Life in Johnstone's Topological Topos 2 -- Topological Algebras

03 Jul 2024 - Tags: life-in-the-topological-topos

In the first post, we introduced Johnstone’s topological topos $\mathcal{T}$ and talked about what its objects look like. We showed how the interpretation of type theory in $\mathcal{T}$ gives us an “intrinsic topology” on any type we construct. We also alluded to the fact that, by working in $\mathcal{T}$ as a universe of sets, we’re able to interact with topological gadgets by forgetting about the topology entirely and just manipulating them naively as we would sets!

In this post, we’ll talk about how that works in the special case of algebraic gadgets, like groups, rings, etc., and use this to prove some interesting theorems about topological groups!

Recall Lawvere’s notion of Functorial Semantics. An Algebraic Theory is presented by some function symbols and equational axioms (we allow constant symbols as 0-ary functions), and this is probably best given through a “definition by examples”.

The usual presentation of the theory of groups is

A set $G$ equipped with function symbols

• $e : G^0 \to G$
• $(-)^{-1} : G^1 \to G$
• $\cdot : G^2 \to G$

satisfying the equational axioms

• $(x \cdot y) \cdot z = x \cdot (y \cdot z)$
• $x \cdot e = x = e \cdot x$
• $x \cdot x^{-1} = e = x^{-1} \cdot x$

Notice that the theory of posets is not algebraic1 and indeed the usual presentation involves a relation symbol $\leq$ (which is not allowed) rather than only function symbols. Similarly, the theory of fields is not algebraic2 and the usual presentation requires an axiom that’s much more complicated than just an equation: $(x = 0) \lor \exists y . xy = 1$.

However, these presentations by functions and equational axioms should really be thought of as presentations. There are superficially quite different presentations which still present the same theory. For instance, here is another presentation of the theory of groups3:

A set $G$ equipped with a function symbol

• $/ : G^2 \to G$

satisfying the equational axiom

• $x / \big ( ((x/x)/y)/z) / ((x/x)/x)/z \big ) = y$

With this in mind it’s natural to want an abstract characterization of an algebraic theory, that is independent of the choice of presentation. In his PhD thesis, Lawvere set this in motion by showing that for any algebraic theory $\mathbb{T}$, there’s a Classifying Category $\mathcal{C}_\mathbb{T}$ so that

$\{ \mathbb{T}\text{-algebras} \} \simeq \{ \text{finite product functors } \mathcal{C}_\mathbb{T} \to \mathsf{Set} \}$

If we have a good understanding of $\mathbb{T}$, then we can get our hands on $\mathbb{C}_\mathbb{T}$ concretely since it’s (opposite) the full subcategory of finitely generated free models!

Important for us is the related result that models of $\mathbb{T}$ in some other finite product category are given exactly by finite product functors from $\mathcal{C}_\mathbb{T}$ into that category!

So, for example, a topological group is the same data is a finite product functor $$\mathcal{C}_\mathsf{Grp} \to \mathsf{Top}$$, while a lie group is the data of a finite product functor $$\mathcal{C}_\mathsf{Grp} \to \mathsf{Diff}$$.

This is what’s going to give us the ability to relate algebras in $\mathcal{T}$ to topological algebras! Let’s see how it works!

First, say we have a group object in $\mathcal{T}$. This is the data of a finite product preserving functor $\mathcal{C}_\mathsf{Grp} \to \mathcal{T}$. But we know from part 1 that the reflector $r : \mathcal{T} \to \mathsf{Seq}$ preserves finite products too! So composing these gives a finite product functor

$\mathcal{C}_\mathsf{Grp} \to \mathcal{T} \to \mathsf{Seq}$

which is a group object in $\mathsf{Seq}$. That is, a (sequential) topological group4!

Conversely, say we have a sequential topological group. Then the embedding $e : \mathsf{Seq} \to \mathcal{T}$ is a right adjoint, and in particular preserves finite products. So again we get a group object in $\mathcal{T}$!

