Homotopy theory

American mathematician

thumb|The two dashed Path (topology)|paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

mathematical group of the homotopy classes of loops in a topological space

British mathematician (1930–1989)

type of continuous map in topology

Can be continuously shrunk to a point

type of topological space

in topology, a continuous function between two points

algebraic construct classifying topological spaces

thumb|The imaginary part of the complex logarithm. Trying to define the complex logarithm on \C-\{0\} gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of \C-\{0\} by a [[helicoid (an example of a Riemann surface).]]

a theorem in topology describing the fundamental group of a space in terms of a cover of the space by two open path-connected subspaces

fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers

subfield of algebraic topology dealing with structures invariant under homotopy equivalence

"one-point union" of a family of topological spaces

quotient space; operation of suspension creates a way of moving up in dimension (dimension n + 1)

right inverse of a fiber bundle map

space of basepoint preserving maps from a circle

topological space with a distinguished point

construction in categorical homotopy theory; contravariant functor from the simplex category to the category of sets

topological space

small category, whose objects are sets of natural numbers of the form {0,1,…,n}, and whose morphisms are nondecreasing functions

one-dimensional vector bundle

In mathematics, in particular homotopy theory, a continuous mapping between topological spaces

variant of type theory incorporating the univalence axiom of Voevodsky

homotopy theory in algebraic topology

topological space with homotopy concentrated in a single degree

binary operation between two pointed spaces: X ⨳ Y = (X × Y) / X ∨ Y

In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.

Property in algebraic topology

In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.

mathematical category with weak equivalences, fibrations and cofibrations

Mathematical theories

topological space equipped with a principal bundle with the property that any principal bundle (with the same fiber group) over a paracompact manifold is isomorphic to a pullback of the principal bundle over this topological space

in mathematics, a topological construction

Homotopy invariant of maps between spheres

property in algebraic topology

concept in topology

how spheres of various dimensions can wrap around each other

construction in algebraic topology that constructs a long (co)exact sequence from a (co)fibration

the study of spectra