Homotopiat

Homotopiat is a term that appears in some mathematical writings to denote a generalized deformation between continuous maps. It is not a standard notion in mainstream topology, and there is no single accepted definition; the phrase is used variably to indicate a family of deformations parameterized by an auxiliary space.

Conceptually, a homotopiat between two maps f: X -> Y and g: X -> Y is described by a continuous map H: X × P -> Y for some topological parameter space P, subject to boundary conditions that relate the start and end maps to f and g in a way that depends on P. Different authors impose different conditions, such as fixing the end of the deformation along a distinguished subspace of P or requiring H(x, p0) = f(x) and H(x, p1) = g(x).

Special cases help situate the idea: if P is the unit interval I, the data specify a standard homotopy between f and g. If P is contracted to a point, the data typically force f and g to coincide (a degenerate case). In many discussions, P is chosen to encode additional symmetry, parameter families, or higher-categorical structure, linking homotopiat notions with parametrized homotopy theory and higher deformation theory.

Because the term is informal, it is important to read the precise definition given in any text that introduces it. It is related to, but distinct from, standard homotopy, parametrized homotopy, and higher-category notions of deformation.

See also: Homotopy, Parametrized homotopy, Higher category theory, Homotopy type theory.