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Internal Group Actions as Enriched Functors

18 Feb 2024 - Tags: featured

Earlier today this month on the Category Theory Zulip, Bernd Losert asked an extremely natural question about how we might study topological group actions via the functorial approach beloved by category theorists. The usual story is to treat a group $G$ as a one-object category $\mathsf{B}G$. Then an action $G \curvearrowright X$ is the same data as a functor $\mathsf{B}G \to \mathsf{Set}$ sending the unique object of $\mathsf{B}G$ to $X$. Is there some version of this story that works for topological groups and continuous group actions?

I wouldn’t be writing this post if the answer were “no”, so let’s get into it! This is a great case study in the ideas behind both internalization and enrichment, and I think it’ll make a great learning tool for future mathematicians wondering why you might care about such things.

Hopefully people find this helpful ^_^.

(Also, I’m especially sorry for the wait on this one, since I know at least one person has been waiting on it for two weeks now! Life got busy, but I’m excited to finally get this posted.)

First, let’s take a second to talk about Internalization.

The idea here is to take a construction that’s usually defined for sets, and interpret it inside some other category. For instance, a group $G$ is usually

satisfying some axioms.

We can internalize this definition into the category $\mathsf{Top}$ of topological spaces by looking at

satisfying the usual axioms. This recovers the usual definition of a topological group.

Similarly we could ask for a manifold $G$ with smooth maps $m,i,e$, and this would recover the definition of a lie group. At the most general, we ask for a Group Object Internal to $\mathcal{C}$. This is the data of:

  • a $\mathcal{C}$-object $G$
  • a $\mathcal{C}$-arrow $m : G \times G \to G$
  • a $\mathcal{C}$-arrow $i : G \to G$
  • a $\mathcal{C}$-arrow $e : 1 \to G$

satisfying the usual axioms1.

As a quick aside, notice the crucial use of the terminal object $1$ and the product $G \times G$ in the above definition. This tells us that we can only define groups internal to a category with finite products2.

Now just like we can internalize a group object, we can also internalize a group action! If $G$ is a group object internal to $\mathcal{C}$ and $X$ is some object of $\mathcal{C}$ (if you like, $X$ is a “set internal to $\mathcal{C}$”) then an Internal Group Action is a $\mathcal{C}$-arrow $\alpha : G \times X \to X$ satisfying the usual axioms.

So then a group action internal to $\mathsf{Top}$ is the usual notion of a continuous group action, and a group action internal to manifolds is a lie group action, etc.

Let’s change tack for a second and talk about the other half of the story.

Now we start with a (symmetric) monoidal closed category. Roughly speaking, this is a category where the set of arrows $\text{Hom}_{\mathcal{C}}(X,Y)$ can be represented by an object of $\mathcal{C}$!

For instance, the category of vector spaces $\mathsf{Vect}$ is monoidal closed since the homset $\text{Hom}(V,W)$ of linear maps is itself a vector space, which we’ll write as $[V,W]$.

Another example is the category of (nice3) topological spaces. The set of continuous functions $\text{Hom}(X,Y)$ can be given the compact-open topology, so that it is itself a topological space $[X,Y]$.

The fact that these categories can “talk about their own homsets” might make you wonder about other categories with structured homsets.

For instance, there are lots of categories in the wild whose homsets are vector spaces! If $R$ is any $k$-algebra, then for $R$-modules $M$ and $N$ the homset $\text{Hom}_{R\text{-mod}}(M,N)$ is a vector space. Similarly, if $G$ is any group, then the homset between any two $G$-representations is actually a vector space! We say that the categories $R\text{-mod}$ and $G\text{-rep}$ are Enriched over $\mathsf{Vect}$.

Similarly, you can ask about categories where each homset is a topological space. It turns out that these give a fantastic first-order approximation4 to the theory of $\infty$-categories!

More generally, we can define a $\mathcal{C}$-Enriched Category to be the data of

  • A set of objects
  • For each pair of objects $x,y$, a $\mathcal{C}$-object $\text{Hom}(x,y)$
  • For each triple of objects, a composition map in $\mathcal{C}$, $\circ : \text{Hom}(y,z) \otimes \text{Hom}(x,y) \to \text{Hom}(x,z)$
  • For each object $x$, a distinguished element5 $\text{id}_x \in \text{Hom}(x,x)$
  • Satisfying the usual axioms.

