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...

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

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

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...

topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored...

mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point...

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

In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain...

in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or...

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

behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants...

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...

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...

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

purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is...

polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Shape theory was invented...

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:...

theory — Galois theory — Game theory — Gauge theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory —...

map 0 to 0. Mathematics portal History of topos theory Homotopy hypothesis Intuitionistic type theory ∞-topos Quasitopos Geometric logic Generalized space...

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

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

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

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

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...

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...

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...

Cobordism Thom space Suspension functor Stable homotopy theory Spectrum (homotopy theory) Morava K-theory Hodge conjecture Weil conjectures Directed algebraic...

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...

wedge product) Internal product, in a monoidal category Product (category theory), a generalization of mathematical products Fibre product or pullback Coproduct...

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...