homotopically

Homotopically is an adverb derived from the term "homotopy," which is a concept in topology, a branch of mathematics. It is used to describe two objects that are equivalent in a certain sense, specifically that one can be continuously deformed into the other. This deformation is known as a homotopy.

In the context of topology, two spaces are said to be homotopically equivalent if there exists a continuous map from one space to the other and a continuous map in the opposite direction such that the composition of these maps is homotopic to the identity maps on the respective spaces. This equivalence is weaker than homeomorphism, which requires a bijective continuous map with a continuous inverse.

The term "homotopically" is often used in various mathematical contexts, such as in homotopy theory, algebraic topology, and differential topology. It is also used in more general contexts to describe situations where one object can be transformed into another in a continuous manner, such as in the study of shapes and deformations in geometry and physics.