Kleisli category

English Etymology Named after the Swiss mathematician Heinrich Kleisli (1930–2011). Noun thumb|right|400px|Commutative diagram of function composition in a Kleisli category. Given a monad , consider a Kleisli category over that monad. Morphisms , , and in correspond to morphisms , , and in , respectively. The composition rule is . The Kleisli category shares the same objects as its underlying category. The morphisms of the Kleisli category (e.g.: and ) are embellished versions of the morphisms of its underlying category (e.g.: f and g), and they are derived from those of the underlying category by means of applying a monad to their codomains. en category theory A category naturally associated to any monad T, and equivalent to the category of free T-algebras. en eponyms