Homotopy Structures for Algebras over a Monad

Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A → A, or path-endofunctor, can be lifted to the category AT of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation λ : T P → P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of ‘algebras’ for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the ‘homotopy operations’ (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to AT, just by verifying the consistency between these operations and λ : T P → P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy.