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...

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental...

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....

In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on...

is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger...

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...

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...

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...

In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial...

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...

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...

computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated...

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...

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting...

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration...

mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties...

Haskell (/ˈhæskəl/) is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for...

topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study...

{\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...

In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted C f {\displaystyle...

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...

cross-section of a bundle. The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for...

in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory...

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states (very roughly speaking) that the ∞-groupoids are spaces. One version...

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...

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...

topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they...

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...

It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial...

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...