nLab
null homotopy

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

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

In the context of algebraic topology, a continuous map f:XYf \colon X \longrightarrow Y between two topological spaces X,YX,Y, is said to be null homotopic if it is homotopic to a constant map gg. This is considered in particular in the context of pointed topological spaces with base point-preserving maps between them, hence for gg the map constant on the base point of YY (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.