Extension Operators and Kleisli Triples
Composition of Kleisli arrows can be expressed succinctly by means of the extension operator (-)* : Hom(X, TY) → Hom(TX, TY). Given a monad 〈T, η, μ〉over a category C and a morphism f : X → TY let
Composition in the Kleisli category CT can then be written
The extension operator satisfies the identities:
where f : X → TY and g : Y → TZ. It follows trivially from these properties that Kleisli composition is associative and that ηX is the identity.
In fact, to give a monad is to give a Kleisli triple, i.e.
- A function ;
- For each object in, a morphism ;
- For each morphism in, a morphism
such that the above three equations for extension operators are satisfied.
Read more about this topic: Kleisli Category
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