homotopical

Homotopical is an adjective used in mathematics to describe theories, constructions, or viewpoints that involve homotopy theory or that are invariant under homotopy. In practice, a process is called homotopical if it respects homotopy equivalence, so that its outcome depends only on the homotopy type of the input rather than on a particular presentation.

In category theory and algebra, a functor is said to be homotopical if it sends weak equivalences to weak equivalences, allowing it to descend to a functor on the homotopy category or its localization. This idea is central to model category theory, where one organizes objects with classes of weak equivalences, fibrations, and cofibrations to study objects up to homotopy.

Homotopical algebra is a program begun by Quillen to apply homotopy-theoretic methods to algebraic problems. It uses model categories to formalize derived or homotopy-invariant versions of constructions such as tensor products, Hom-objects, and module categories, leading to derived categories and other localization techniques.

Higher or derived perspectives replace ordinary categories with ∞-categories or related frameworks, providing a language in which homotopical ideas can be encoded more flexibly. These methods have widespread use in topology, algebraic geometry (for example in derived or spectral algebraic geometry), and homological algebra, where invariants are examined up to homotopy.

See also: model category, Quillen functor, weak equivalence, derived functor.