A Pseudomonad is …

The aim of this post is to generalise the notion of monad to incorporate new examples, by weakening the requirement for the usual monad equations to hold exactly to instead be up to isomorphism. The resulting object is known as a pseudomonad. As we have seen several different perspectives on the notion of monad, all of which we will adapt to this new, more flexible setting. The aim is to give a high level overview of the key structures and the relationships between them, whilst skimming over some technical details.

Weakening the notion of monad

The most general definition of monad we have discussed is as a structure in a (strict) 2-category, consisting of:

  • An object \mathcal{C}
  • A 1-cell \mathbb{T} : \mathcal{C} \rightarrow \mathcal{C}
  • A unit 2-cell \mathsf{Id} \Rightarrow \mathbb{T}
  • A multiplication 2-cell \mathbb{T} \circ \mathbb{T} \Rightarrow \mathbb{T}

such that the unit and multiplication satisfy three equations.

In fact, we generalised this definition to the weaker setting of bicategories, but this was not strictly necessary. As every bicategory is biequivalent to a 2-category, we got “nothing new” by doing so. Of course, if the natural setting of your work is a bicategory, you may beg to differ, at least from the point of view of convenience.

Our aim is to weaken the definition above, so that the three equations defining a monad are only required to hold up to isomorphism. To do so, we require a notion of morphisms between 2-cells. That is, we need 3-cells, which points us in the direction of tricategories. This is a recurring theme, to weaken a definition we need to go up a dimension to provide the room to express the weakness.

Roughly speaking, we can arrive at the notion of tricategory by starting with categories enriched in 2-categories, and then weakening all the equations to coherent isomorphisms. That all sounds a bit scary, so it would be nice to restrict to strict 3-categories, by analogy with the simplifying step from bicategories to 2-categories. Unfortunately, this plan is flawed, as it is not the case that every tricategory is triequivalent to a strict 3-category. Annoying…

Fortunately all is not lost. Every tricategory is triequivalent to what is known as a Gray-category. A Gray-category is a tricategory in which composition is unital and associative on the nose, but interchange only holds up to isomorphism. They can also be seen as categories enriched over a suitable monoidal category \mathbf{Gray}. This allows us to work in a relatively simple setting, without throwing anything fundamental away.

In a Gray category, a pseudomonad consists of the data described above for a monad, along with three invertible 3-cells:

𝗂𝖽⇛μ⋅𝕋∘η𝗂𝖽⇛μ⋅η∘𝕋μ⋅𝕋∘μ⇛μ⋅μ∘𝕋\mathsf{id} \Rrightarrow \mu \cdot \mathbb{T} \circ \eta \qquad \mathsf{id} \Rrightarrow \mu \cdot \eta \circ \mathbb{T} \qquad \mu \cdot \mathbb{T} \circ \mu \Rrightarrow \mu \cdot \mu \circ \mathbb{T}

satisfying two coherence equations.

The Monoid Perspective

We have already encountered the standard monad meme:

A monad is a monoid in the category of endofunctors.

We would hope to get a statement of a similar form for pseudomonads. Mostly this will be an exercise in terminology.

A Gray-monoid is a one-object Gray-category. This is a variation of the notion of monoidal category, and so it seems natural to refer to a pseudomonad in a Gray-monoid as a pseudomonoid. As we can always restrict to the one-object sub-Gray-category induced by the 0-cell our pseudomonad is defined upon, simply by definition we arrive at

A pseudomonad is a pseudomonoid in the Gray-monoid of endo 1-cells.

We did not really do anything here apart from introduce new names for things we have already encountered, but the notion of pseudomonoid provides a different emphasis. As Gray-categories were a simplifying device, we can inevitably extend the definition to one-object tricategories, or equivalently to monoidal bicategories. An alternative motivation for pseudomonoids is the following:

Example: In the monoidal bicategory of categories, functors and natural transformations, with products as the monoidal structure, a pseudomonoid is the same thing as a monoidal category.

We can also pick out more exotic gadgets.

Example: In the monoidal bicategory of monoidal categories, lax monoidal functors and monoidal natural transformations, with products as the monoidal structure, a pseudomonoid is the same thing as a duoidal category.

It is then natural to search for variations of pseudomonoids, living in corresponding variations of monoidal bicategories, corresponding to other types of monoidal categories, such as with braiding or symmetries.

The Functorial Semantics Perspective

We can form what is known as the simplicial category \Delta, with:

  • Objects: Finite ordinals
  • Morphisms: Order preserving functions

This category is a strict monoidal category with respect to ordinal sums. Every strict monoidal category can be seen as a 1-object monoidal category, referred to as its suspension. This 2-category can be thought of as the generic monad. We then have an observation of Lawvere that:

A monad is a strict 2-functor from the suspension of the ordinal category.

Or possibly more clearly:

A monad is a strict 2-functor from the generic monad.

These 2-functors stamp out copies of the generic monad in their codomain 2-categories.

It would be nice if we could capture pseudomonads in a similar manner. This is indeed the case, as shown by Lack, using a construction adapting the simplicial category construction to yield a Gray-monoid, with a suspension we might describe as the generic pseudomonad, such that:

A pseudomonad is a Gray-functor from the generic pseudomonad.

Here, Gray-functors are the usual notion from enriched category theory.

The Enriched Perspective

We have seen:

A monad is a one-object enriched category.

This gave yet another perspective on the notion of monad. Can we get something similar for pseudomonads?

Recall that a strict 2-category is a category enriched in the monoidal category of categories and functors, with products as the monoidal structure. We cannot conveniently describe bicategories in the same way, as we need to weaken certain equalities to isomorphisms. This sounds reminiscent of the situation above where we wanted to weaken the notion of monad. Again, in order to incorporate the additional weakness we need to go up a dimension, and enrich in the monoidal bicategory of categories, functors and natural transformations, with products again as the monoidal structure. Of course we cannot use the ordinary notion of enriched category, but there is a notion of a category weakly enriched in a monoidal bicategory or tricategory \mathcal{V}, in which associativity and unitality hold up to specified coherent invertible 3-cells. These structures are referred to as \mathcal{V}-bicategories, as the motivating \mathsf{Cat}-bicategories are the usual notion of bicategory.

With this little detour out of the way, the coherence properties of \mathcal{V}-bicategories and pseudomonads coincide, in that:

A pseudomonad is the same thing as a one object \mathcal{V}-bicategory.

Summary

We have seen several different perspectives on the notion of pseudomonad which pleasingly parallel ideas for ordinary monads. This involved incorporating some possibly intimidating weaker higher-dimensional structures. As the developments for monads and pseudomonads run parallel, hopefully this helps transfer intuitions to the more complex setting.

Some references that I found useful:

  • Pseudomonads – “Distributive Laws for Pseudomonads” by Marmolejo, along with other work of the same author.
  • The functorial semantics perspective – “A Coherent Approach to Pseudomonads” by Lack.
  • My source for weak enrichment was “Enriched Categories as a Free Cocompletion” by Garner and Shulman.

The NLab was very useful for background on pseudomonoids.

Remark: This post developed out of background reading for some ongoing work, and so is slightly outside my useful comfort zone. I welcome comments and feedback to improve the presentation and intuitions, or correct errors.

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