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) - 09:41, 30 July 2024
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 the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....
82 KB (7,972 words) - 00:30, 24 August 2024
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on...
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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) - 07:10, 6 August 2024
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...
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branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology...
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of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter...
6 KB (862 words) - 03:51, 27 March 2024
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial...
11 KB (1,721 words) - 01:40, 6 January 2024
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the...
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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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Path (topology) (redirect from Path homotopy)
also define paths and loops in pointed spaces, which are important in homotopy theory. If X {\displaystyle X} is a topological space with basepoint x...
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Numerical algebraic geometry (redirect from Homotopy continuation)
computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated...
10 KB (1,283 words) - 06:20, 11 March 2024
mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences...
6 KB (1,051 words) - 14:31, 3 January 2023
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting...
6 KB (843 words) - 03:37, 30 April 2024
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration...
10 KB (1,853 words) - 22:06, 16 July 2024
Haskell (redirect from Homotopy Haskell)
Haskell (/ˈhæskəl/) is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for...
49 KB (4,557 words) - 10:46, 27 August 2024
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
Fibration (section Homotopy lifting property)
{\displaystyle p\colon E\to B} satisfies the homotopy lifting property for a space X {\displaystyle X} if: for every homotopy h : X × [ 0 , 1 ] → B {\displaystyle...
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Mapping cone (topology) (redirect from Homotopy cofiber)
In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted C f {\displaystyle...
8 KB (1,311 words) - 00:39, 17 August 2024
Spectrum (topology) (redirect from Stable homotopy category)
technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing...
21 KB (3,451 words) - 18:25, 26 March 2024
Obstruction theory (redirect from Obstruction (homotopy))
cross-section of a bundle. The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for...
8 KB (1,085 words) - 08:53, 4 April 2024
Glossary of algebraic topology (redirect from Chain homotopy equivalence)
in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory...
52 KB (7,629 words) - 12:07, 26 July 2024
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states (very roughly speaking) that the ∞-groupoids are spaces. One version...
6 KB (572 words) - 00:28, 5 August 2024
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...
7 KB (868 words) - 08:22, 10 April 2020
terms. A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra and the derived category of a ring are both...
29 KB (4,503 words) - 21:16, 26 April 2024
Directed algebraic topology (redirect from Directed homotopy)
directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental...
18 KB (2,358 words) - 17:23, 12 August 2023
topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they...
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In algebraic topology, a fiber-homotopy equivalence is a homotopy equivalence between fibers of maps into a space B from spaces D and E (that is, a map...
3 KB (672 words) - 00:14, 31 December 2023
CW complex (category Homotopy theory)
It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial...
23 KB (3,419 words) - 18:36, 23 August 2024