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Simplicial Diagram
In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all = \ and the order-preserving functions). Formally, a simplicial diagram in a category or an ∞-category ''C'' is a contraviant functor from the simplex category to ''C''. Thus, it is the same thing as a simplicial object but is typically thought of as a sequence of objects in ''C'' that is depicted using multiple arrows :\cdots \, \underset\rightrightarrows \, U_2 \, \underset\rightrightarrows \, U_1 \, \rightrightarrows\, U_0 where U_n is the image of /math> from \Delta in ''C''. A typical example is the Čech nerve of a map U \to X; i.e., U_0 = U, U_1 = U \times_X U, \dots. If ''F'' is a presheaf with values in an ∞-category and U_ a Čech nerve, then F(U_) is a cosimplicial diagram and saying F is a sheaf exactly means that F(X) is the limit of F(U_) for each U \to X in a Grothendieck topology. See also: simplicial presheaf. I ... [...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 invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Category
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition The simplex category is usually denoted by \Delta. There are several equivalent descriptions of this category. \Delta can be described as the category of ''non-empty finite ordinals'' as objects, thought of as totally ordered sets, and ''(non-strictly) order-preserving functions'' as morphisms. The objects are commonly denoted = \ (so that is the ordinal n+1 ). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Object
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. 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 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. 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. Simplicial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Čech Nerve
In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings. It captures many of the interesting topological properties in an algorithmic or combinatorial way. Basic definition Let I be a set of indices and C be a family of sets (U_i)_. The nerve of C is a set of finite subsets of the index set ''I''. It contains all finite subsets J\subseteq I such that the intersection of the U_i whose subindices are in J is non-empty:'', Section 4.3'' :N(C) := \bigg\. In Alexandrov's original definition, the sets (U_i)_ are open subsets of some topological space X. The set N(C) may contain singletons (elements i \in I such that U_i is non-empty), pairs (pairs of elements i,j \in I such that U_i \cap U_j \neq \emptyset), triplets, and so on. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Presheaf
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site. Examples Example: Consider the étale site of a scheme ''S''. Each ''U'' in the site represents the presheaf \operatorname(-, U). Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). Example: Let ''G'' be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf BG. For example, one might set B\operatorname = \varinjlim B\operatorname. These types of examples appe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action Groupoid
In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action :X \times G \to X, we get the groupoid \mathcal (= a category whose morphisms are all invertible) where *objects are elements of X, *morphisms from x to y are the actions of elements g in G such that y = xg, *compositions for x \overset\to y and y \overset\to z is x \overset\to z. A groupoid is often depicted using two arrows. Here the above can be written as: :X \times G \,\overset\underset\rightrightarrows\, X where s, t denote the source and the target of a morphism in \mathcal; thus, s(x, g) = x is the projection and t(x, g) = xg is the given group action (here the set of morphisms in \mathcal is identified with X \times G). In an ∞-category Let C be an ∞-category and G a groupoid object in it. Then a group action or an action groupoid on an object ''X'' in ''C'' is the simplicial diagram :\cdots \, \underset\rightrightarrows \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Groupoid
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division) or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the ''dividend'') by 3 (the ''divisor''), the ''quotient'' is 6 (with a remainder of 2) in the first sense and 6+\tfrac=6.66... (a repeating decimal) in the second sense. In metrology (International System of Quantities and the International System of Units), "quotient" refers to the general case with respect to the units of measurement of physical quantities. ''Ratios'' is the special case for dimensionless quotients of two quantities of the same kind. Quotients with a non-trivial dimension and compound units, especially when the divisor is a duration (e.g., "per second"), are known as ''rates''. For example, densi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Stack
In algebraic geometry, a quotient stack is a stack (mathematics), stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a Scheme (mathematics), scheme or a algebraic variety, variety by a Group (mathematics), group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks. Definition A quotient stack is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' an ''S''-scheme on which ''G'' group-scheme action, acts. Let the quotient stack [X/G] be the fibered category, category over the category of ''S''-schemes, where *an object over ''T'' is a torsor (algebraic geometry), principal ''G''-bundle P\to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stackification
In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks. Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category ''C''; for example, ''C'' can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology. Informal definition Let ''F'' be a category and suppose it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmented Simplex Category
Augment or augmentation may refer to: Language *Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages * Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns in certain Bantu languages *Augment, a name sometimes given to the verbal ''ō-'' prefix in Nahuatl grammar Technology * Augmentation (obstetrics), the process by which the first and/or second stages of an already established labour is accelerated or potentiated by deliberate and artificial means *Augmentation (pharmacology), the combination of two or more drugs to achieve better treatment results *Augmented reality, a live view of a physical, real-world environment whose elements are ''augmented'' by computer-generated sensory input *Augmented cognition, a research field that aims at creating revolutionary human-computer interactions *Augment (Tymshare), a hypertext system derived from Douglas Engelbart's oN-Line System, renamed "Augment ... [...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 invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
