Kleisli category

English

Etymology

Named after the Swiss mathematician Template:W (1930–2011).

Noun

Commutative diagram of function composition in a Kleisli category. Given a monad $(T,\eta ,\mu )=(T:𝒞\to 𝒞,\eta :{\text{id}}_{𝒞}\to T,\mu :T^2\to T)$, consider a Kleisli category $𝒦$ over that monad. Morphisms $\tilde{f}:A\to B$, $\tilde{g}:B\to C$, and $\tilde{h}=\tilde{g}\circ \tilde{f}:A\to C$ in $𝒦$ correspond to morphisms $f:A\to TB$, $g:B\to TC$, and $h:A\to TC$ in $𝒞$, respectively. The composition rule is $\tilde{g}\circ \tilde{f}={({\mu }_{\text{cod}(g)}\circ Tg\circ f)}^{\sim }$. The Kleisli category shares the same objects as its underlying category. The morphisms of the Kleisli category (e.g.: $\tilde{f}$ and $\tilde{g}$) 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.

1. Template:Lb A category naturally associated to any monad T, and equivalent to the category of free T-algebras.