nLab
homotopy class

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

A a homotopy class is an equivalence class under homotopy:

For f:XYf \;\colon\; X \to Y a continuous function between topological spaces which admit the structure of CW-complexes, its homotopy class is the morphism in the classical homotopy category that is represented by ff. The sets of such homotopy classes if often denoted [X,Y][X,Y] or similar.

Similarly, if ff is a base-point preserving function between pointed topological spaces admitting the structure of CW-complexes, then its homotopy class (“pointed homomotopy class”) represents a morphism in the homotopy category of pointed homotopy types. The sets of such pointed homotopy classes if often denoted [X,Y] *[X,Y]_\ast or similar.

Examples

Homotopy groups and Cohomotopy sets

If the domain XS nX \simeq S^n is an n-sphere, then the homotopy classes of maps f:S nXf \colon S^n \to X form the nnth homotopy group of YY.

If the codomain is an n-sphere YS nY \simeq S^n, then the homotopy classes of maps f:XS nf \colon X \to S^n form the Cohomotopy set of XX.

Last revised on April 5, 2021 at 02:22:49. See the history of this page for a list of all contributions to it.