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Lift (mathematics)
In category theory, a branch of mathematics, given a morphism ''f'': ''X'' → ''Y'' and a morphism ''g'': ''Z'' → ''Y'', a lift or lifting of ''f'' to ''Z'' is a morphism ''h'': ''X'' → ''Z'' such that . We say that ''f'' factors through ''h''. Lifts are ubiquitous; for example, the definition of fibrations (see Homotopy lifting property) and the valuative criteria of separated and proper maps of schemes are formulated in terms of existence and (in the last case) uniqueness of certain lifts. In algebraic topology and homological algebra, tensor product and the Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Tor functor and the Ext functor. Covering space A basic example in topology is lifting a path in one topological space to a path in a covering space. For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lifting Diagram
Lifting may refer to: *Manual handling of loads *Raising objects upwards, for example with lifting equipment *Weightlifting, lifting weights for exercise and sport, including: **Olympic weightlifting, an Olympic sport that tests explosive strength **Powerlifting, a sport that tests limit strength **Weight training, a way of increasing strength * An undesirable type of movement in the sport of racewalking *Shoplifting, an unnoticed theft of goods from an open retail establishment * Facelifting, a type of cosmetic surgery * Lift, a morphism in mathematics * Lifting theory, a notion in measure theory * Lifting scheme (wavelets) * Lambda lifting Lambda lifting is a meta-process that restructures a computer program so that functions are defined independently of each other in a global scope. An individual ''lift'' transforms a local function (subroutine) into a global function. It is a tw ..., meta-process that defines functions independently of each other in a global scope * Taking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tor Functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group. In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let ''R'' be a ring. Write ''R''-Mod for the category of left ''R''-modules and Mod-''R'' for the category of right ''R''-modules. (If ''R'' is comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Predicate Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Topological Spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces. As a concrete category Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor to the category of sets which assigns to each topological space the underlyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in Perspective (graphical)#Renaissance, perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function (topology)
In mathematics, a continuous function is a function (mathematics), function such that a small variation of the argument of a function, argument induces a small variation of the Value (mathematics), value of the function. This implies there are no abrupt changes in value, known as ''Classification of discontinuities, discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on Intuition, intuitive notions of continuity and considered only continuous functions. The (ε, δ)-definition of limit, epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real number, real and complex number, complex numbers. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeremy Gray
Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics. Biography Gray studied mathematics at the University of Oxford from 1966 to 1969, and then at Warwick University, obtaining his PhD in 1980 under the supervision of Ian Stewart and David Fowler. He has worked at the Open University since 1974, and became a lecturer there in 1978. He also lectured at the University of Warwick from 2002 to 2017, teaching a course on the history of mathematics. Gray was a consultant on the television series, '' The Story of Maths'',''To Infinity and Beyond'' 27 October 2008 21:00 BBC Four a co-production between the Open University and the BBC. He edits Archive for History of Exact Sciences. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2012 he became a fellow of the American Mathematical Society. Books Gray has been awarded prizes for his contributions to mathematics, includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Space
In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If p : \tilde X \to X is a covering, (\tilde X, p) is said to be a covering space or cover of X, and X is said to be the base of the covering, or simply the base. By abuse of terminology, \tilde X and p may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space. Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Bernhard Riemann, Riemann as domains on which naturally multivalued function, multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
