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Homotopy Sheaf
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 appea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Actions
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-group (category Theory)
In mathematics, an ''n''-group, or ''n''-dimensional higher group, is a special kind of ''n''-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of under the moniker 'gr-category'. The general definition of n-group is a matter of ongoing research. However, it is expected that every topological space will have a ''homotopy '' at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group \pi_n, or the entire Postnikov tower for n=\infty. Examples Eilenberg-Maclane spaces One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces K(A,n) since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group G can be turned into an Eilenberg-Maclane space K(G,1) through a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubical Set
In topology, a branch of mathematics, a cubical set is a set-valued contravariant functor on the category of (various) ''n''-cubes. Cubical sets have been often considered as an alternative to simplicial sets in combinatorial topology, including in the early work of Daniel Kan and Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inau .... They have also been developed in computer science, in particular in concurrency theory and in homotopy type theory. See also * Simplicial presheaf References * nLabCubical set * Rick JardineCubical sets Lecture 12 in "Lectures on simplicial presheaves" https://web.archive.org/web/20110104053206/http://www.math.uwo.ca/~jardine/papers/sPre/index.shtml Topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerbe
In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them. "Gerbe" is a French (and archaic English) word that literally means wheat sheaf. Definitions Gerbes on a topological space A gerbe on a topological space S is a stack \mathcal of groupoids over S that is ''locally non-empty'' (each point p \in S has an open neighbourhood U_p over which the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obstruction Theory
Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway obstruction * Bowel obstruction, a blockage of the intestines. * Gastric outlet obstruction * Distal intestinal obstruction syndrome * Congenital lacrimal duct obstruction * Bladder outlet obstruction Politics and law * Obstruction of justice, the crime of interfering with law enforcement * Obstructionism, the practice of deliberately delaying or preventing a process or change, especially in politics * Emergency Workers (Obstruction) Act 2006 Science and mathematics * Obstruction set in forbidden graph characterizations, in the study of graph minors in graph theory * Obstruction theory, in mathematics * Propagation path obstruction ** Single Vegetative Obstruction Model Sports * Obstruction (baseball), when a fielder illegally hinders a baser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Limit
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, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Equivalence (homotopy Theory)
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 equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercovering
In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to X in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover is to instead of only working with n-fold intersections of the sets of the given open cover \mathcal U, to allow the pairwise intersections of the sets in \mathcal U=\mathcal U_0 to be covered by an open cover \mathcal U_1, and to let the triple intersections of this cover to be covered by yet another open cover \mathcal U_2, and so on, iteratively. Hypercoverings have a central role in étale homotopy and other areas where homotopy theory is applied to algebraic geometry, such as motivic hom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reedy Category
In mathematics, especially category theory, a Reedy category is a category ''R'' that has a structure so that the functor category from ''R'' to a model category ''M'' would also get the induced model category structure. A prototypical example is the simplex category or its opposite. It was introduced by Christopher Reedy in his unpublished manuscript. Definition A Reedy category consists of the following data: a category ''R'', two wide WIDE or Wide may refer to: * Wide (cricket), a type of illegal delivery to a batter *Wide and narrow data Wide and narrow (sometimes un-stacked and stacked, or wide and tall) are terms used to describe two different presentations for tabular data ... (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees. Note some authors such as nlab require each factorizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective And Projective Model Structure
In higher category theory in mathematics, injective and projective model structures are special model structures on functor categories into a model category. Both model structures ''do not have'' to exist, but there are conditions guaranteeing their existence. An important application is for the study of limits and colimits, which are functors from a functor category and can therefore be made into Quillen adjunctions. Definition Let \mathcal be a small category and \mathcal be a model category. For two functors F,G\colon \mathcal\rightarrow\mathcal, a natural transformation \eta\colon F\Rightarrow G is composed of morphisms \eta_X\colon FX\rightarrow GX in \operatorname\mathcal for all objects X in \operatorname\mathcal. For those it hence be studied if they are fibrations, cofibrations and weak equivalences, which might lead to a model structure on the functor category \operatorname(\mathcal,\mathcal). * ''Injective cofibrations'' and ''injective weak equivalences'' are the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
