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Mapping Cylinder
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f between topological spaces X and Y is the quotient :M_f = (( ,1times X) \amalg Y)\,/\,\sim where the \amalg denotes the disjoint union, and ∼ is the equivalence relation generated by :(0,x)\sim f(x)\quad\textx\in X. That is, the mapping cylinder M_f is obtained by gluing one end of X\times ,1/math> to Y via the map f. Notice that the "top" of the cylinder \\times X is homeomorphic to X, while the "bottom" is the space f(X)\subset Y. It is common to write Mf for M_f, and to use the notation \sqcup_f or \cup_f for the mapping cylinder construction. That is, one writes :Mf = ( ,1times X) \cup_f Y with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone Cf, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations. Basic properties The bottom ''Y'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Cylinder
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f between topological spaces X and Y is the quotient :M_f = (( ,1times X) \amalg Y)\,/\,\sim where the \amalg denotes the disjoint union, and ∼ is the equivalence relation generated by :(0,x)\sim f(x)\quad\textx\in X. That is, the mapping cylinder M_f is obtained by gluing one end of X\times ,1/math> to Y via the map f. Notice that the "top" of the cylinder \\times X is homeomorphic to X, while the "bottom" is the space f(X)\subset Y. It is common to write Mf for M_f, and to use the notation \sqcup_f or \cup_f for the mapping cylinder construction. That is, one writes :Mf = ( ,1times X) \cup_f Y with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone Cf, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations. Basic properties The bottom ''Y'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Pushout
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfconsidered as an object in the homotopy category of diagrams F \in \text(\textbf^I), (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone\begin \underset(F)&: * \to \textbf\\ \underset(F)&: * \to \textbf \endwhich are objects in the homotopy category \text(\textbf^*), where * is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category \text(\textbf) since the latter homotopy functor category has functors which picks out an object in \text and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colimit (category Theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coequalizer
In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a colimit of the diagram consisting of two objects ''X'' and ''Y'' and two parallel morphisms ''f'', ''g'' : ''X'' → ''Y''. More explicitly, a coequalizer can be defined as an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that ''q'' ∘ ''f'' = ''q'' ∘ ''g''. Moreover, the pair (''Q'', ''q'') must be universal in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism ''u'' : ''Q'' → ''Q''′ such that ''u'' ∘ ''q'' = ''q''′. This information can be captured by the following commutative diagram: As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (this is why, by abuse of language, one sometimes speaks of " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pushout (category Theory)
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' with a common domain. The pushout consists of an object ''P'' along with two morphisms ''X'' → ''P'' and ''Y'' → ''P'' that complete a commutative square with the two given morphisms ''f'' and ''g''. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are P = X \sqcup_Z Y and P = X +_Z Y. The pushout is the categorical dual of the pullback. Universal property Explicitly, the pushout of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''i''1 : ''X'' → ''P'' and ''i''2 : ''Y'' → ''P'' such that the diagram : commutes and such that (' ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Colimit
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfconsidered as an object in the homotopy category of diagrams F \in \text(\textbf^I), (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone\begin \underset(F)&: * \to \textbf\\ \underset(F)&: * \to \textbf \endwhich are objects in the homotopy category \text(\textbf^*), where * is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category \text(\textbf) since the latter homotopy functor category has functors which picks out an object in \text and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace Topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced topology, or the trace topology). Definition Given a topological space (X, \tau) and a subset S of X, the subspace topology on S is defined by :\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace. That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X, \tau). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X, \tau). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map :\iota: S \hookrightarrow X is continuous. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories). Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a " map" is a continuous function, possibly with some extra constraints. Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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