# nLab null homotopy

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## 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.

## References

Textbook accounts:

Last revised on February 10, 2021 at 09:12:20. See the history of this page for a list of all contributions to it.