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Continuous deformation between two continuous functions
being called a homotopy (/həˈmɒtəpiː/ hə-MOT-ə-pee; /ˈhoʊmoʊˌtoʊpiː/ HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition
Homotopy
Type theory in logic and mathematics
In mathematical logic and computer science, homotopy type theory (HoTT) includes various lines of development of intuitionistic type theory, based on the
Homotopy_type_theory
How spheres of various dimensions can wrap around each other
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.
Homotopy_groups_of_spheres
Algebraic construct classifying topological spaces
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental
Homotopy_group
Branch of mathematics
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
Homotopy_theory
Mathematical theory of topological spaces
topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored
Rational_homotopy_theory
Concept in algebraic topology
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
Homotopy_sphere
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
Regular_homotopy
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:
Algebraic_homotopy
Application of homotopy to algebraic varieties
mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties
A¹_homotopy_theory
Mathematical group of the homotopy classes of loops in a topological space
is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger
Fundamental_group
Concept in math
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
Homotopy_category
Additive category in homological algebra
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
Homotopy category of chain complexes
Homotopy_category_of_chain_complexes
Hypothesis in mathematical category theory
category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces.
Homotopy_hypothesis
Branch of mathematics
topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study
Algebraic_topology
Property in algebraic topology
the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension
Homotopy_extension_property
In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of
Homotopy_Lie_algebra
1982 studio album by Nurse with Wound
Homotopy to Marie is the fifth album by Nurse with Wound, released in 1982. Although Nurse with Wound had generated considerable interest across their
Homotopy_to_Marie
topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they
Simple-homotopy_equivalence
Concepts in algebraic topology
algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle
Homotopy_colimit_and_limit
Mathematical object
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
Spectrum_(topology)
Branch of mathematics
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic"
Chromatic_homotopy_theory
Topological subject
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain
Stable_homotopy_theory
Index of articles associated with the same name
In mathematics, especially algebraic topology, the homotopy dimension of a topological space does not have a fixed meaning. However, it can refer to the
Homotopy_dimension
computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated
Numerical_algebraic_geometry
Concept in algebraic topology
{\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
Fibration
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 →
Simplicial_homotopy
Generalization of category theory
category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as the
Higher_category_theory
Technique to solve differential equations
The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method
Homotopy_analysis_method
Simplicial set constructed from the objects and morphisms of a small category
Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the
Nerve_(category_theory)
Topological construction defined up to homotopy
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration
Homotopy_fiber
Continuous, position-preserving mapping from a topological space into a subspace
For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Let X be a topological
Retraction_(topology)
Mapping which preserves all topological properties of a given space
deformations, such as the homeomorphism between a trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous
Homeomorphism
Partial differential equation technique
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial
Homotopy_principle
Results on triad homotopy groups
Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces. This connectivity result may be expressed more precisely
Blakers–Massey_theorem
Homological construction
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
Derived_category
Homotopy theory in algebraic topology
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting
Homotopy_lifting_property
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
Weak equivalence (homotopy theory)
Weak_equivalence_(homotopy_theory)
Mathematical construction used in homotopy theory
purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is
Simplicial_set
Analogue of homotopy type for algebraic varieties
mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly
Étale_homotopy_type
Type of topological space
It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial
CW_complex
Topological operation of turning a sphere inside-out without creasing
surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while
Sphere_eversion
On composition of paths in matroids
In mathematics, Tutte's homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly that
Tutte_homotopy_theorem
a fiber-homotopy equivalence is a map over a space B that has homotopy inverse over B (that is if h t {\displaystyle h_{t}} is a homotopy between the
Fiber-homotopy_equivalence
Topological construction on a map between spaces
In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted C f {\displaystyle
Mapping_cone_(topology)
Mathematical theories
constructing a section of a bundle. The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for
Obstruction_theory
Branch of mathematics
The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological
Topology
British mathematician (1904–1960)
as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in British India
J._H._C._Whitehead
directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental
Directed_algebraic_topology
that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so forth. They are ubiquitous
Homotopy_associative_algebra
Russian mathematician, writer, and film director
and independent filmmaker. He is a specialist in homological algebra, homotopy theory, and group theory. He was an invited speaker at the 7th European
Roman_Mikhaylov
mathematics, an A∞-operad is a type of operad used in algebraic topology and homotopy theory to describe algebraic structures where the property of associativity
A∞-operad
In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built
Puppe_sequence
On the homotopy groups of the infinite symmetric product of a connected CW complex
In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as
Dold–Thom_theorem
conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism)
Borel_conjecture
Gives a homomorphism from homotopy groups to homology groups
the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism
Hurewicz_theorem
Mathematics glossary
in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory
Glossary of algebraic topology
Glossary_of_algebraic_topology
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
Ring_spectrum
Mathematical category
associated to the site underlying a topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse
Topos
Theorem in vector calculus
