Kleisli category

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