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Ran Space
In mathematics, the Ran space (or Ran's space) of a topological space ''X'' is a topological space \operatorname(X) whose underlying set is the set of all nonempty finite subsets of ''X'': for a metric space ''X'' the topology is induced by the Hausdorff distance. The notion is named after Ziv Ran. Definition In general, the topology of the Ran space is generated by sets : \ for any disjoint open subsets U_i \subset X, i = 1, ..., m. There is an analog of a Ran space for a scheme: the Ran prestack of a quasi-projective scheme ''X'' over a field ''k'', denoted by \operatorname(X), is the category whose objects are triples (R, S, \mu) consisting of a finitely generated ''k''-algebra ''R'', a nonempty set ''S'' and a map of sets \mu: S \to X(R), and whose morphisms (R, S, \mu) \to (R', S', \mu') consist of a ''k''-algebra homomorphism R \to R' and a surjective map S \to S' that commutes with \mu and \mu'. Roughly, an ''R''-point of \operatorname(X) is a nonempty finite set ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Chiral Homology
In mathematics, chiral homology, introduced by Alexander Beilinson and Vladimir Drinfeld, is, in their words, "a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine \mathcal_X-scheme (i.e., the space of global solutions of a system of non-linear differential equations)." Jacob Lurie's topological chiral homology gives an analog for manifolds. See also *Ran space * Chiral Lie algebra *Factorization homology In algebraic topology and category theory, factorization homology is a variant of topological chiral homology, motivated by an application to topological quantum field theory and cobordism hypothesis In mathematics, the cobordism hypothesis, due to ... References *{{cite book, last1=Beilinson, first1=Alexander, authorlink1=Alexander Beilinson, last2=Drinfeld, first2=Vladimir, authorlink2=Vladimir Drinfeld , title=Chiral algebras, date=2004, publisher=American Mathematical Society, isbn=0-8218-3528-9, chapter=Chap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Type
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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. Bimodules over ''A'' are essentially the same as modules over the enveloping algebra of ''A'', so in particular ''A'' and ''M'' can be considered as ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-algebra th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosheaf
In topology, a branch of mathematics, a cosheaf with values in an ∞-category ''C'' that admits colimits is a functor ''F'' from the category of open subsets of a topological space ''X'' (more precisely its nerve) to ''C'' such that *(1) The ''F'' of the empty set is the initial object. *(2) For any increasing sequence U_i of open subsets with union ''U'', the canonical map \varinjlim F(U_i) \to F(U) is an equivalence. *(3) F(U \cup V) is the pushout of F(U \cap V) \to F(U) and F(U \cap V) \to F(V). The basic example is U \mapsto C_*(U; A) where on the right is the singular chain complex of ''U'' with coefficients in an abelian group ''A''. Example:http://www.math.harvard.edu/~lurie/282ynotes/LectureIX-NPD.pdf If ''f'' is a continuous map, then U \mapsto f^(U) is a cosheaf. See also *sheaf (mathematics) In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weakly Contractible
In mathematics, a topological space is said to be weakly contractible if all of its homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...s are trivial. Property It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible. Example Define S^\infty to be the inductive limit of the spheres S^n, n\ge 1. Then this space is weakly contractible. Since S^\infty is moreover a CW-complex, it is also contractible. See Contractibility of unit sphere in Hilbert space for more. The Long Line is an example of a space which is weakly contractible, but not contractible. This does not contradict Whitehead theorem since the Long Line does not have the homotopy type of a CW-complex. Another prominent example for this phenomenon is the War ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Complex
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has played a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
