Noun
homotopy theory (countable and uncountable, plural homotopy theories)
(mathematics) The systematic study of the situation of maps' having a homotopy between them; the study of equivalence classes (called homotopy classes) of maps.
Homotopy theory originated as a topic in algebraic topology, but is now studied as an independent discipline in its own right, and as a tool in fields including category theory and algebraic geometry.