Groups which are only defined up to conjugation

Question

I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples I have in mind:

"The" absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ which depends on a choice of algebraic closure and so is only well-defined up to conjugation. Here the usual approach is to study finite dimnesional representations of this group, since the isomorphism class of the representation is a conjugation invariant.

The (outer) automorphism group of "the" hyperfinite $\mathrm{II}_1$ factor. Again this depends on a choice of hyperfinite $\mathrm{II}_1$ factor, and so is only well-defined up to conjugation. This group is simple, so the kinds of techniques you use to study the absolute Galois group won't work.

As pointed out in comments, and in email by Qiaochu, one way to think about these examples is that instead of a group you have a connected groupoid. If you pick a basepoint a connected groupoid gives a group, but without picking a basepoint you have something somewhat different from a group.

But these two examples have something additional in common which is that one doesn't have easy direct access to the points. Or even if you do find a particular way to write down a base point, there are many natural choices of basepoint and there's no constructive way to write down the path between basepoints.

Is there a good way to think or talk about this which will make more concrete the intuiton that you're banning yourself from ever fixing basepoints or fixing a path between two points?

(The original version of this question was NARQ, so I've edited and rewritten it to try to improve it.)

Answer 1

For reference, here's roughly the content of the email I sent to Noah:

The short answer is "connected groupoids, regarded as objects in the homotopy category of groupoids."

Recall that the category of groupoids is really a $2$-category: it has objects groupoids, morphisms functors, and $2$-morphisms natural transformations. The homotopy category of groupoids, $\text{Ho}(\text{Gpd})$, is the category whose objects are groupoids and whose morphisms are natural isomorphism classes of functors. Concretely, there's a subcategory given by one-object groupoids, which are groups, and morphisms between these are pointwise conjugacy classes of group homomorphisms (so in particular it's morphisms, not groups, that we're only regarding as well-defined up to conjugacy, but as category theorists we know that it's the morphisms that are important anyway). For example, in this category

• $\text{Hom}(\mathbb{Z}, G)$ returns the conjugacy classes, rather than the elements, of $G$.
• $\text{Aut}(G)$ returns the outer automorphism group of $G$.

This category is an algebraic model of what happens when you try to get a continuous map $f : X \to Y$ between topological spaces to induce a map on fundamental groups: to do this you need to pick a basepoint $x \in X$ so that you get a map $\pi_1(X, x) \to \pi_1(Y, f(x))$, and the homomorphism you get by varying the basepoint along a path depends on the homotopy class of the path, unless you only work up to conjugacy. More precisely, the homotopy category of $1$-types is equivalent to the homotopy category of groupoids, with the equivalence given by taking the fundamental groupoid. (To get the ordinary category of groups we need to use pointed connected $1$-types.)

In the case of Galois theory, the connected groupoid in question is the groupoid whose objects are the algebraic closures of a field and whose morphisms are isomorphisms between these, just as in the topological case the fundamental groupoid is the groupoid whose objects are the points of a space and whose morphisms are homotopy classes of paths between these.

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The case of Galois theory exhibits the difficulty Noah talks about in the edit if we don't allow ourselves enough choice to construct algebraic closures. In this case I think the correct replacement of, say, $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the category of finite extensions of $\mathbb{Q}$. The "absolute Galois groupoid" can be recovered from this category via Grothendieck's Galois theory, as the groupoid whose objects are fiber functors and whose morphisms are natural isomorphisms, but I think it takes some choice to write down fiber functors as well so one can just refrain from doing so and do other stuff instead.

In particular, it should be possible to write down, say, $\text{Rep}(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ from the above data without ever explicitly constructing the group itself. (Analogy: I can write down the category of local systems on a space $X$ with values in $\text{Vect}$ without choosing a basepoint for $X$. It should in fact be possible to describe what a local system on a locale ought to be, even if the locale has no points! Perhaps this is the sort of thing you want.)

Answer 2

I think I can say something about why this problem rears its head particularly in the study of $\operatorname{Gal}( \overline{\mathbb Q}|\mathbb Q)$. As Qiaochu and others have pointed out, a group-up-to-conjugation is more or less a connected groupoid, and this interpretation meets up with the topological intuition. As HJRW points out, the fact that the Galois group appears to us as a groupoid does not distinguish it much from other symmetry groups. Those are naturally groupoids, and we must fix a basepoint to speak of elements of the group.

Note also that there is no difficulty in fixing a basepoint of the Galois goup! Simply take the algebraic closure of $\mathbb Q$ inside $\mathbb C$. Grothendieck's dessins d'enfants are essentially a way of studying the Galois group via this basepoint.

Instead, I think a key problem is that there is more than one good way of choosing the basepoint. It might be better to work with a more general notion of a basepoint. A basepoint is a functor from the one-object one-morphism category to the groupoid. More generally, we can consider functors from an arbitrary category to the groupoid.

For each prime $p$, we have a functor from the groupoid of algebraic closures of $\mathbb Q_p$ to the groupoid of algebraic closures of $\mathbb Q$ - just take all the algebraic elements. Since the Galois group of $\mathbb Q_p$ is not so hard to understand (a least compared to the Galois group of $\mathbb Q$), having one of these is a lot like having a basepoint.

Why is this problematic? Because different phenomena are most naturally studied using different basepoints. Remember that, inverse Galois problem aside, we mostly don't study the Galois group as an abstract group. Instead, we study how it relates to other things we care about: Number fields are subgroups of the Galois group, but more importantly, we can relate that subgroup to the arithmetic of that number field. Almost everywhere a Galois group appears in number theory, it's being related to lots of other interesting things. These interesting things usually have aspects that only make sense at a given place. So to study the Galois group, we really need to study it at each place idependently. While these different basepoints exist perfectly constructively, the paths between them do not, and these paths are crucial to get an understanding of the Galois group.

So I think there is no mathematical difference with the group of symmetries of the cube, but a metamathematical difference. I know nothing about von Neumann algebras, but I would be interested to hear if there is a similar picture for the hyperfinite $\operatorname{II}_1$ factor.

Finally, note that one reason Galois representations are so important is that we have a good tool for constructing Galois representations, that being etale cohomology of algebraic varieties. This is why we consider Galois representations over $\mathbb Q_l$, rather than another field. We also have powerful tools like L-functions and automorphic forms for describing properties of Galois representations. These all make Galois representations a fruitful field, whereas dessin d'enfants has not been so fruitful because there are not so many powerful tools - but there is nothing intrinsic about the origin of the Galois group that forces it to be that way. Maybe we just haven't discovered the right tools yet!