In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic...
19 KB (3,171 words) - 07:04, 19 September 2024
logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation...
39 KB (4,433 words) - 11:07, 19 August 2024
mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties...
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being called a homotopy (/həˈmɒtəpiː/, hə-MO-tə-pee; /ˈhoʊmoʊˌtoʊpiː/, HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition...
23 KB (3,271 words) - 19:34, 3 September 2024
topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored...
25 KB (3,945 words) - 18:51, 26 January 2024
mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point...
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In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental...
20 KB (3,417 words) - 21:07, 23 November 2023
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain...
4 KB (669 words) - 23:26, 17 August 2023
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....
82 KB (7,977 words) - 00:48, 18 September 2024
in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or...
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behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants...
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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...
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Spectrum (topology) (redirect from Spectrum (homotopy theory))
evaluating the cohomology theory in degree k {\displaystyle k} on a space X {\displaystyle X} is equivalent to computing the homotopy classes of maps to the...
21 KB (3,451 words) - 18:25, 26 March 2024
cross-section of a bundle. The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension...
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Simplicial set (redirect from Simplicial homotopy theory)
purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is...
23 KB (3,327 words) - 19:12, 4 March 2024
In mathematics, algebraic homotopy is a research program on homotopy theory proposed by J.H.C. Whitehead in his 1950 ICM talk, where he described it as:...
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polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Shape theory was invented...
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theory — Galois theory — Game theory — Gauge theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory —...
38 KB (4,356 words) - 22:54, 24 August 2024
Topos (redirect from Topos theory)
map 0 to 0. Mathematics portal History of topos theory Homotopy hypothesis Intuitionistic type theory ∞-topos Quasitopos Geometric logic Generalized space...
33 KB (4,369 words) - 16:45, 17 September 2024
Fundamental group (redirect from First homotopy group)
is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger...
53 KB (8,076 words) - 11:44, 12 September 2024
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting...
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Cobordism Thom space Suspension functor Stable homotopy theory Spectrum (homotopy theory) Morava K-theory Hodge conjecture Weil conjectures Directed algebraic...
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Algebraic topology (section Homotopy groups)
topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study...
19 KB (2,081 words) - 18:42, 13 April 2024
CW complex (category Homotopy theory)
was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial complexes...
23 KB (3,419 words) - 18:36, 23 August 2024
Postnikov system (category Homotopy theory)
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups...
19 KB (3,645 words) - 16:45, 22 May 2024
mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was...
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Ring spectrum (redirect from Ring spectrum (homotopy theory))
In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map μ: E ∧ E → E and a unit map η: S → E, where S is the sphere...
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Bott periodicity theorem (category Theorems in homotopy theory)
much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can...
13 KB (1,836 words) - 15:27, 3 May 2024
Product (section Homotopy theory)
wedge product) Internal product, in a monoidal category Product (category theory), a generalization of mathematical products Fibre product or pullback Coproduct...
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and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other...
13 KB (1,749 words) - 11:06, 19 August 2024