nLab homotopy class

Definition

A a homotopy class is an equivalence class under homotopy:

For $f \;\colon\; X \to Y$ 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 $f$. The sets of such homotopy classes if often denoted $[X,Y]$ or similar.

Similarly, if $f$ 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 $[X,Y]_\ast$ or similar.

Examples

Homotopy groups and Cohomotopy sets

If the domain $X \simeq S^n$ is an n-sphere, then the homotopy classes of maps $f \colon S^n \to Y$ form the $n$th homotopy group of $Y$.

If the codomain is an n-sphere $Y \simeq S^n$, then the homotopy classes of maps $f \colon X \to S^n$ form the Cohomotopy set of $X$.

Related concepts

• homotopy category

• cohomology

• isomorphism class