Algebra: Chapter 0 — Joke 1.1 – Left Adjoint to Forgetful Thinking

In Aluffi’s textbook “Algebra: Chapter 0” (if I’m not being redundant enough), he makes a joke defining a mathematical group as a one-object groupoid. If not just to show how fun I am at parties, I want to argue that this is the right definition of a group! Hear me out; I’m not trying to make a case for category theory to be taught too early in one’s mathematical journey.

So what is a group? If you asked me in my second year of undergrad, I would have regurgitated the definition I learned in class:

Definition 1: A group is a set $G$ equipped with a binary operation $\circ : G\times G\to G$ called the group multiplication, and a distinguished element $1_G\in G$ called the group identity subject to:

(associativity) for $a,b,c\in G$ , we have $a\circ(b\circ c)=(a\circ b)\circ c$
(unity) for $a\in G$ , we have $1_G\circ a=a=a\circ 1_G$
(inverse) for $a\in G$ , there exists some $a^{-1}\in G$ such that $a\circ a^{-1}=1_G=a^{-1}\circ a$

This is the standard definition of an abstract group. The issue is that it does not really answer the question… so what is a group? The above algebraic definition obscures the underlying intention, and it’s hard to believe the ubiquity of groups in mathematics from such an innocent-looking set of axioms.

The first misleading detail is that a group is defined as a structured set, similar to a vector space being a set $V$ equipped with some operations and other relevant structure. This makes it seem like the elements of a group are points in some abstract space, an impression that is then reinforced by some of the simplest examples of groups: the group $\mathbb{Z}$ of integers with addition, or the group $\mathbb R^\times$ of nonzero real numbers under multiplication, both of which make the elements of a group appear to be nothing more than “points.”

A group is supposed to be the abstraction of a group of symmetries. A prototypical example is the family of dihedral groups, which collect the rotations and reflections that preserve the orientation of a regular polygon. Another prototypical example—perhaps the main one—is even called the symmetric group, which is the collection $S_n$ of permutations of $n$ letters. In both cases, the groups are collections of reversible actions on an object.

Consider as a final example the general linear group $\mathop{\mathrm{GL}}_n(\mathbb R)$ . On one hand, this can be defined as the set of invertible $n\times n$ real matrices under multiplication, but this makes the group seem more like a toy example rather than something with real and useful applications (e.g., in physics). This group is actually very similar to the symmetric group, because it is precisely the collection of linear symmetries (called linear automorphisms) on $n$ -dimensional space $\mathbb{R}^n$ . In particular, by setting $n=1$ , we find that the previous example $\mathbb{R}^\times$ of nonzero real numbers is really the group of symmetries of the real number line under rescaling (e.g., if you scale the real number line by a factor of two, you get a “twice as long” infinite line, which holistically is the same as the original real number line).

Amidst his many polarised opinions on teaching mathematics, Vladimir Arnold makes a similar argument about the definition of groups. “Algebraists teach that [a group] is supposedly a set with two operations [composition and inversion] that satisfy a load of easily-forgettable axioms… We get a totally different situation if we start off not with the group but with the concept of a transformations [symmetries].” I’m not sure about calling three axioms a “load” per se, but it sounds like he would have preferred something more like the following:

Definition 2: A group is the collection $\mathop{\mathrm{Aut}} X$ of automorphisms (structure-preserving symmetries) for some “object” $X$ .

It’s easier to verify that our prototypical examples are groups under this definition. For instance, $S_n = \mathop{\mathrm{Aut}}\{1,\dots,n\}$ is the collection of set-theoretic automorphisms (i.e., bijections) on an $n$ -element set, and $\mathop{\mathrm{GL}}_n(\mathbb R) = \mathop{\mathrm{Aut}}\mathbb{R}^n$ as we have already discussed. The only hiccough is that it’s not a priori clear what the automorphisms of some arbitrary “object” is. This is where category theory comes in handy.

A category is precisely the abstraction of having a collection of (mathematical) objects and maps between them (called morphisms). Given a category $\mathcal{C}$ and an object $X$ of $\mathcal{C}$ , an automorphism of $X$ is just an invertible morphism $X\to X$ —a way of transforming $X$ that can be undone. Thus, we can obtain symmetric groups from the category $\mathbf{Set}$ of sets and functions, and general linear groups from the category $\mathbf{Vect}_{\mathbb{R}}$ of real vector spaces and linear transformations.

So what’s a groupoid? A groupoid is just a category whose morphisms are all invertible. In particular, if the groupoid only has one object $X$ , then its only morphisms are the automorphisms of $X$ . Voilà: Joke 1.1!

Well, not quite. There is a subtle difference between Definition 2 and Joke 1.1: in fancy words, Joke 1.1 is the delooping of Definition 2. More precisely, Definition 2 only concerns itself with the automorphisms of some object $X$ , whereas Joke 1.1 says that the group should be the collection of automorphisms of $X$ along with $X$ itself. Big deal, right? Well, there is some additional conceptual benefit to remembering the object on which the automorphisms act.

