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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] |
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] |
Join (topology)
In topology, a field of mathematics, the join of two topological spaces A and B, often denoted by A\ast B or A\star B, is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in A to every point in B. Definitions The join is defined in slightly different ways in different contexts Geometric sets If A and B are subsets of the Euclidean space \mathbb^n, then: A\star B\ :=\ \,that is, the set of all line-segments between a point in A and a point in B. Some authors restrict the definition to subsets that are ''joinable'': any two different line-segments, connecting a point of A to a point of B, meet in at most a common endpoint (that is, they do not intersect in their interior). Every two subsets can be made "joinable". For example, if A is in \mathbb^n and B is in \mathbb^m, then A\times\\times\ and \\times B\times\ are joinable in \mathbb^. The figure above shows an example for m=n=1, where A and B are line-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups\cdots \to \pi_(B) \to \pi_n(\text(f)) \to \pi_n(A) \to \pi_n(B) \to \cdotsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangleC(f)_\bullet 1\to A_\bullet \to B_\bullet \xrightarrowgives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber. Construction The homotopy fiber has a simple description for a continuous map f:A \to B. If we replace f by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, (C^)^ = C. Examples * An example comes from reversing the direction of inequalities in a partial order. So if ''X'' is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by :: ''x'' ≤op ''y'' if and only if ''y'' ≤ ''x''. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/up-set, ideal/ filter etc. This order theoretic duality is in turn a special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a " well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nerve (category Theory)
In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space, called the classifying space of the category ''C''. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory. Motivation The nerve of a category is often used to construct topological versions of moduli spaces. If ''X'' is an object of ''C'', its moduli space should somehow encode all objects isomorphic to ''X'' and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Category
In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track of data surrounding a fixed object X in some category \mathcal. There is a dual notion of undercategory, which is defined similarly. Definition Let \mathcal be a category and X a fixed object of \mathcalpg 59. The overcategory (also called a slice category) \mathcal/X is an associated category whose objects are pairs (A, \pi) where \pi:A \to X is a morphism in \mathcal. Then, a morphism between objects f:(A, \pi) \to (A', \pi') is given by a morphism f:A \to A' in the category \mathcal such that the following diagram commutes\begin A & \xrightarrow & A' \\ \pi\downarrow \text & \text &\text \downarrow \pi' \\ X & = & X \endThere is a dual notion called the undercategory (also called a coslice category) X/\mathcal whose objects are pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Adjoint
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 X in \mathcal and Y in \mathcal a bijection between the respective morphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ... [...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] |
Mapping Telescope
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'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Colimit With Composition Filled In
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
