The Kleisli Adjunction

The Kleisli category is an interesting construction, but as with the Eilenberg-Moore category, it has a deeper significance.

For a monad $\mathbb{T}$ on category $\mathcal{C}$ , there is a functor $U_{\mathbb{T}} : \mathcal{C}_{\mathbb{T}} \rightarrow \mathcal{C}$ with:

$U_{\mathbb{T}}(A) = \mathbb{T}(A) \qquad U_{\mathbb{T}}(f) = \mu \circ \mathbb{T}(f)$

There is also a functor $F_{\mathbb{T}} : \mathcal{C} \rightarrow \mathcal{C}_{\mathbb{T}}$ with:

$F_{\mathbb{T}}(A) = A \qquad F_{\mathbb{T}}(f) = \eta \circ f$

(Verifying that both these are legitimate functors is a useful exercise in applying the various axioms.)

In fact, $F_{\mathbb{T}} \dashv U_{\mathbb{T}}$ . To see this, we must show a natural bijection between:

Kleisli morphisms $F_{\mathbb{T}}(A) \rightarrow B$ .
$\mathcal{C}$ -morphisms $A \rightarrow U_{\mathbb{T}}(B)$ .

Recalling that Kleisli morphisms $A \rightarrow B$ are $\mathcal{C}$ -morphisms $A \rightarrow \mathbb{T}(B)$ , and expanding definitions:

A Kleisli morphism $F_{\mathbb{T}}(A) \rightarrow B$ is a $\mathcal{C}$ -morphism $A \rightarrow \mathbb{T}(B)$ .
A $\mathcal{C}$ -morphism $A \rightarrow U_{\mathbb{T}}(B)$ is a $\mathcal{C}$ -morphism $A \rightarrow \mathbb{T}(B)$ .

So the bijection holds trivially. I think we can probably guess what the monad induced by this adjunction is going to turn out to be, but lets check the details anyway.

Endofunctor: The endofunctor is $U_{\mathbb{T}} \circ F_{\mathbb{T}}$ . On objects this is $U_{\mathbb{T}}(F_{\mathbb{B}}(A)) = U_{\mathbb{T}}(A) = \mathbb{T}(A)$ . On morphisms $U_{\mathbb{T}}(F_{\mathbb{B}}(f)) = U_{\mathbb{T}}(\eta \circ f) = \mathbb{T}(\mu \circ \mathbb{T}(\eta \circ f)) = \mathbb{T}(f)$ .
Unit: The unit is the transpose of $\mathsf{id}_{F_{\mathbb{T}}(A)} : A \rightarrow A$ , which is $\eta_A$ .
Multiplication: The multiplication at $A$ is $U_{\mathbb{T}}(\epsilon_{F_{\mathbb{T}}(A)})$ , where $\epsilon_{\mathbb{F}(A)}$ is the transpose of $\mathsf{id}_{U_{\mathbb{T}}(F_{\mathbb{T}}(A))} : U_{\mathbb{T}}(F_{\mathbb{T}}(A)) \rightarrow U_{\mathbb{T}}(F_{\mathbb{T}}(A))$ . Expanding definitions slightly, $\epsilon_{\mathbb{F}(A)}$ is $\mathsf{id}_{\mathbb{T}(A)}$ . Then $U_{\mathbb{T}}(\mathsf{id}_{\mathbb{T}(A)})$ is $\mu \circ \mathbb{T}(\mathsf{id}_{\mathbb{T}(A)})$ , which is simply $\mu$ .

Summing up, for a monad $\mathbb{T}$ , the the monad induced by the Kleisli adjunction $F_{\mathbb{T}} \dashv U_{\mathbb{T}}$ is $\mathbb{T}$ itself.

We have now seen two different constructions, the Eilenberg-Moore and Kleisli constructions, and their corresponding adjunctions. Both adjunctions yield the original monad. The natural question we shall pursue next is the relationship between these two constructions and other adjunctions that induce the same monad.

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