In fact, the adjunction $r \dashv e$ gives us an adjunction $$r_* \dashv e_*$$ between the functor categories, and since $r \circ e \cong \text{id}_\mathsf{Seq}$, this is true at the level of functor categories too!

So the category of sequential topological groups is a reflective subcategory of the category of groups in $\mathcal{T}$, and the reflector is exactly what you expect: Just take the topological reflection of the underlying object of $G$!

There’s nothing special about groups here, and so we learn that for any algebraic theory, the category of sequential models is a reflective subcategory of the category of models in $\mathcal{T}$. Thus, any question we have about (sequential) topological models can be answered in $\mathcal{T}$ without losing information, and anything we prove about models in $\mathcal{T}$ immediately gives us results about topological models by reflecting (though in this direction we possibly lose information about the proofs of convergence).

This is all kind of abstract right now, so let’s do a very down-to-earth example:

In $\mathcal{T}$, a subset of $G$ is any monic $X \hookrightarrow G$ (that is, any continuous injection). In particular, $X$ does not need to have the subspace topology!

With this in mind, a subgroup of $G$ is just a continuous injection $H \hookrightarrow G$ whose image is a subgroup in the usual sense. Of course, this pulls back (by injectivity) to a unique group structure on $H$ rendering the inclusion a homomorphism.

Now here’s a typical (very easy!) theorem/construction:

Let $X$ be any subset of $G$, a group. Then there’s a smallest subgroup $\langle X \rangle \leq G$ containing $X$.

$\ulcorner$ If $X$ is any subset, we define

$\langle X \rangle = \bigcap \{ H \leq G \mid X \subseteq H \}.$

This is a subgroup containing $X$, and any other subgroup containing $X$ is part of the intersection, rendering this the smallest such subgroup. $\lrcorner$

Notice that this proof is constructive in the sense that it doesn’t use LEM or Choice5. In particular, this proof works in every topos, and thus in $\mathcal{T}$.

But what does this exceptionally simple proof tell us about topological groups? Well subsets and subgroups are continuous injections, so this tells us that6

Let $X \hookrightarrow G$ be any continuous injection into a topological group $G$. Then there’s a topological group $\langle X \rangle$ with a continuous injection $\langle X \rangle \hookrightarrow G$ so that

1. $X \hookrightarrow G$ factors through $\langle X \rangle$
2. $\langle X \rangle$ is initial with this property

We can actually build such an $H$ by externalizing the proof of this theorem too! Subsets are interpreted as general monics into $G$, and the “intersection” of two monics externalizes to their pullback. So the desired $\langle X \rangle$ is exactly the pullback of the family of all continuous injections $H \hookrightarrow G$ factoring the inclusion from $X$7.

In case $G$ is sequentially hausdorff, this is on-the-nose correct. In case $G$ isn’t, then $\langle X \rangle$ might live in $\mathsf{Kur}$ instead of $\mathsf{Seq}$. But that’s ok! We can just hit it with the reflector to get an “honest” topological group with the same universal property (among the continuous injections whose domain is also “honest”).

Now, it’s entirely possible that you would have come up with such a theorem yourself. After all, a moment’s thought shows that $\langle X \rangle$ is “just” the usual subgroup generated by $X$, equipped with the finest topology rendering $X \hookrightarrow \langle X \rangle$ continuous.

The utility of the topos theoretic language is in doing more complicated constructions, where we’re still allowed to manipulate everything as though they’re sets, and we can be safe in the knowledge that, at the end of the day, we can cash out our theorem for one about topological spaces! It frees us from the burden of carrying around topologies all the time.

For a more complicated example, one can show that the category of abelian groups in a (grothendieck) topos is always AB5. In particular, the category of abelian groups in $\mathcal{T}$ is abelian and has enough injectives, so we can do homological algebra to it! Contrast this with the category of abelian groups in $\mathsf{Top}$, which is famously not abelian!

