Kleisli Category - Extension Operators and Kleisli Triples

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:

\begin{align}\eta_X^* &= \mathrm{id}_{TX}\\ f^*\circ\eta_X &= f\\ (g^*\circ f)^* &= g^* \circ f^*\end{align}

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.

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