Kleisli category

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Kleisli category

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)

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

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