Homotopical Algebra

In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases. The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categories.

Homotopy theory takes homotopy in a rather general sense including not only classical homotopy, but also algebraic homotopy, homotopical algebra in the narrow sense (model categories and variants) and (∞,1)-categories.