nLab null homotopy

Definition

In the context of algebraic topology, a continuous map $f \colon X \longrightarrow Y$ between two topological spaces $X,Y$, is said to be null homotopic if it is homotopic to a constant map $g$. This is considered in particular in the context of pointed topological spaces with base point-preserving maps between them, hence for $g$ the map constant on the base point of $Y$ (which is the zero morphism in this context).

In the context of homological algebra, a null homotopy is a chain homotopy from (or to) the zero map.

Generally in abstract homotopy theory, a null homotopy in an pointed (infinity,1)-category is a 2-morphism to (or from) a zero morphism.

This general concept subsumes the previous two cases via the (infinity,1)-categories presented by the classical model structure on pointed topological spaces or a model structure on chain complexes, respectively.

Related concepts

• homotopy category of chain complexes

• contractible chain complex

References

Textbook accounts:

• James Munkres, Section 51 of: Introduction to Topology

• Charles Weibel, Section 1.4 of: An Introduction to Homological Algebra