Eilenberg-Moore limits and colimits

For every monad \mathbb{T} : \mathcal{C} \rightarrow \mathcal{C}, we have seen that the Eilenberg-Moore construction yields a new category \mathcal{C}^{\mathbb{T}}. So far, we don’t have a lot of general facts about Eilenberg-Moore categories we can exploit. In fact, these categories are typically very nice, with lots of good properties that a category theorist might hope for being inherited from their base categories. In this post, we will consider (co)completeness of Eilenberg-Moore categories.

Creation of limits and colimits

A functor

U : \mathcal{A} \rightarrow \mathcal{B}

is said to create limits of shape \mathcal{D} if for every diagram

D : \mathcal{D} \rightarrow \mathcal{A}

the limit \lim_{U \circ D} exists, with limit cone:

\lim_{U \circ D} \xrightarrow{\lambda_I} UD(I)

and there exists a unique D-cone

L \xrightarrow{\lambda'_I} D(I)

living above the limit cone, that is for all I

U(\lambda'_I) = \lambda_I

Furthermore, (L, \lambda') is the limit of the diagram D.

The point of knowing that a functor creates limits is that assuming we understand those limits in the codomain category, we the know both:

  1. Limits of the same shape exist in the domain category.
  2. We can explicitly calculate those limits from those in the base category.

There is an obvious dual notion of a functor that creates colimits.

Limits in Eilenberg-Moore Categories

For a monad \mathbb{T}, recall that there is a forgetful functor:

U^{\mathbb{T}} : \mathcal{C}^{\mathbb{T}} \rightarrow \mathcal{C}

We have seen that this functor has a left adjoint, and so will preserve all limits that exist in \mathcal{C}^{\mathbb{T}} for standard reasons. It turns out, we can say a lot more than that. In fact U^{\mathbb{T}} creates all limits that exist in \mathcal{C}.

Lets look at what this means in more detail. The action of U^{\mathbb{T}} on algebra morphisms is trivial:

U^{\mathbb{T}}(h) = h

Therefore, for diagram:

D : \mathcal{D} \rightarrow \mathcal{C}^{\mathbb{T}}

if \lim_{U^{\mathbb{T}} \circ D} exists, with limit cone:

\lim_{U^{\mathbb{T}} \circ D} \xrightarrow{\lambda_I} U^{\mathbb{T}}D(I)

then there exists a unique Eilenberg-Moore algebra structure \alpha on \lim_{U^{\mathbb{T}} \circ D} such that each \lambda_I is an algebra homomorphism:

(lim_{U^{\mathbb{T}} \circ D}, \alpha) \rightarrow D(I)

and this is the limit cone in \mathcal{C}^{\mathbb{T}}.

This result is fairly easy to prove. The universal property of the limit of U^{\mathbb{T}} \circ D completely determines what \alpha can be. Verifying it is an Eilenberg-Moore algebra, and satisfies the required universal property in \mathcal{C}^{\mathbb{T}} requires further applications of the universal property of the limit in the base category.

This theorem gives us a very good handle on limits in Eilenberg-Moore categories:

  1. \mathcal{C}^{\mathbb{T}} is as complete as the base category \mathcal{C}.
  2. We can calculate these limits explicitly, using our understanding of limits in the base category.

Colimits in Eilenberg-Moore Categories

The situation for colimits is not as good as that for limits, but it’s still pretty nice. There are standard theorems that address some important cases.

Firstly, U^{\mathbb{T}} creates every colimit which exists in the base category and is preserved by \mathbb{T} and \mathbb{T} \circ \mathbb{T}. Again, we get very nice behaviour, but for a slightly strange looking restricted class of colimits. The proof is similar to that for limits, exploiting the universal property of colimits in the base category to find a unique algebra structure map such that the cocone morphisms in the base category are algebra morphisms. The additional preservation assumptions are used to get colimits of the required shape to complete the argument.

You may ask where we are going to find colimits that satisfy these assumptions. There are special colimits called absolute colimits, which are preserved by every functor. These colimits are important in the theory of monads, particularly what are known as monadicity theorems, which we have only briefly touched upon so far. Applying the result above tells us that U^{\mathbb{T}} creates all absolute colimits which exist in the base category.

If you’re wondering how on earth there can be colimits which are preserved by every functor, or how we might prove such a thing, it is instructive to think about what functors preserve. As functors are homomorphisms of categories, they preserve commuting diagrams. It turns out there are certain limits which are characterised by certain diagrams commuting, and therefore are automatically preserved by every functor. We will discuss concrete examples, specifically certain special classes of coequalizers, in later posts when we look at monadicity theorems in more detail.

Another important results is that \mathcal{C}^{\mathbb{T}} is cocomplete if the base category is, and \mathcal{C}^{\mathbb{T}} has coequalizers. This is because in Eilenberg-Moore categories, we can then construct coproducts using these coequalizers, and that establishes cocompleteness. Using this observation, Linton established that for \mathsf{Set} monads, Eilenberg-Moore categories are cocomplete. Note these results are weaker than those we saw before, as there is no suggestion that these colimits are created by U^{\mathbb{T}}, and so the situation is getting wilder. There are more axiomatic versions of Linton’s result, which can then be applied in settings beyond \mathsf{Set} to establish cocompleteness of Eilenberg-Moore categories.

Conclusion

As a warm up application of the results above, and the connection between monads and universal algebra, we can conclude that:

  1. Categories of algebraic structures such as monoids, groups, rings and so on are all complete and cocomplete.
  2. The limits in these algebraic categories are created by the forgetful functor to \mathsf{Set}.
  3. Certain well behaved colimits in these algebraic categories are also created by the forgetful functor.

We will see some more exciting applications of these results later, once we have discussed some other machinery.

This is only one aspect of the niceness of Eilenberg-Moore categories. There are further tame conditions under which the Eilenberg-Moore category is either regular or locally presentable for example. Certain special classes of monads, such as commutative monads imply further structure on the Eilenberg-Moore category. We shall return to these topics later.

Further reading: The results in this post are all very well known, and can be found in all the standard sources, such as Categories for the Working Mathematician, Toposes, Triples and Theories, or volume 2 of the Handbook of Categorical Algebra.

3 thoughts on “Eilenberg-Moore limits and colimits”

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