homotopy

American  

[huh-mot-uh-pee, hoh-] / həˈmɒt ə pi, hoʊ- /

noun

Mathematics.

homotopies plural

1. the relation that exists between two mappings in a topological space if one mapping can be deformed in a continuous way to make it coincide with the other.

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Other Word Forms

Noun Inflected Forms

• homotopies
  plural

Etymology

Origin of homotopy

1915–20; homo- + -topy (< Greek tóp ( os ) place + -y ³, or < New Latin -topia )

Example Sentences

See Examples For:

homotopy homotopies

The researchers provide a unified skyrmion-hopfion homotopy classification and offer an insight into the diversity of topological solitons in three-dimensional chiral magnets.

From Science Daily ● Nov. 22, 2023

But contractability guarantees that any two composite paths are homotopic, any two homotopies relating two composite paths are connected by a higher homotopy, and so on.

From Scientific American ● Sep. 14, 2021

He added another invariant, known as the fundamental group, and believed that if a manifold had the same homotopy and fundamental group as a sphere, it had to be a sphere.

From Scientific American ● Jun. 4, 2017

Vladimir Voevodsky revolutionized algebraic geometry and is best known for developing the new field of ‘motivic homotopy theory’.

From Nature

In Voevodsky’s motivic homotopy theory, familiar classical geometry was replaced by homotopy theory — a branch of topology in which a line may shrink all the way down to a point.

From Nature

For example, in π ∞ X the objects and arrows are the points and the paths—no longer considered up to wiggling—while the higher-dimensional transformations encode the higher homotopies.

From Scientific American ● Sep. 14, 2021

These deformations, which are technically called homotopies, are necessary for the composition of paths to define an associative operation, as is required by a category.

From Scientific American ● Sep. 14, 2021

But contractability guarantees that any two composite paths are homotopic, any two homotopies relating two composite paths are connected by a higher homotopy, and so on.

From Scientific American ● Sep. 14, 2021