Exactly, it has to do with a group’s action on other things. What good is a group of symmetries if we can’t realise these symmetries on other objects? For example, there is a canonical action of $S_n$ on $n$ -dimensional space $\mathbb{R}^n$ , where a permutation $\sigma\in S_n$ acts on a vector in $\mathbb{R}^n$ by permuting the vector’s coordinates. (In fact, since this action is linear, we obtain a representation of $S_n$ .) Of course, there is a nuts-and-bolts way of axiomatising group actions (and group representations). There is also a somewhat slicker definition using group homomorphisms (maps of groups that preserve the identity and multiplication):

Definition 3: Let $G$ be a group, and $X$ an object of some category, then an action of $G$ on $X$ is a group homomorphism $G\to\mathop{\mathrm{Aut}}X$ .

You get classical group actions if $X$ is a set, and you get group representations if $X$ is a vector space. This is fine, of course, and gets the idea of a group action across perfectly well. However—and perhaps I am a bit biased—the definition in Joke 1.1 is just a bit slicker still! We just need an intuitive grasp of a bit more categorical jargon: given two categories $\mathcal{C}$ and $\mathcal{D}$ , a functor $F:\mathcal{C}\to\mathcal{D}$ can be thought of as a meaningful way of “representing” the objects of $\mathcal{C}$ as objects of $\mathcal{D}$ (in a way that respects how objects of $\mathcal{C}$ map between each other via morphisms). This gives us the following:

Proposition 4: Let $G$ be a group, and $\mathcal{C}$ a category. Denote by $\mathbf BG$ the group viewed as a one-object groupoid (with unique object $*$ ). Then, a functor $F:\mathbf BG\to\mathcal C$ is precisely an action of $G$ on the object $X := F(*)$ in $\mathcal C$ .

In other words, a group action on an object is a realisation of an abstract group as symmetries of the target object. As an added terminological bonus, the action is faithful (i.e., different elements of $G$ act differently on $X$ ) precisely if the corresponding functor is faithful (i.e., injective on morphisms) by definition. If you think I’m being pedantic, check out what happens when you extend to natural transformations:

Proposition 4 (upgraded): Let $G$ be a group, and $\mathcal{C}$ a category. Then, the category whose objects are those of $\mathcal{C}$ equipped with a $G$ -action, and whose morphisms are those of $\mathcal{C}$ equivariant under these $G$ -actions, is precisely the functor category $\mathop{\mathrm{Func}}(\mathbf BG,\mathcal C)$ .

Let me just say that I’m not arguing that category theory should be taught before group theory; I don’t think it’s worthwhile to learn category theory if your only useful examples are essentially just $\mathbf{Set}$ and $\mathbf{Vect}_\Bbbk$ . I’m just trying to emphasise that, no matter how groups are introduced to you, they really are one-object groupoids!

“But Zach,” you’re probably wondering, “what’s an example of a bad definition of a group then?” Before I answer this question (note that this is just my opinion), here is another definition of groups that I don’t think is actually that bad.

Definition 5: A group is given by $\pi_0\Omega X$ for some pointed topological space $X$ .

In more words, Definition 5 takes a group to be the collection of loops around a fixed basepoint of a space $X$ , where loops are considered “the same” if you can continuously deform one (on $X$ ) to obtain the other (called a homotopy between the loops). While this might seem foolish, this definition might actually be more closely aligned with how we visualise symmetries on an object!

Consider the dihedral groups: when you picture the symmetries of some polygon (e.g., reflection), you probably picture this as an action happening “over time” (the polygon is smoothly rotating about the axis of reflection). Thus, consider the (imprecise) space $X$ of the infinitude of possible orientations of your polygon in 3D space, and pick the basepoint to be the “neutral” configuration. Then, visualising a symmetry of your polygon boils down to tracing out a certain path in $X$ starting and ending at the neutral configuration (i.e., an element of $\Omega X$ ). Homotopy equivalence then just ensures that any two visualisations of the same symmetry are really the same, so that $\pi_0\Omega X$ recovers the dihedral group, as desired! (This space gives me “moduli stack” vibes…)

Side note: If you accept the Homotopy Hypothesis, then Definition 5 really isn’t that different from Definition 2. Indeed, the Homotopy Hypothesis allows us to think of $X$ as an $\infty$ -groupoid (up to homotopy). If $x\in X$ is the marked point, then the loop space $\Omega X$ is just the $\infty$ -groupoid $\mathop{\mathrm{Hom}}_X(x,x)$ , and therefore $\pi_0\Omega X \cong \pi_0\left(\mathop{\mathrm{Hom}}_X(x,x) \right) = \mathop{\mathrm{Hom}}_{\Pi_1X}(x,x)$ is just the collection of (auto)morphisms $x\to x$ in the 1-truncation $\Pi_1X$ (which is a groupoid).

So, what do I think is a bad definition?

Definition 6: A group is a group object in $\mathbf{Set}$ .

Why? Well, if you use the “internal group” definition of a group object, then you’re just writing the ordinary nuts-and-bolts definition of a group in a needlessly complicated way. What’s worse, you have to define inversion as a function in this case, but having inverses should be a property, not additional structure! (You get “saved” by how inverses are necessarily unique.) If you use the presheaf definition (which on the plus side avoids all seemingly arbitrary axioms of a group, and in general categories is quite elegant), then you’re being circular: you’re saying that a group is a set whose representable presheaf factors through the category of groups!

Moral of today’s story: don’t can a definition just because it seems “needlessly complicated.” There, joke ruined. You’re welcome $\heartsuit$ .