And what will be relevant for us, a $\mathcal{C}$-enriched groupoid, which moreover has an inverse map $i : \text{Hom}(x,y) \to \text{Hom}(y,x)$ showing that every arrow is an isomorphism.

Note that every (symmetric monoidal closed) $\mathcal{C}$ is enriched over itself in a canonical way. We take the $\mathcal{C}$-category whose objects are objects of $\mathcal{C}$, and define $\text{Hom}(x,y)$ to be the $\mathcal{C}$-object $[x,y]$. This specializes to the right notion for vector spaces and topological spaces which were examples earlier in this section.

We’ll also need the notion of a $\mathcal{C}$-enriched functor, which is exactly what you might expect given the above definition.

Given two $\mathcal{C}$-enriched categories $\mathbf{A}$ and $\mathbf{B}$, an Enriched Functor $F : \mathbf{A} \to \mathbf{B}$ sends objects of $\mathbf{A}$ to objects of $\mathbf{B}$. Moreover, for every pair of objects in $\mathbf{A}$ there should be a $\mathcal{C}$-arrow \(F_{x,y} : \text{Hom}_{\mathbf{A}}(x,y) \to \text{Hom}_{\mathbf{B}}(Fx,Fy)\) which are compatible with identities and composition.

For example, a $\mathsf{Vect}$-enriched functor is just a functor so that the map $\text{Hom}(x,y) \to \text{Hom}(Fx,Fy)$ is moreover a linear map (recall our homsets are vector spaces). Similarly, a $\mathsf{Top}$-enriched functor is a functor so that the maps on homsets $\text{Hom}(x,y) \to \text{Hom}(Fx,Fy)$ are continuous.

Now we have all the pieces we’ll need to prove

Theorem: Fix a cartesian closed category $\mathcal{C}$.

There is a natural bijection between group objects $G$ internal to $\mathcal{C}$ and $1$-object groupoids $\mathsf{B}G$ enriched over $\mathcal{C}$.

Moreover, for a fixed group object $G$, there is a bijection between internal $G$-actions and enriched functors $\mathsf{B}G \to \mathcal{C}$.

$\ulcorner$ Say that we have a group object $G$ internal to a (cartesian closed) category $\mathcal{C}$. Then let’s build a $\mathcal{C}$-enriched category, $\mathsf{B}G$ with a single object $\star$, where $\text{Hom}(\star,\star) = G$. Of course, we write $\text{id}_\star = e$, and composition is multiplication.

Note that $G$ is an object of $\mathcal{C}$, and the identity/composition/inverse maps are $\mathcal{C}$-arrows. So this really is a $\mathcal{C}$-enriched groupoid with one object!

Conversely, say we have a one-object $\mathcal{C}$-enriched groupoid $\mathcal{G}$. Then $\text{Hom}_\mathcal{G}(\star,\star)$ had better be an object of $\mathcal{C}$, and it’s easy to check that composition and inverse in $\mathcal{G}$ gives this object an internal group structure!

So the data of an enriched $1$-object groupoid is exactly the data of an internal group!

Now, what is a $\mathcal{C}$-enriched functor $F : \mathsf{B}G \to \mathcal{C}$?

We have to send $\star$ to some object of $\mathcal{C}$, say $X$. Then we need a $\mathcal{C}$-morphism $\text{Hom}(\star,\star) \to \text{Hom}(X,X)$. But by the definitions of $\mathsf{B}G$ and $\mathcal{C}$ (enriched over itself) this is the data of a $\mathcal{C}$-arrow $G \to [X,X]$.

Now we use cartesian closedness! This arrow transposes (uncurries) to a $\mathcal{C}$-arrow $G \times X \to X$, and one can check that the identity and composition preservation for the functor corresponds exactly to the axioms for $G \times X \to X$ to be a group action internal to $\mathcal{C}$.

Of course, walking backwards through the above discussion shows that an internal group action $G \times X \to X$ in $\mathcal{C}$ is exactly the data of a $\mathcal{C}$-enriched functor $\mathsf{B}G \to \mathcal{C}$ sending $\star \mapsto X$! $\lrcorner$

Your category-theorist senses should be tingling after reading the statement of the previous theorem!

