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 C_T 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 η_X 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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Famous quotes containing the word extension:

“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)