or "homotopy"; the latter omit condition [TLH3]. So from now on we refer to homotopy (homotope) in the sense of theorem 2-1 as a tubular homotopy (resp
Stokes'_theorem
Class of topological space
of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces L ( 5 ; 1 ) {\displaystyle
Lens_space
before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent
Localization_of_a_category
Manifold with the same rational homotopy groups as a sphere
topology, a rational homotopy n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homotopy groups as the n {\displaystyle
Rational_homotopy_sphere
Functional programming language
Haskell (/ˈhæskəl/) is a general-purpose, statically typed, purely functional programming language with type inference and lazy evaluation. Haskell pioneered
Haskell
BG. When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise
Universal_bundle
Describes a periodicity in the homotopy groups of classical groups
mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which
Bott_periodicity_theorem
already cellular on a subcomplex A of X, then we can furthermore choose the homotopy to be stationary on A. From an algebraic topological viewpoint, any map
Cellular approximation theorem
Cellular_approximation_theorem
Techniques in topology used to produce one finite-dimensional manifold from another
some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy
Surgery_theory
Concept in homotopy theory
particular homotopy theory, a continuous mapping between topological spaces i : A → X {\displaystyle i:A\to X} is a cofibration if it has the homotopy extension
Cofibration
Describes the fundamental group in terms of a cover by two open path-connected subspaces
of nonabelian second relative homotopy groups, and in fact of homotopy 2-types. The second part applies a Higher Homotopy van Kampen Theorem for crossed
Seifert–Van_Kampen_theorem
Concept in algebraic topology
the chain complex. The resulting homology groups are the same for all homotopy equivalent spaces, which is the reason for their study. These constructions
Singular_homology
Topological space with only one nontrivial homotopy group
Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological
Eilenberg–MacLane_space
Differentiable function whose derivative is everywhere injective
regular homotopy is thus a homotopy through immersions. Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s
Immersion_(mathematics)
Offers a substitute for the absence of excision in homotopy theory
In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let ( X ; A
Homotopy_excision_theorem
Mathematical category with weak equivalences, fibrations and cofibrations
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences'
Model_category
Standard that diagrams must satisfy up to isomorphism
in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or
Coherency_(homotopy_theory)
British mathematician (1923–2010)
November 2010) was a British mathematician, noted for his contributions to homotopy theory and for code-breaking during World War II. He was born in Brondesbury
Peter_Hilton
Type of group in mathematics
the homotopy groups stabilize, and πk(O(n + 1)) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups
Orthogonal_group
Wiki for mathematics, physics, and philosophy
philosophy with a focus on methods from type theory, category theory, and homotopy theory. The nLab espouses the "n-point of view" (a deliberate pun on Wikipedia's
NLab
Topologically stable solution of a partial differential equation
which is explained by having the soliton belong to a different topological homotopy class or cohomology class than the base physical system. More simply: it
Topological_defect
Mathematical theory of data types
is an active area of research, one direction being the development of homotopy type theory. The first computer proof assistant, called Automath, used
Type_theory
American mathematician (born 1980)
of San Diego who works in category theory and higher category theory, homotopy theory, logic as applied to set theory, and computer science. Shulman did
Michael Shulman (mathematician)
Michael_Shulman_(mathematician)
Continuous function whose domain is a closed unit interval
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
Path_(topology)
Theorem in homotopy theory
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms
Whitehead_theorem
Mathematical theory
In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum
Sphere_spectrum
Tool in homological algebra
spaces. In the case of singular homology, a homotopy between continuous maps f, g : X → Y induces a chain homotopy between the chain maps corresponding to
Chain_complex
Concept in mathematics
A representation up to homotopy has several meanings. One of the earliest appeared in physics, in constrained Hamiltonian systems. The essential idea is
Representation_up_to_homotopy
American mathematician
University. Hopkins' work concentrates on algebraic topology, especially stable homotopy theory. It can roughly be divided into four parts (while the list of topics
Michael_J._Hopkins
Russian mathematician (1966–2017)
September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to
Vladimir_Voevodsky
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
Simple_homotopy_theory
American mathematician (1940–2011)
higher algebraic K-theory in 1972. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving problems in
Daniel_Quillen
Cohomology with real coefficients computed using differential forms
{\displaystyle d} restricted to closed forms has a local inverse called a homotopy operator. Since it is also nilpotent, it forms a dual chain complex with
De_Rham_cohomology
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
group Θ n {\displaystyle \Theta _{n}} of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is
Exotic_sphere
In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with
Equivariant stable homotopy theory
Equivariant_stable_homotopy_theory
In mathematics, especially category theory, the homotopy category of an ∞-category C is the category where the objects are those in C but the hom-set from
Homotopy category of an ∞-category
Homotopy_category_of_an_∞-category
Quotient of a weakly contractible space by a free action
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e
Classifying_space
widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental
Fundamental_groupoid
Concept in mathematics
configuration space of two points in R n {\displaystyle \mathbf {R} ^{n}} is homotopy equivalent to the sphere S n − 1 {\displaystyle S^{n-1}} . The configuration
Configuration space (mathematics)
Configuration_space_(mathematics)
American mathematician
American mathematician who has contributed to higher category theory and homotopy theory. Much of her work, including her PhD thesis, concerns model structures
Emily_Riehl
Topics referred to by the same term
Look up product in Wiktionary, the free dictionary. Product may refer to: Product (business), an item that can be offered to a market to satisfy the desire
Product
HOMOTOPY
HOMOTOPY
HOMOTOPY
HOMOTOPY
Girl/Female
Arabic, Muslim
Silver
Boy/Male
Tamil
Gyanendra | ஜà¯à®žà®¾à®¨à¯‡à®¨à¯à®¤à¯à®°Â
Knowledge
Boy/Male
Tamil
Prabakaran | பà¯à®°à®ªà®•à®°à®£
Male
Danish
, peace ruler.
Boy/Male
Tamil
King, Arjun
Girl/Female
Tamil
Agrima | அகà¯à®°à¯€à®®à®¾
Leadership
Girl/Female
American, German, Italian
Whimsical; Unpredictable; Fanciful; Ruled by Whim; Impulsive
Boy/Male
Tamil
Narendar | நரேநà¯à®¤à¯à®°
Leader of all human beings, King of men, The king
Girl/Female
Arabic, Bengali, Hindu, Indian, Kannada, Muslim, Telugu, Traditional
Goddess
Boy/Male
Indian, Punjabi, Sikh
Union with Naam
HOMOTOPY
HOMOTOPY
HOMOTOPY
HOMOTOPY
HOMOTOPY