Kleisli category

English

Etymology

Named after the Swiss mathematician Heinrich Kleisli (1930–2011).

Noun

Commutative diagram of function composition in a Kleisli category. Given a monad $(T,\eta ,\mu )=(T:{\mathcal {C}}\rightarrow {\mathcal {C}},\ \eta :{\mbox{id}}_{\mathcal {C}}\rightarrow T,\ \mu :T^{2}\rightarrow T)$, consider a Kleisli category ${\mathcal {K}}$ over that monad. Morphisms ${\tilde {f}}:A\rightarrow B$, ${\tilde {g}}:B\rightarrow C$, and ${\tilde {h}}={\tilde {g}}\circ {\tilde {f}}:A\rightarrow C$ in ${\mathcal {K}}$ correspond to morphisms $f:A\rightarrow TB$, $g:B\rightarrow TC$, and $h:A\rightarrow TC$ in ${\mathcal {C}}$, 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.

Kleisli category (plural Kleisli categories)

1. (category theory) A category naturally associated to any monad T, and equivalent to the category of free T-algebras.