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Mapping Cone (topology)
In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a Quotient space (topology), quotient space and denoted C_f. Alternatively, it is also called the homotopy cofiber and also notated Cf. Its dual, a fibration, is called the homotopy fiber, mapping fiber. The mapping cone can be understood to be a mapping cylinder Mf with the initial end of the cylinder collapsed to a point. Mapping cones are frequently applied in the homotopy theory of pointed spaces. Definition Given a continuous function, map f\colon X \to Y, the mapping cone C_f is defined to be the quotient space of the mapping cylinder (X \times I) \sqcup_f Y with respect to the equivalence relation \forall x,x' \in X, (x, 0) \sim \left(x', 0\right)\,, (x, 1) \sim f(x). Here I denotes the unit interval [0, 1] with its standard Topological space, topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1. Visually, one takes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion Map
In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota(x)=x. An inclusion map may also be referred to as an inclusion function, an insertion, or a canonical injection. A "hooked arrow" () is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (However, some authors use this hooked arrow for any embedding.) This and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f between objects X and Y, if there is an inclusion map \iota : A \to X into the domain X, then one can form the restriction f\circ \iota of f. In many instances, one can also construct a canonical inclusion into the codomain R \to Y known as the range of f. Applications of inclusion maps Inclusion maps tend to be homomorphisms of algebraic structures; thus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as group (mathematics), groups and ring (mathematics), rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an Additive identity, (often denoted as 0) and an identity with respect to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a nonempty topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any path-connected space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have Group isomorphism, isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface (mathematics), surface), and some point in it, and all the loops both starting and ending at this point—path (topology), paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then alo ... [...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] |
Adjoint Functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects c in \mathcal and d in \mathcal, a bijection between the respective morphism sets :\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Currying
In mathematics and computer science, currying is the technique of translating a function that takes multiple arguments into a sequence of families of functions, each taking a single argument. In the prototypical example, one begins with a function f:(X\times Y)\to Z that takes two arguments, one from X and one from Y, and produces objects in Z. The curried form of this function treats the first argument as a parameter, so as to create a family of functions f_x :Y\to Z. The family is arranged so that for each object x in X, there is exactly one function f_x. In this example, \mbox itself becomes a function that takes f as an argument, and returns a function that maps each x to f_x. The proper notation for expressing this is verbose. The function f belongs to the set of functions (X\times Y)\to Z. Meanwhile, f_x belongs to the set of functions Y\to Z. Thus, something that maps x to f_x will be of the type X\to \to Z With this notation, \mbox is a function that takes objects from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pushout
A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is considered a school dropout. In typical use, the category of pushouts excludes students who have been formally expelled from school for violating rules (e.g., for being violent). Students may be pushed out of school because their presence in the school creates difficulty in meeting some goal of the school. For example, in the case where funding for the school is dependent upon scholastic achievement of the students, if the school can get rid of low-performing students, average test scores on academic performance tests will go up, thus increasing funding. Schools may pushout truant students, who formally enroll in classes, but then refuse to attend. In some low-performing schools in Chicago combined dropout/pushout rates have exceeded 25% in o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y. f(y(x)) \equiv g(x) It is such a fundamental process that it is often passed over without mention. However, it is not just functions that can be "pulled back" in this sense. Pullbacks can be applied to many other objects such as differential forms and their cohomology classes; see * Pullback (differential geometry) * Pullback (cohomology) Fiber-product The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibered Product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a 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 square. The dual concept of the pullback is the ''pushout''. Universal property Explicitly, a pullback of the morphisms f and g consists of an object P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