Sure there’s a bijection of group/group actions, but what about the arrows!?

As a cute exercise, prove that this theorem upgrades to an equivalence6 of categories between

\[\left \{ \begin{array}{c} \text{group objects internal to $\mathcal{C}$ with} \\ \text{internal group homs as arrows} \end{array} \right \} \simeq \left \{ \begin{array}{c} \text{$1$-object groupoids enriched over $\mathcal{C}$ with} \\ \text{enriched functors as arrows} \end{array} \right \}\]

and for fixed $G$

\[\left \{ \begin{array}{c} \text{Internal actions $G \times X \to X$ in $\mathcal{C}$ with} \\ \text{internal $G$-equivariant arrows} \end{array} \right \} \simeq \left \{ \begin{array}{c} \text{Enriched functors $\mathsf{B}G \to \mathcal{C}$ with} \\ \text{enriched natural transformations as arrows} \end{array} \right \}\]

Part of the puzzle is how to define some of these notions (such as “internal $G$-equivariant arrows”). You might find it helpful to read the preexisting definition of an enriched natural transformation.

Let’s take a second to meditate on the difference between “internalization” and “enrichment”. This difference is usually invisible, since for “ordinary” categories we’re always working both internal to $\mathsf{Set}$7 and enriched over $\mathsf{Set}$. That is, our categories are always sets with some structure, and our homsets are always… well, sets!

When you have some gadget and you think “Gee! I sure wish this gadget automatically had the structure of a $\mathcal{C}$-object!”8, you want to work internally to $\mathcal{C}$. Doing this means that pretending that $\mathcal{C}$ is the universe of sets, and the $\mathcal{C}$-arrows are the universe of functions, and then just doing whatever we usually do but “inside $\mathcal{C}$”.

Figuring out exactly how to do this is the purview of much of categorical logic. We can construct standard ways of interpreting set theoretic constructions (such as “${x \in \mathbb{N} \mid \exists y. y^2 = x}$”, etc.) inside a (sufficiently structured) category $\mathcal{C}$. Then there’s a routine, but slightly annoying9, method for cashing out these set theoretic constructions for an “internal” version in $\mathcal{C}$! You can read all about this here or here. One of the reasons so many people care about topos theory is because a topos is a category with so much structure that we can actually internalize any concept we want inside it!

What about enrichment? This is useful when you have an otherwise “normal” category, but your homsets have ~bonus structure~ that you want to respect. For instance, your homsets might be abelian groups, or vector spaces, topological spaces, or chain complexes! Then enriched category theory tells you that, say, yoneda’s lemma still works when you ask that everything in sight respects this ~bonus structure~. This turns out to be the start of an incredibly interesting subject called formal category theory10.

To see how well you understand internal versus enriched things, here’s a cute exercise:

Write out, in some detail, the definition of a category internal to $\mathsf{Cat}$ (the category of categories). Then write out, in some detail, the definition of a category enriched over $\mathsf{Cat}$ (with $\times$ as its monoidal structure).

Both of these concepts are extremely useful in lots of ongoing research in algebra, logic, and applied category theory! A category internal to $\mathsf{Cat}$ is a double category (See Evan Patterson’s excellent blog post on the subject). A category enriched over $\mathsf{Cat}$ is a 2-category, these show up very naturally, as I’ll hopefully show in an upcoming blog post!

Ok, this blog post became something much longer than I originally intended (to nobody’s surprise), but let’s have one more cute puzzle before we go.

Another common way group actions get treated is as a group homomoprhism $G \to \text{Aut}(X)$, where $\text{Aut}(X)$ is the group of automorphisms of $X$. Is there some way to make this perspective fit in with the internal/enriched perspectives we’ve been working with so far?

Again, the answer is yes, but now we need to work with a cartesian closed category with all finite limits.

Given a cartesian closed category with finite limits $\mathcal{C}$, and an object $X \in \mathcal{C}$, can you build a group object $\underline{\text{Aut}}(X)$ internal to $\mathcal{C}$ so that the global elements $1 \to \underline{\text{Aut}}(X)$ are in bijection with the usual automorphism group $\text{Aut}(X)$ (which is just a set)?

