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Homotopy Category
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra. The naive homotopy category The category of topological spaces Top has objects the topological spaces and morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps ''f'': ''X'' → ''Y'' are considered the same in the naive ho ... [...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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Homotopy Equivalence
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated homotopy category of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms. It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations. Topological spaces Model categories were defined by Quillen as an axiomatization of homotopy theory that applies to topological spaces, but also to many other categories in algebra and geometry. The example that started the subject is the category of topological spaces with Serre fibrations as fibrations and weak homotopy equivalences as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representable Functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. Definition Let C be a locally small category and let Set be the category of sets. For each object ''A'' of C let Hom(''A'',–) be the hom functor that maps object ''X'' to the set Hom(''A'',''X''). A functor ''F'' : C → Set is said to be representable if it is naturally isomorphic to Hom(''A'',–) for some object ''A'' of C. A representation of ''F'' is a pair (''A'', Φ) where :Φ : Hom(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular 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] |
Singular Homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of the standard ''n''-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation – mapping each ''n''-dimensional simplex to its (''n''−1)-dimensional boundary – induces the singular chain complex. The singular homology is then the homology of the chain complex. The resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point. The integers are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all real numbers, continuing forever in each direction, and also numbers that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers. In advanced mathematics, the number line can be called as a real line or real number line, formally defined as the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum. Just like ... [...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] |
CW Complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Full Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's Theorem
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping. Statement In more detail, let ''X'' and ''Y'' be topological spaces. Given a continuous mapping :f\colon X \to Y and a point ''x'' in ''X'', consider for any ''n'' ≥ 1 the induced homomorphism :f_*\colon \pi_n(X,x) \to \pi_n(Y,f(x)), where π''n''(''X'',''x'') denotes the ''n''-th homotopy group of ''X'' with base point ''x''. (For ''n'' = 0, π0(''X'') just means the set of path components of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category. Introduction and motivation A category ''C'' consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace ''C'' by another category ''C in which certain morphisms are forced to be isomorphisms. This process is called localization. For example, in the category of ''R''- modules (for some fixed commutative ring ''R'') the multiplication by a fixed element ''r'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
