pi(x)) is bijective. Explicitly, thus, f : x \to y is cartesian if given *g: z \to y and *u : \pi(z) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-category
In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors. The concept of a strict 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 1967 by Jean Bénabou. A (2, 1)-category is a 2-category where each 2-morphism is invertible. Definitions A strict 2-category By definition, a strict 2-category ''C'' consists of the data: * a class of 0-''cells'', * for each pairs of 0-cells a, b, a set \operatorname(a, b) called the set of 1-''cells'' from a to b, * for each pairs of 1-cells f, g in the same hom-set, a set \operatorname(f, g) called the set of 2-''cells'' from f to g, * ''ordinary compo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Morphism
In mathematics, especially homotopy theory, a cartesian fibration is, roughly, a map so that every lift exists that is a final object among all lifts. For example, the forgetful functor :\textrm \to \textrm from the category of pairs (X, F) of schemes and quasi-coherent sheaves on them is a cartesian fibration (see ). In fact, the Grothendieck construction says all cartesian fibrations are of this type; i.e., they simply forget extra data. See also: fibred category, prestack. The dual of a cartesian fibration is called an op-fibration; in particular, not a cocartesian fibration. A right fibration between simplicial sets is an example of a cartesian fibration. Definition Given a functor \pi : C \to S, a morphism f : x \to y in C is called \pi-cartesian or simply cartesian if the natural map :(f_*, \pi) : \operatorname(z, x) \to \operatorname(z, y) \times_ \operatorname(\pi(z), \pi(x)) is bijective. Explicitly, thus, f : x \to y is cartesian if given *g: z \to y and *u : \pi(z) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Universe
In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' are both elements of ''U'', then \ is an element of ''U''. # If ''x'' is an element of ''U'', then ''P''(''x''), the power set of ''x'', is also an element of ''U''. # If \_ is a family of elements of ''U'', and if is an element of ''U'', then the union \bigcup_ x_\alpha is an element of ''U''. A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry. Grothendieck’s o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
