Adjunctions of Kleisli Type

In this short post, we will focus on a very simple but useful fact about adjunctions. This fact is well known to experts, and crops up regularly in the literature. It might be less familiar to those learning the subject, but that is the situation we aim to address.

Adjunctions of Kleisli Type

Recall that for a monad $\mathbb{T}$ on some category $\mathcal{C}$ , there is a functor from the base category to the Kleisli category:

$F_{\mathbb{T}} : \mathcal{C} \rightarrow \mathcal{C}_{\mathbb{T}}$

with right adjoint

$U_{\mathbb{T}} : \mathcal{C}_{\mathbb{T}} \rightarrow \mathcal{C}$

forming what we refer to as the Kleisli adjunction.

The question we are interested in is if we are given a functor

$F : \mathcal{C} \rightarrow \mathcal{D}$ ,

when is it the left adjoint of the Kleisli adjunction? It is worth pausing to think for a second about what exactly this should mean. The details will become clearer as we proceed.

It is obvious we should require a right adjoint

$U : \mathcal{D} \rightarrow \mathcal{C}$ .

Clearly this is not sufficient on its own, as we can easily find adjunctions $F \dashv U$ as above, where the category $\mathcal{D}$ is not equivalent to a Kleisli category. One obvious example would be to consider the Eilenberg-Moore adjunction, as the Eilenberg-Moore category $\mathcal{C}^{\mathbb{T}}$ is rarely equivalent to the Kleisli category. (Although there are examples where this is the case).

So what else should we require to pin things down correctly? It turns out we don’t need very much at all, just to examine a bit more carefully some machinery we’ve seen before. Recall that the Kleisli adjunction is initial amongst adjunctions inducing a particular monad. That is, for monad $\mathbb{T} = U \circ F$ , there is a comparison functor

$K_{\mathbb{T}} : \mathcal{C}_{\mathbb{T}} \rightarrow \mathcal{D}$

which is the unique homomorphism from the Kleisli adjunction to the adjunction $F \dashv U$ . This functor is exactly the device we need to answer questions like this. This functor is always full and faithful, which is very good start. Furthermore, the action on objects is very simple:

$K_{\mathbb{T}}(C) = F(C)$

From here, we can make the following simple observations about the functor $K_{\mathbb{T}}$ :

It is an isomorphism of categories if and only if it is bijective on objects, which is true if and only if $F$ is bijective on objects.
It is an equivalence of categories if and only if it is essentially surjective on objects, which is true if and only if $F$ is essentially surjective on objects.

Summing up, for a functor $F : \mathcal{C} \rightarrow \mathcal{D}$ , with right adjoint $U$ , inducing monad $\mathbb{T} = U \circ F$ :

The category $\mathcal{D}$ is canonically isomorphic to $\mathcal{C}_{\mathbb{T}}$ if $F$ is bijective on objects. This situation is sometimes referred to as an adjunction of Kleisli type.
The category $\mathcal{D}$ is canonically equivalent to $\mathcal{C}_{\mathbb{T}}$ if $F$ is essentially surjective on objects.

The word canonically in the statements above simply means that the isomorphism / equivalence is the comparison functor.

Conclusion

This has been a somewhat technical post about a simple fact. I have run across this simple fact three times is the space of a month reading various papers, so it seemed an opportune time to publicise this somewhat “expert friendly” little fact.

A nice concise explanation of this situation is given in the answer to this MathOverFlow post given by Peter LeFanu Lumsdaine.

Adjunctions of Kleisli Type

Conclusion

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