Kleisli category

A Kleisli category is a category that naturally arises from a monad . It is named after the Swiss mathematician Heinrich Kleisli .

definition

Let be a category and a monad, with as endofunctor and , as the monoid operations specified on it . The Kleisli category that belongs to and is referred to below as . The objects and morphisms in it are ${\displaystyle C}$ ${\displaystyle M=(T,\mu ,\eta )}$ ${\displaystyle T\colon C\to C}$ ${\displaystyle \mu \colon T^{2}\to T}$ ${\displaystyle \eta \colon 1\to T}$ ${\displaystyle C}$ ${\displaystyle M}$ ${\displaystyle C_{M}}$

• ${\displaystyle \operatorname {Ob} (C_{M})=\operatorname {Ob} (C)}$, such as
• ${\displaystyle \operatorname {Mor} _{C_{M}}(X,Y)=\operatorname {Mor} _{C}(X,T(Y))}$.

Identity morphisms and concatenation are

• ${\displaystyle \operatorname {id} _{A}=\eta _{A}}$ and
• ${\displaystyle f\circ _{C_{M}}g=\mu \circ _{C}T(f)\circ _{C}g}$.

Examples

• Correspondence forms a Kleisli category. The endo-functor on Set is the formation of the power quantity ,, with . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}(f)(A)=\{f(a)|a\in A\}}$

literature

• Saunders Mac Lane : Categories for the Working Mathematician (= Graduate Texts in Mathematics . 5). 2nd edition. Springer, New York et al. 1998, ISBN 0-387-98403-8 .