This is one of the big motivations for Condensed Mathematics. Indeed, in $\mathsf{Top}$, the continuous bijection of abelian groups $(\mathbb{R},\text{discrete}) \to (\mathbb{R},\text{euclidean})$ is not an isomorphism. Yet the kernel and cokernel are both trivial! In both condensed mathematics and the topological topos, this is remedied by a more complicated cokernel. Remember that the colimits preserved by the embedding $\mathsf{Seq} \to \mathcal{T}$ are only those that look like covers.

In fact, we can compute the cokernel as the coequalizer of the inclusion map and the constant $0$ map. In the topos, this is the sheafififcation of the colimit of presheaves, which are computed pointwise. So the underlying set of the cokernel is the colimit of the underlying sets is ${ 0 }$. But the convergent sequences is the colimit of

$\{ \text{eventually constant sequences} \} \rightrightarrows \{ \text{convergent sequences} \}$

where one map is just the inclusion, and the other sends every eventually constant sequence to the constant $0$ sequence.

So the cokernel has a single point ${ 0 }$, but there’s a proof that the constant $0$ sequence converges for every equivalence class of convergent sequences differing by an eventually constant sequence!

Keeping track of these proofs (which themselves form an abelian group) is exactly what we need to do to algebraically detect that $(\mathbb{R}, \text{discrete}) \to (\mathbb{R}, \text{euclidean})$ isn’t an isomorphism!

As an aside, I don’t understand condensed mathematics well enough to know how it differs from math in the topological topos. Just looking at definitions, I know it’s based on test maps from all compact hausdorff spaces instead of test maps from only $$\mathbb{N}_\infty$$. This probably means it’s closely related to compactly generated spaces in much the way that $\mathcal{T}$ is related to sequential spaces. I’m sure there’s a reason to prefer this, but I don’t know what it is8. The moral to keep in mind is that the power of doing algebra in a topos that handles the topology for you is currently being used to great effect in applying homological algebra to analytic situations where it previously couldn’t go!

Alright, I told you this one was going to be more leisurely than the last one! Now that we’ve seen some applications of $\mathcal{T}$ to topological algebra, and we’ve seen some basic externalization, let’s move on to part 3 and really get familiar with the internal logic and how it relates to the real world!

1. You can show this with categorical techniques. For instance, the category of models of any algebraic theory is always regular, while the category of posets isn’t

2. The category of models for any algebraic theory always has an initial object, yet the category of fields doesn’t!

3. See McCune’s Single Axioms for Groups and Abelian Groups with Various Operations.

This operaetion is related to the “usual” operations by $x / y = x \cdot y^{-1}$.

4. Remember, though, that the product on $\mathsf{Seq}$ is different from the product on $\mathsf{Top}$. This never matters in practice, and the $\mathsf{Seq}$ product agrees with the product in the “convenient category” of compactly generated spaces, but if you want an honest group object in $\mathsf{Top}$, you’ll want $G$ to be locally compact.

5. It’s not predicative, but that’s fine for a topos. And regardless, if you know enough to complain about predicativity, you know enough to give a predicative version of this proof :P.

6. Indeed, it says something slightly stronger than this! In the case of non (sequentially) hausdorff spaces, there might be extra subsets that are merely kuratowski limit spaces! The theorem says we’re actually allowed to take $X$ to be such a subspace as well!

7. The diligent reader will note there are a proper class of such arrows, so this pullback as written isn’t defined. Of course, the domain of any such arrow has at most $|G|$ many elements, and there’s only a set worth of topologies we can put on one of these domains. So up to isomorphism there’s only a set worth of arrows, and we’re good to go!

8. Peter Scholze actually says a few words about why condensed sets are easier to work with than objects of $\mathcal{T}$ in a comment to his answer to this MO question. I still don’t really see it, but that’s probably because I haven’t spent a lot of time (or any time) working with condensed sets.