Definition. Kleisli category of a promonad [mde-0008]
Definition. Kleisli category of a promonad [mde-0008]
Let \((T, \eta , \mu )\) be a promonad on a category \(ℂ\). Its Kleisli category \(\textsf {kl}(T)\) has
- Objects: those of \(ℂ\)
- Morphisms: \(\textsf {kl}(T)(A;B) := T(A;B)\)
- Identities: given by \(\eta _{A,A}(\textsf {id}_{A})\)
- Composition: given by \(\mu \).
Note that when \(T\) is a representable profunctor, i.e. \(T(A;B) = ℂ(A; FB)\) for some functor \(F\), this is just the usual Kleisli category for a monad.