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Simplicial Homotopy Class
In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if :f, g: X \to Y are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map :h: X \times \Delta^ \to Y such that the restriction of h along X \simeq X \times \Delta^ \overset\hookrightarrow X \times \Delta^ is f and the restriction along 1 is g; se In particular, f(x) = h(x, 0) and g(x) = h(x, 1) for all ''x'' in ''X''. Using the adjunction :\operatorname(X \times \Delta^1, Y) = \operatorname(\Delta^1 \times X, Y) = \operatorname(\Delta^1, \underline(X, Y)), the simplicial homotopy h can also be thought of as a path in the simplicial set \underline(X, Y). A simplicial homotopy is in general not an equivalence relation. However, if \underline(X, Y) is a Kan complex (e.g., if Y is a Kan complex), then a homotopy from f : X \to Y to g : X \to Y is an equivalence relation. Indeed, a Kan complex is an ∞-groupoid; i.e., every ... [...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] |
Kan Complex
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets. Definitions Definition of the standard n-simplex For each ''n'' ≥ 0, recall that the standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ( Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n-simplex: the convex subspace of \mathbb^ consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negative and sum to 1. Definition of a horn For each ''k'' ≤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
