Eilenberg-Moore Objects

This time, our aim is to understand the Eilenberg-Moore construction better. To do so, we will look for a generalisation of this construction for monads in an arbitrary 2-category $\mathbf{K}$ . Working at this level of abstraction will provide results that apply in other settings, for example monads in the sense of enriched category theory. Even if all you care about is ordinary monads, this perspective forces us towards a universal property characterisation of the usual Eilenberg-Moore category, rather than our previous focus on an explicit construction.

Remark: Moving to a 2-categorical setting has a certain “intimidation factor”. It may help to recall that 0, 1 and 2 cells in $\mathbf{K}$ are analogous to categories, functors and natural transformations in $\mathbf{Cat}$ . In particular, 1 and 2-cells can be composed vertically and horizontally just as functors as natural transformations can. For readers aware of such issues, we will be working entirely with strict 2-categories and 2-functors, so there are no distracting coherence isomorphisms to worry about either.

Eilenberg-Moore locally

For a monad $(\mathcal{C},\mathbb{T},\eta,\mu)$ in $\mathbf{K}$ , where should we start looking for an analogue of the conventional Eilenberg-Moore construction? In that case, we built a well-chosen 0-cell in the 2-category $\mathbf{Cat}$ , as we know these are simply categories, which we understand well. As we now work with an arbitrary 2-category $\mathbf{K}$ , we’ve no idea what the 0-cells are, so we cannot persist with this plan directly.

A natural plan of attack is to reduce this problem to one we’re more familiar with. To do so, for a 0-cell $\mathcal{X}$ , we consider the representable 2-functor

$\mathbf{K}(\mathcal{X},-) : \mathbf{K} \rightarrow \mathbf{Cat}$

This maps structures in $\mathbf{K}$ , which we don’t understand very well, into $\mathbf{Cat}$ , for which we have more understanding. As monads are defined in terms of equations between structures in a 2-category, they are preserved by 2-functors. Therefore

$\mathbf{K}(\mathcal{X},\mathbb{T})$

is an ordinary monad on the category $\mathbf{K}(\mathcal{X},\mathcal{C})$ . Concretely, the functor part of this monad acts as:

$U : \mathcal{X} \rightarrow \mathcal{C} \;\mapsto\; \mathbb{T} \circ U : \mathcal{X} \rightarrow \mathcal{C}$

The components of the unit and multiplication at $U : \mathcal{X} \rightarrow \mathcal{C}$ are $\eta \circ U$ and $\mu \circ U$ . We can then consider the Eilenberg-Moore category

$\mathbf{K}(\mathcal{X},\mathcal{C})^{\mathbf{K}(\mathcal{X},\mathbb{T})}$

Here we find something familiar, the Eilenberg-Moore algebras are left actions for the monad $\mathbb{T}$ in $\mathbf{K}$ . Of course we must generalise our original definition of left action from the 2-category $\mathbf{Cat}$ to an arbitrary 2-category $\mathbf{K}$ , but this is routine.

We now need to abstract a bit further, as we have defined this construction for an arbitrary 0-cell $\mathcal{X}$ . To remove this arbitrary choice, we make this component a parameter, and consider the 2-functor:

$\mathbf{K}(-,\mathcal{C})^{\mathbf{K}(-,\mathbb{T})} : \mathbf{K}^{op} \rightarrow \mathbf{Cat}$

A standard next step for such a functor is to ask when it is representable, that is when there is a 0-cell $\mathcal{C}^{\mathbb{T}}$ and 2-natural isomorphism:

$\mathbf{K}(\mathcal{X}, \mathcal{C}^{\mathbb{T}}) \cong \mathbf{K}(\mathcal{X}, \mathcal{C})^{\mathbf{K}(\mathcal{X},\mathbb{T})}$

If such a representation exists, $\mathcal{C}^{\mathbb{T}}$ is referred to as the Eilenberg-Moore object for the monad $\mathbb{T}$ , and establishes its universal property. We can think of this as internalising the collection of left monad actions, in a similar way to the relationship between hom sets and exponentials for example.

To unpick the universal property a bit further, write $\mathbf{\theta}$ for the 2-isomorphism. We consider the image of the identity 1-cell under $\mathbf{\theta}$ into:

$\mathbf{K}(\mathcal{C}^{\mathbb{T}}, \mathcal{C})^{\mathbf{K}(\mathcal{C}^\mathbb{T},\mathbb{T})}$

This is a left $\mathbb{T}$ -action

$\chi : \mathbb{T} \circ U^\mathbb{T} \Rightarrow U^\mathbb{T}$

This action is said to be a universal left action, and fully defines $\mathbf{\theta}$ . The action of $\mathbf{\theta}$ on 1-cells is:

$H : X \rightarrow \mathcal{C}^\mathbb{T} \;\mapsto\; \chi \circ H$

and on 2-cells:

$\sigma : H \Rightarrow H' \;\mapsto\; U^\mathbb{T} \circ \sigma$

Unravelling these details is a worthwhile exercise if you want to develop understanding representable 2-functors and 2-naturality, generalising a similar argument ordinary representable functors.