Then, once you’ve defined $\underline{\text{Aut}}(X)$, can you show that an internal group action $G \times X \to X$ is the same data as an internal group hom $G \to \underline{\text{Aut}}(X)$?

If you find this exercise hard, maybe that’s incentive to learn some categorical logic! The category $\mathcal{C}$ has enough structure11 for its internal language to support a definition

\[\{ f : X \to X \mid \exists g : X \to X . fg = \text{id}_X \land gf = \text{id}_X \}\]

which we can then cash out for an honest object $\underline{\text{Aut}}(X)$ in $\mathcal{C}$.

Since the usual proof that this set is a group is constructive, we get for free that any object $\underline{\text{Aut}}(X)$ in any $\mathcal{C}$ is actually a group object in $\mathcal{C}$! Moreover, the usual proof that an action $G \curvearrowright X$ is a group hom $G \to \text{Aut}(X)$ is constructive. So we learn, for free, that in $\mathcal{C}$ an internal group action is the same thing as an internal group hom $G \to \underline{\text{Aut}}(X)$!

Thanks for sticking around! This was a super fun post to write since it touches on a LOT of aspects of “more advanced” category theory that people might struggle with at first. I would normally give more of an outro, but I have some friends coming over in a half hour and I really want to get this posted!

Stay warm, and stay safe, all! We’ll talk soon ^_^

  1. Note that the “usual axioms” can all be expressed as equalities between composites of these functions. For instance, the inverse law says that the composite

    \[\begin{array}{ccccccc} G & \overset{\Delta}{\longrightarrow} & G \times G & \overset{1_G \times i}{\longrightarrow} & G \times G & \overset{m}{\longrightarrow} & G\\ g & \mapsto & (g,g) & \mapsto & (g, g^{-1}) & \mapsto & g \cdot g^{-1} \end{array}\]

    is the same arrow as

    \[\begin{array}{ccccc} G & \overset{!}{\longrightarrow} & 1 & \overset{e}{\longrightarrow} & G \\ g & \mapsto & \star & \mapsto & e \end{array}\]

    The elementwise definitions in the lower lines are primarily for clarity in showing what these composites “are really doing”, but they can be made precise using the language of “generalized elements”. 

  2. There turns out to be a deep connection between “algebraic theories” (like groups, rings, etc) and categories with finite products, which I want to write about someday. This is the start of the story of categorical logic, which is near and dear to my heart.

    One can view this whole game of “internalization” as a subfield of categorical logic, where we focus in on the things we normally do to sets, and precisely say

    1. what categories can interpret various set-theoretic constructions
    2. how to figure out what the “right way” to internalize a given construction is. This turns out to be totally algorithmic!

  3. There’s a couple things I could mean by “nice” here. See here for some options, but if pressed I’d probably say compactly generated spaces. Keep in mind that this makes the compact-open topology I linked to incorrect in some edge cases, but morally what I’ve said is right, and I think it’s literally true for compactly generated hausdorff spaces. 

  4. In fact, every $\infty$-category is equivalent (in the appropriate sense) to a category enriched in simplicial sets. See here, for instance.

    The claim then follows from the fact that the category of simplicial sets (up to homotopy) is equivalent (again, in an appropriate sense) to the category of topological spaces (up to homotopy). 

  5. Of course, by “element” here I mean a global element. That is, a map from the monoidal unit $\text{id}_x : I \to \text{Hom}(x,x)$. 

  6. Isomorphism? 

  7. Ignoring size issues 

  8. Which for some reason I’m hearing in Tom Mullica’s voice 

  9. In exactly the same way that, say, gaussian elimination, is routine but annoying. 

  10. Of course, there’s other more exotic ways to use enriched categories where the hom-objects aren’t structured sets! See, for instance, the famous Lawvere Metric Spaces. These are still extremely interesting, but are of a different flavor to the enriched categories that fit into the story I’m trying to tell. 

  11. Notice that, in this definition, the $g$ in the existential quantifier is provably unique. This is super important because it means we can make this definition using only finite limits. More complicted existential quantifiers require more structure on our category, namely regularity. For more information see the nlab pages on cartesian logic and regular logic