Talk  2Categorical Descent and (Essentially) Algebraic Theories
14 Nov 2023  Tags: mytalks
A few weeks ago I gave a talk at the CT Octoberfest 2023 about some work I did over the summer that I’m really proud of. Unfortunately, while writing up the result I found a 1999 paper by Pedicchio and Wood that proves the same theorem (with roughly the same proof), so I wasn’t able to publish. Thankfully, the work is still extremely interesting, and I was more than happy to talk about it at a little online conference for other category theorists ^_^.
Recall an algebraic theory is something like groups, rings, modules, etc. It’s a structure that can be defined as a set (or possibly multiple sets) with some operations defined on it (allowing constants as $0$ary operations) and equations specifying the behavior of those operations.
An essentially algebraic theory is something like categories. It’s a structure that can be defined as a set (or possibly multiple sets) with some operations defined on it, etc. The main superpower we get in the essentially algebraic world over the algebraic one is partially defined functions. Now our operations don’t have to be defined everywhere, they are allowed to be defined on subsets of the sorts. As long as those subsets are definable by equations!
For instance, the theory of categories is essentially algebraic since we have
 Sets $O$ and $A$ (the sets of objects and arrows)
 operations $\text{dom}, \text{cod} : A \to O$ taking an arrow to its domain/codomain
 an operation $\text{id} : O \to A$ taking an object to the identity arrow at that object
 an operation \(\circ : \{ (f,g) \in A \times A \mid \text{dom}(f) = \text{cod}(g) \}\)
 satisfying certain equational axioms, like $\text{dom}(\text{id}(x)) = x = \text{cod}(\text{id}(x))$, $(f \circ g) \circ h = f \circ (g \circ h)$, etc.
Notice that composition isn’t defined on the whole set $A \times A$. It’s only partially defined! But the set where it’s defined is easy to understand – it’s defined by an equation in the other functions ($\text{dom}(f) = \text{cod}(g)$).
Contrast this with fields, which have a partially defined inverse operation \(()^{1} : \{ x \in k \mid x \neq 0 \} \to k\). There is no way to write the domain of inversion as an equation^{1}.
Now, essentially algebraic theories are extremely nice, for lots of reasons I outlined in my talk (and mentioned on the nlab page I linked earlier), but they’re not quite as nice as honest algebraic theories.
For instance, the underlying set of a quotient of groups is a quotient of the underlying set. If we have a surjection $G \twoheadrightarrow H$, then there’s an equivalence relation^{2} $\theta$ on $UG$ (the underlying set of $G$) so that $UH \cong (UG) \big / \theta$.
This is no longer the case for models of an essentially algebraic theory! That is, the underlying set of a quotient might not be a quotient of the underlying set^{3}.
For example, consider the following category:
Notice its set of arrows (ignoring identities) is ${ f, g }$.
Now if we quotient to set $Y_1 = Y_2$, we get a new category
But now that $Y_1 = Y_2$, $f$ and $g$ are composable! So we had better add a composite!
So after quotienting, our underlying set of arrows (again, ignoring identities) is ${ f, g, gf }$, which isn’t a quotient of the set we started with! Also, note the role that partial operations played in this. The reason we got ~bonus elements~ in our underlying set is because after quotienting the domain for the partial operation got bigger, so we had to freely add stuff to make sure we were closed under composition.
Another reason to care about algebraic theories over essentially algebraic ones is that algebraic theories can be interpreted in any finite product category, while essentailly algebraic theories make use of all finite limts! This shows up even for “real mathematicians”, since the category $\mathsf{Diff}$ of smooth manifolds doesn’t have finite limits! So we can define a lie group as a group object in $\mathsf{Diff}$ (since the theory of groups is algebraic) but we can’t define a lie groupoid as a groupoid object in $\mathsf{Diff}$ (since the theory of groupoids is merely essentially algebraic)^{4}!
With this in mind, it’s natural to ask when we can recognize an algebraic theory amongst the essentially algebraic ones. It turns out we can, and the process requires a fair amount of category theory!
We’ve already touched on the relationship between
 \[\{ \text{algebraic theories} \} \leftrightsquigarrow \{ \text{finite product categories} \}\]
 \[\{ \text{essentially algebraic theories} \} \leftrightsquigarrow \{ \text{finite limit categories} \}\]
But it turns out the relationship goes much deeper! Indeed, one can show that the “sets” in the above bullets actually represent $2$categories, and that the correspondences are (contravariant) biequivalences!
Given a finite product (resp. finite limit) category $\mathcal{C}$, we treat it as an (essentially) algebraic theory, and say its category of models is the category of finite product (resp. finite limit) functors $\mathcal{C} \to \mathsf{Set}$.
In fact, we can go further! Given a finite product (resp. finite limit) categories $\mathcal{C}$ and $\mathcal{V}$, we say that the cateogry of $\mathcal{C}$ models in $\mathcal{V}$ is the category of finite product (resp. finite limit) functors $\mathcal{C} \to \mathcal{V}$.
Conversely, given a category of models for some (essentially) algebraic theory, its category of ($\mathsf{Set}$valued) finitely generated free algebras^{5} (resp. finitely presented algebras) has finite coproducts (resp. finite colimits). So if we take the opposite category of this, we get a category with finite products (resp. finite limits)!