Example: The 2-category $\mathbf{Cat}$ has Eilenberg-Moore objects for every monad. This is also the case in the enriched category setting of $\mathcal{V}-\mathbf{Cat}$ for a base of enrichment $\mathcal{V}$ with (enough) equalizers. In both cases, the Eilenberg-Moore objects are given by the usual construction of the Eilenberg-Moore category. The universal left action is given by

$U^\mathbb{T} \circ \epsilon : \mathbb{T} \circ U^{\mathbb{T}} \Rightarrow U^\mathbb{T}$

where $\epsilon$ is the counit of the free / forgetful adjunction for the Eilenberg-Moore category.

That Eilenberg-Moore objects in the settings of ordinary and enriched category theory are the usual constructions is encouraging.

As further evidence of the usefulness of this perspective, if a monad $\mathbb{T}$ has an Eilenberg-Moore object, then other aspects of monad theory can be developed directly from the universal property. Of immediate interest are:

There is a 1-cell $F^\mathbb{T} : \mathcal{C} \rightarrow \mathcal{C}^\mathbb{T}$ such that $\mathbb{T}$ is induced by an adjunction $F^\mathbb{T} \dashv U^\mathbb{T}$ in $\mathbf{K}$ .
There is a unique comparison 1-cell in $\mathbf{K}$ exhibiting this adjunction as the terminal such adjunction.

We avoid spelling out the details, but they follow fairly directly from the universal property, and details can be found in Street’s paper.

Eilenberg-Moore globally

In an earlier post, we introduced the 2-category $\mathbf{\mathsf{Mnd}}(\mathbf{K})$ . Sketching the details:

0-cells monads in $\mathbf{K}$ .
1-cells Eilenberg-Moore laws between a pair of monads.
2-cells are 2-cells from $\mathbf{K}$ which suitably commute with the Eilenberg-Moore laws.

The full details can be found in an earlier discussion about distributive laws.

There is a 2-functor:

$\mathbf{K} \rightarrow \mathbf{\mathsf{Mnd}}(\mathbf{K})$

sending a 0-cell to its corresponding identity monad. This functor has a right 2-adjoint $\mathbf{\mathsf{Alg}}$ exactly when there is a 2-natural isomorphism:

$\mathbf{K}(\mathcal{X}, \mathbf{\mathsf{Alg}}(\mathcal{C}, \mathbb{T})) \cong \mathbf{\mathsf{Mnd}}(\mathbf{K})((\mathcal{C}, \mathsf{Id}), (\mathcal{C}, \mathbb{T}))$

This doesn’t immediately look familiar, but if we unravel the definitions, we find that Eilenberg-Moore laws of type:

$\mathbb{T} \circ H \Rightarrow \mathbb{H} \circ \mathsf{Id}$

are the same thing as left $\mathbb{T}$ -actions, and so we are simply requiring the representability discussed in the previous section globally. Therefore, this right adjoint exists exactly when there is an Eilenberg-Moore object for every monad at $\mathbf{K}$ . In this case, we say that $\mathbf{K}$ admits the construction of algebras.

Intuitively, we might have anticipated that Eilenberg-Moore objects arise as part of an adjunction, as they are involved in an extremal resolution of monads.

Example: $\mathbf{Cat}$ admits the construction of algebras. This extends to the enriched setting, where $\mathcal{V}-\mathbf{Cat}$ for a sufficiently complete base of enrichment.

Duality

For our 2-category $\mathbf{K}$ , we can consider its various duals. Eilenberg-Moore objects in $\mathbf{K}^{op}$ , where we reverse 1-cells are what are known as Kleisli objects. These appropriately generalise the Kleisli construction to a 2-categorical setting, and the resulting universal property of Kleisli objects involves a universal right monad action. The duals $\mathbf{K}^{co}$ and $\mathbf{K}^{coop}$ , where we reverse 2-cells, recover the corresponding theory for comonads, involving what might be called universal left and right co-actions of a comonad.

Example: $\mathbf{Cat}$ admits the construction of algebras for all 4 of its duals, given by the usual Eilenberg-Moore and Kleisli constructions for monads and comonads.

Conclusion

By considering the 2-categorical generalisation of the Eilenberg-Moore construction, we have encountered further aspects of left monad actions:

They are the appropriate 2-categorical generalisation of ordinary Eilenberg-Moore algebras.
They are themselves algebras for a suitably defined monad.
They are also a special case of Eilenberg-Moore laws.
Left and right monad actions are fundamental to giving universal properties for the Eilenberg-Moore and Kleisli constructions.

It is beautiful that the theory can be developed at this level of abstraction, and that the results for the Kleisli construction and comonads fall out via duality, providing a deeper mathematical relationship between all these constructions.

Further reading: The ideas in this post originate in Street’s wonderful paper “Formal Theory of Monads”. There are lots of other interesting ideas in this paper, and I strongly recommend reading it. Also useful for background is Kelly and Street “Review of the elements of 2-categories”, and Lack’s “2-categories companion”. Other potential sources on Eilenberg-Moore objects are the nlab, and MacDonald and Sobral’s “Aspects of monads” chapter of “Categorical Foundations” edited by Pedicchio and Tholen.

Acknowledgements: This post was prompted by questions and suggestions on mastodon by users @varkor@mathstodon.xyz and @maxsnew@types.pl. I am very grateful for these comments, as they pointed to some gaps in my understanding, and encouraged me to explore further.

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Eilenberg-Moore locally

Eilenberg-Moore globally

Duality

Conclusion

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