homotopy theory

homotopy theory (countable and uncountable, plural homotopy theories)

1. (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.

   • 1988, Alex Heller, Homotopy Theories, American Mathematical Society, .
   • 1989, Hans Joachim Baues, Algebraic Homotopy, Cambridge University Press, page 371:

     This is our main result, deduced from the axioms of a cofibration category, which leads to many new theorems on the homotopy classification problems in topology and in various algebraic homotopy theories.

   • 2010, Julia E. Bergner, “A Survey of ${\displaystyle (\infty ,1)}$-Categories”, in John C. Baez, J. Peter May, editors, Towards Higher Categories, Springer,, page 69:

     They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories.

• abstract homotopy theory
• A¹ homotopy theory
• rational homotopy theory

• fundamental group
• homotopic
• homotopy class
• homotopy group
• homotopy hypothesis