The Idea of Monadicity

Naively, for a given category , we might wonder if it is equivalent to the Eilenberg-Moore category of a monad. This would help us deduce properties of , such as (co)completeness, using facts we know about the Eilenberg-Moore construction.

Giving this a moments thought, it doesn’t sound quite right, as to deduce properties of an Eilenberg-Moore category, we need more information, for example about the base category. In order to reach a mathematically useful notion, we need to pin down exactly what we intend with regard to this equivalence.

We would like to know what the base category we’re interested in is, and how it relates to , so it is natural to consider functors of the form.

Intuitively this is a candidate forgetful functor from an Eilenberg-Moore category to its base category. If there is any hope of being equivalent to an Eilenberg-Moore category, should have a left adjoint

.

This gives us a bit more to work with, as adjunction will give us a monad

We are now interested in when there is an equivalence:

To specify this equivalence more precisely, recall we have already seen the Eilenberg-Moore comparison functor, which relates resolutions of the same monad. Recall this is the unique functor

that commutes with the free and forgetful functors, in that:

and .

We have now arrived at the right notion. A right adjoint functor is monadic if the comparison functor is an equivalence of categories. If there exist such a , we say that is monadic over .

Monadicity Theorems

Unsurprisingly, a monadicity theorem gives conditions under which a functor is monadic. The key result is known as Beck’s monadicity theorem, which gives precise conditions for monadicity. There are many variations, often with sufficient assumptions that are more convenient to establish in practice. There are also a family of related results, which don’t specifically establish monadicity, but rather other nice properties of the comparison functor.

In fact, if you have a specific functor with a left adjoint, it is often easier to establish monadicity by directly proving the required equivalence. So what’s the point of monadicity theorems? Their main application is when you are working more abstractly, aiming to show that under suitable assumptions that all functors of a certain type are monadic. This situation is similar to the use of adjoint functor theorems. Again, these are rarely applied to establish that a particular functor has an adjoint, but rather that in an abstract setting every functor satisfying certain axioms always has an adjoint.

In our discussions, we will focus on when the comparison functor is an equivalence. MacLane considers the stronger situation where the comparison functor is an isomorphism. We will pay this perhaps less categorically natural formulation less attention. It is however important to be aware that there are monadicity theorems of this type as well when looking at the wider literature.

If instead of the Eilenberg-Moore construction, we consider the Kleisli construction, we can ask similar questions. The situation is much simpler in this case, as was previously discussed.

Conclusion

The notions of monadicity and monadicity theorems are fundamental tools. Despite this, as they are used in more abstract work, they tend to be perceived as a more advanced topic. This is probably reinforced by the fact that the details of monadicity theorems are a little bit technical looking, with certain special classes of colimits having a prominent role. In reality, the details only involve elementary ideas, and are not particularly challenging. We will discuss the specifics of monadicity theorems, related results about comparison functors, and their applications in future posts.