nLab higher homotopy

Definition

Higher homotopies are homotopies of homotopies, existing in higher homotopy types.

The generalization of mathematical structures to the situation where identities/laws hold only up to coherent higher homotopies are now often known as higher structures.

Basic examples in higher algebra (“brave new algebra”) are $A_\infty$ algebras, $E_\infty$-algebras and $L_\infty$-algebras.

Related concepts

• homotopy theory

• identity type

• higher structures

  • higher category theory, higher topos theory

  • higher geometry

  • higher observational type theory

  • higher gauge transformation

  • higher differential geometry

  • higher Lie theory

  • higher prequantum geometry

  • higher complex analytic geometry

  • higher algebra, homotopical algebra

  • higher linear algebra

  • higher category theory and physics

References

Some history in the context of higher structures:

• Johannes Huebschmann: Origins and breadth of the theory of higher homotopies in: Higher Structures in Geometry and Physics – In Honor of Murray Gerstenhaber and Jim Stasheff, Progress in Mathematics 287, Birkhäuser (2011) 25-38 [arXiv:0710.2645, doi:10.1007/978-0-8176-4735-3]