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Bundle (mathematics)
In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: ''E'' → ''B'' with ''E'' and ''B'' sets. It is no longer true that the preimages \pi^(x) must all look alike, unlike fiber bundles, where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic. Definition A bundle is a triple where are sets and is a map. p 11. * is called the total space * is the base space of the bundle * is the projection This definition of a bundle is quite unrestrictive. For instance, the empty function defines a bundle. Nonetheless it serves well to introduce the basic terminology, and every type of bundle has the basic ingredients of above with restrictions on and usually there is additional structure. For ea ... [...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] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comma Category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by William Lawvere, F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some Limit (category theory), limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13). Definition The most general comma ca ... [...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] |
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] |
Pullback (category Theory)
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit (category theory), limit of a diagram (category theory), diagram consisting of two morphisms and with a common codomain. The pullback is written :. Usually the morphisms and are omitted from the notation, and then the pullback is written :. The pullback comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative diagram, commutative square. The Dual (category theory), dual concept of the pullback is the ''Pushout ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concrete Category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets. A concrete category, when defined without reference to the notion of a category, consists of a class of ''objects'', each equipped with an ''underlying set''; and for any two objects ''A'' and ''B'' a set of functions, called ''homomorphisms'', from the underlying set of ''A'' to the underlying set of ''B''. Furthermore, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epimorphism
In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category ''C'' is a monomorphism in the dual category ''C''op). Many authors in abstract algebra and universal algebra define an epimorphism simply as an ''onto'' or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given ... [...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] |
Tangent Bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frame Bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal ''\mathrm(k,\mathbb)''-bundle (where ''k'' is the rank of ''E''). The frame bundle of a smooth manifold is the one associated with its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Definition and construction Let ''E \to X'' be a real vector bundle of rank ''k'' over a topological space ''X''. A frame at a point ''x \in X'' is an ordered basis for the vector space ''E_x''. Equivalently, a frame can be viewed as a linear isomorphism :p : \mathbf^k \to E_x. The set of all frames at ''x'', denoted ''F_x'', has a natural right action by the general linear group ''\mathrm(k,\mathbb)'' of invertible ''k \times k'' matrices: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
