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
A a homotopy class is an equivalence class under homotopy:
For 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 . The sets of such homotopy classes if often denoted or similar.
Similarly, if 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 or similar.
If the domain is an n-sphere, then the homotopy classes of maps form the th homotopy group of .
If the codomain is an n-sphere , then the homotopy classes of maps form the Cohomotopy set of .
Last revised on November 1, 2022 at 08:43:42. See the history of this page for a list of all contributions to it.