It’s then not so hard to show that these operations are mutually inverse^{6}!
Now if we have an algebraic theory $\mathbb{A}$, that corresponds to a finite product category $\mathcal{A}$ where the category of $\mathbb{A}$models is the functor category $\mathsf{FinProd}(\mathcal{A}, \mathsf{Set})$.
To view this as an essentially algebraic theory, we want to find a finite limit category $\mathcal{E}$ which has the same models. That is, so that for every finite limit category $\mathcal{V}$:
\[\mathsf{FinLim}(\mathcal{E}, \mathcal{V}) \cong \mathsf{FindProd}(\mathcal{A}, U \mathcal{V})\]where $U$ is the forgetful functor from finite limit categories to finite product categories.
This makes it clear that $\mathcal{E}$ should be the free finite limit completion of the finite product category $\mathcal{A}$ we started with! Since we already have products, all we have to do is freely add equalizers!
With this in mind, we see how to rephrase our problem of recognizing the algebraic theories among the essentially algebraic ones!
Fix an essentially algebraic theory $\mathbb{E}$, with (finite limit) classifying category $\mathcal{E}$.
Then $\mathbb{E}$ is actually algebraic if and only if $\mathcal{E}$ is equivalent to the free equalizer completion of a finite product category $\mathcal{A}$!
This means if we want to recognize the algebraic theories, we just need a way to recognize the essential image of the equalizer completion functor!
Thankfully, there’s a very heavy hammer we can use to understand the image of a left adjoint: Comonadic Descent!
I don’t want to say too much about (co)monadic descent here, mainly because I’m going to a friend’s concert tonight and I’ve already written quite a lot about it in my recent preprint. But here’s the short story. We have a diagram of categories
where $\mathsf{FinLim}_{\mathsf{Eq} \ U}$ is the category of coalgebras for the $\mathsf{Eq}\ U$ comonad, and the usual BarrBeck yoga shows that everything in the image of $\mathsf{Eq}$ has a canonical coalgebra structure, which is where the top map (which I’m abusively also calling $\mathsf{Eq}$) comes from.
The adjunction $\mathsf{Eq} \dashv U$ is called comonadic exactly when this top map is an equivalence. In particular, this means we can recognize the image of $\mathsf{Eq}$ in $\mathsf{FinLim}$ as those categories admitting an $\mathsf{Eq} \ U$ coalgebra structure!
It turns out to not be too hard to prove that this adjunction is comonadic by using Beck’s famed (Co)monadicity Theorem! This comes down to some combinatorics^{7} involving Pitt’s explicit construction of the equalizer completion, first published in Bunge and Carboni’s The Symmetric Topos, which solves our problem!
Pedicchio and Wood, in their ‘99 paper A Simple Characterization of Theories of Varieties, give a nice characterization of the image of $\mathsf{Eq}$ as those categories with enough “effective projectives”^{8}.
Let me say a quick word about the “2categorical” in the title. In the last section, to make use of the descent machinery, we had to work with 1categories $\mathsf{FinLim}$ and $\mathsf{FinProd}$. That is, with stict such categories. Of course, this result should really be 2categorical in nature, working with all such categories, and we should be using a 2categorical version of comonadic descent to prove the theorem… Unfortunately I don’t know of one!
I was kind of hoping that someone would ask about this during the talk – after all I put “2categorical” in the title, but didn’t mention 2categories at all! But my talk was the first talk of the day^{9}, so it makes sense that people would have been nice and not asked that.
Regardless, I’m pretty sure an australian would have a reference for this kind of descent (and I might ask about it in the category theory zulip after posting this), because there’s no way I’m the first person to want to use it!
All in all, I’m happy with how the talk went. It was one of the shorter talks I’ve given, and I wanted to assume the audience didn’t know a ton of logic. I think I did a good job giving the flavor of the theorem, and some reasons to care about it, without necessarily getting bogged down in the details of the proof. Hopefully I didn’t come off as too upset that I was 24 years late to publish it myself!
As usual, here’s a copy of the slides, abstract, and recording. I’ll also encourage people to take a look at some of the other talks from the conference (which you can find here). There were a ton of interesting ones^{10} and if you like category theory I’m sure you’ll find something you enjoy!
Thanks again for reading. Stay warm, and try not to let the November darkness weigh on you too much! Talk soon ^_^
$2$Categorical Descent and (Essentially) Algebraic Theories
An essentially algebraic theory is an algebraic theory that moreover allows certain partially defined operations. Since algebraic theories enjoy certain nice properties that essentially algebraic theories don’t, it’s natural to ask if we can recognize when an essentially algebraic theory is actually algebraic. In the language of functorial semantics, this amounts to recognizing when a finite limit category is the free completion of a finite product category, and the problem can be solved by considering a 2categorical descent theory. This was independent work, but writing it up I learned that the same result can already be found in a 1999 paper of Pedicchio and Wood. This seems to be less well known than it should be, and I hope this talk brings attention to this fascinating subject.
The slides are here, and a recording is below:

