homotopy class in nLab
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
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Definitions
Paths and cylinders
Homotopy groups
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Contents
Definition
A a homotopy class is an equivalence class under homotopy:
For f:X→Yf \;\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 ff. The sets of such homotopy classes if often denoted [X,Y][X,Y] or similar.
Similarly, if ff 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] *[X,Y]_\ast or similar.
Examples
Homotopy groups and Cohomotopy sets
If the domain X≃S nX \simeq S^n is an n-sphere, then the homotopy classes of maps f:S n→Yf \colon S^n \to Y form the nnth homotopy group of YY.
If the codomain is an n-sphere Y≃S nY \simeq S^n, then the homotopy classes of maps f:X→S nf \colon X \to S^n form the Cohomotopy set of XX.
Last revised on November 1, 2022 at 08:43:42. See the history of this page for a list of all contributions to it.