homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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(also nonabelian homological algebra)
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.
Textbook accounts:
James Munkres, Section 51 of: Introduction to Topology
Charles Weibel, Section 1.4 of: An Introduction to Homological Algebra
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