In fact, we can prove this with category theory! It’s a theorem that the category of all models for an essentially algebraic theory has an initial object. But there isn’t an initial object in the category of fields! So no matter how clever we are, there won’t be an essentially algebraic axiomatization of fields. ↩

In fact there’s much more to be said here. The equivalence relation $\theta$ will be a congruence (meaning it’s compatible with the algebraic structure), and the study of such congruences is historically one of the biggest topics in universal algebra. I won’t say more here, but trust me that there’s much more to say. If you’re interested, I recommend Burris and Sankappanavar’s book, freely available here. ↩

This should be believable for a few reasons. Indeed, the “underlying set” functor is a right adjoint, so we shouldn’t expect it to play nicely with any kind of colimit (like a quotient).
Moreover, $U$ playing nicely with congruences is one of the defining features of an algebraic theory! This is the key criterion for monadicity! ↩

If you haven’t seen them before, it may come as a surprise that “real mathematicians” care about lie groupoids, since they sound quite abstract. But they’re really not esoteric at all! They model orbifolds, which are manifolds with certain mild singularities. They arise incredibly naturally when studying, say, manifolds with a group action. ↩

This is the slightest of fibs. The situation for finite product categories is actually slightly more sublte than I’m letting on, basically because a finite product category might not be cauchy complete. If you know you know, if you don’t, then trust me that it doesn’t really matter. If you’re interested in the details, you can find them in Adámek, Vitale, and Rosický’s book Algebraic Theories. ↩

But they’re only mutually inverse up to equivalence! If we start with a finite limit category $\mathcal{C}$, and then look at the opposite of the finitely presented objects in the functor category $[\mathcal{C},\mathsf{Set}]$, then we merely get something equivalent to $\mathcal{C}$!
In particular, we don’t get something isomorphic to $\mathcal{C}$, and we definitely don’t get something equal to $\mathcal{C}$! Some readers will likely say that concepts of “isomorphism” and “equality” aren’t even defined in a 2category (they would say bicategory, of course), but that’s not a quibble I want to have right now.
What matters is that there’s something honestly 2categorical happening here, and we need that language to make this notion of “sameness” precise in the same way we need 1categories to make the notion of “isomorphism” precise. I’m literally so close to finishing a blog post on “2categories and why you should care” that goes into this in more depth, but I wanted to say a word about it here. After all, rambling footnotes are a feature of this blog! ↩

As many theorems do, at the end of the day. Thankfully our combinatorics is also pretty polite. ↩

Though they work with the opposite of the categories we work with, so for us they’re more likely to be “effective injectives” after dualizing. ↩

Which put it at 6am my time… Thankfully I’ve recently become a morning person, so I only had to wake up an hour earlier than usual. ↩

And I haven’t even had time to watch them all yet! ↩