A large part of the richness of ordinary monad theory comes with the interplay between monads and adjunctions. Now we have encountered relative monads, a natural next step is to look for connections with adjunctions. That is the purpose of todays post.

## Kleisli Triples and Adjunctions

As relative monads are formulated as a generalisation of Kleisli triples, it makes sense to use the relationship between adjunctions and that formulation of ordinary monads as a starting point, before attempting to generalise to the relative setting.

As we are favouring monads in Kleisli form, it is worth thinking about which formulation of adjunctions is going to fit comfortably. There are many equivalent choices, we can consider adjunctions in terms of:

- Cup and cap natural transformations satisfying snake equations.
- A pair of functors satisfying a certain unique fill-in condition.
- A natural bijection between hom sets .
- …

As a monad in Kleisli form has an extension operation:

,

which is a mapping between hom sets, this seems to connect well with the natural bijection between hom sets formulation of adjunctions. So lets assume we have an adjunction, witnessed by a natural isomorphism:

We wish to show that this induces a monad, using the extension form. The action on objects is:

For the unit map, we note that has type

so we take the components of our unit as

.

Finally, we need to identify a suitable extension operation. The following composite has the right type:

where is the map that applies the functor to a given hom set. More explicitly, we take:

.

We have three axioms to check. Firstly, we calculate:

.

Secondly,

,

where we use the naturality of in the only non-trivial step.

Finally:

Again, the only non-trivial step uses naturality. With this proof for ordinary monads under our belts, we’re ready to look at relative monads.

## Relative adjunctions

We now want to tweak things to try and recover a -relative monad, from a similar situation to before. Previously, we constructed the unit as the image of the identity under a map of type:

For a -relative monad, the component of the unit at has type . This suggests we require a natural isomorphism of type:

where , and . If we have such an isomorphism, we can follow an almost identical recipe to before to form a -relative monad with:

- Object map .
- Unit components .
- Extension operation .

The proofs of the three axioms are identical to the previous ones, up to trivial changes in type information.

Our calculations lead us to consider isomorphisms of hom sets of the form:

If we have such an natural isomorphism we say that has a left -relative adjoint . We could also consider the situation:

in which case we say that has a right -relative adjoint . Both situations are known as **relative adjunctions**.

**Example**: A functor is full and faithful if and only if is its -relative left adjoint, as that is the case when we have a natural isomorphism:

It follows that every full and faithful functor induces a relative monad.

We also have the obvious source of examples.

**Example**: Every ordinary adjunction is a -relative adjunction.

We can also build new relative adjoints in a routine way.

**Example**: If we have relative adjunction:

then for any of appropriate type:

so is -relative left adjoint to . Using the previous example, given an ordinary adjunction , we get relative adjunctions:

so is -relative adjoint to .

## Some words of caution

We have seen that left -relative adjoints yield -relative monads. Unfortunately relative adjunctions are not as well behaved as ordinary adjunctions, for example:

- The situation is asymmetric. Left -relative adjoints induce -relative monads but not comonads, and unit maps of type , but no counit. On the other hand, right -relative adjoints induce -relative comonads but not monads, and counit maps of type , but no unit.
- A functor determines its -relative left adjoint, but the -relative left adjoint does not fully determine the right adjoint.
- There is an equivalent formulation of relative adjunctions in terms of absolute Kan lifts. This might sound like it would lead to a tidier setting, more suitable for formal category theory. Unfortunately, it does not capture the right notion if we move to the setting of enriched category theory.

## Conclusion

By examining the construction of Kleisli triples from ordinary adjunctions, we identified a variation on the notion of adjunction, that induces relative monads. We have seen a small number of examples of relative adjunctions so far. Readers of this blog are probably (hopefully) asking themselves the following questions:

- Does every relative monad arise via an adjunction?
- Is there a suitable generalisation of the Kleisli construction?
- Is there a suitable generation of the Eilenberg-Moore construction?

Positive answers to these questions would provide stronger evidence that relative adjunctions are a reasonable concept. We shall explore these topics in future posts.

**Further reading**: Ulmers “Properties of Dense and Relative Adjoint Functors” is a good place to start for more background, and is surprisingly readable for an older category theory paper.

**Acknowledgements**: Nathanael Arkor helpfully pointed out a blunder in an earlier version of this post, where I had muddled up Kan extensions and lifts in a remark.