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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''. 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 in the projective plane is a continuous map from the unit interval ,1 We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have :\begin f\colon\, & ,1\to \mathbb^2 &&\ \text \\ g\colon\, ... [...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 of weights for various ends **Olympic weightlifting, an Olympic sport that tests explosive strength **Powerlifting, a sport that tests limit 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 into a global function. It is a two step process ..., meta-process that defines functions independently of each other in a global scope * Taking an inference rule in propositional logic and adapting it for pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Lifting Property
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space ''E'' to another one, ''B''. It is designed to support the picture of ''E'' "above" ''B'' by allowing a homotopy taking place in ''B'' to be moved "upstairs" to ''E''. For example, a covering map has a property of ''unique'' local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces. The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting. Formal definition Assume from now on all maps are continuous functions from one topological space to another. Given a map \pi\colon E \to B, and a space X\,, one says that (X, \pi) has the homotopy lifting pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Predicate Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—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, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. 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 is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of a ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ext Functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let ''R'' be a ring and let ''R''-Mod be the category of modules over ''R''. (One can take this to mean either left ''R''-modules or right ''R''-modules.) For a fixed ''R''-module ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Tensor-hom Adjunction
In mathematics, the tensor-hom adjunction is that the tensor product - \otimes X and hom-functor \operatorname(X,-) form an adjoint pair: :\operatorname(Y \otimes X, Z) \cong \operatorname(Y,\operatorname(X,Z)). This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint. General statement Say ''R'' and ''S'' are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules): :\mathcal = \mathrm_S\quad \text \quad \mathcal = \mathrm_R . Fix an (''R'',''S'')-bimodule ''X'' and define functors ''F'': ''D'' → ''C'' and ''G'': ''C'' → ''D'' as follows: :F(Y) = Y \otimes_R X \quad \text Y \in \mathcal :G(Z) = \operatorname_S (X, Z) \quad \text Z \in \mathcal Then ''F'' is left adjoint to ''G''. This means there is a natural isomorphism :\operatorname_S (Y \otimes_R X, Z) \cong \operatorname_R (Y , \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hom Functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Formal definition Let ''C'' be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes). For all objects ''A'' and ''B'' in ''C'' we define two functors to the category of sets as follows: : The functor Hom(–, ''B'') is also called the '' functor of points'' of the object ''B''. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(''A'', –) and Hom(–, ''B'') are related in a natural manner. For any pair of morphisms ''f'' : ''B'' → ''B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for and , a basis of V \otimes W is formed by all tensor products of a basis element of and a basis element of . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (see Universal property). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Theorem
In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: * Alexandrov's uniqueness theorem of three-dimensional polyhedra * Black hole uniqueness theorem * Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. * Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients. * Division theorem, the uniqueness of quotient and remainder under Euclidean division. * Fundamental theorem of arithmetic, the uniqueness of prime factorization. * Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. * Picard– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
