homotopy theory

plural homotopy theories

1. countableuncountablemathematicsThe 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.

   3 more examples

   1988, Alex Heller, Homotopy Theories, American Mathematical Society, https://books.google.com.au/books?id=hkXUCQAAQBAJ&printsec=frontcover&dq=%22homotopy+theories%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiAzeb7qKfsAhUayTgGHeEyBZQQ6AEwAHoECAAQAg#v=onepage&q=%22homotopy%20theories%22&f=false.

   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.

   1989, Hans Joachim Baues, Algebraic Homotopy, Cambridge University Press, page 371

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

   2010, Julia E. Bergner, “A Survey of (∞,1)-Categories”, in John C. Baez, J. Peter May, editors, Towards Higher Categories, Springer,, page 69