Quirky Facts: Group objects in the category of groups
04 Dec 2017Recently I was thinking some more about (abelian) groups and I was led to the question of what the group objects in the category of groups are. This is a funny question in and of itself and the answer also has a sense of humor, hence a new series “Quirky facts”. I don’t just work on amusing concequences of axioms of algebra, honestly… But it is fun and provides excellent blog fodder.
So a group object in some category (that has a terminal object and products) is just a few morphisms that give an object the structure of a group. By way of some examples: groups are group objects in the category of sets, lie groups the group objects in smooth manifolds, algebraic groups the group objects in the category of algebraic varieties, the list goes on.
Thinking about this too late at night one is led to the question: what is a group object in the category of groups? Is it all the groups? Some subset perhaps? Are there more somehow (like how a set can often be given two or more group structures)?
In any event I encourage the reader to try and work it out for a while before reading on, it’ll be worth it, I promise!
With that said, let’s begin, let $G$ be a group object in the category of groups, so we have $\times\colon G \times G \to G$ and $e\colon \{1\} \to G$ and $i\colon G \to G\text{,}$ all group homomorphisms. $G$ is itself a group, so we’ll denote its own product by $\cdot$. The first thing to note is that as the group object identity map $e$ must be a group homomorphism, the identity element for $\times$ must be the same as the underlying group operation $\cdot$ as $\{1\}\ni 1 \mapsto 1 \in G$ via $e$.
Now we have a set $G$ with essentially two group operations on it $\cdot, \times\text{,}$ the fact that $\times$ has to be a group morphism and that the product group structure on $G \times G$ is given by $(a_1,b_1)\cdot(a_2,b_2) = (a_1a_2,b_1b_2)$ means that
for any $a_1,a_2,b_1,b_2 \in G\text{.}$ As this is symmetric in $\cdot , \times$ this also says that the group $(G, \times)$ has a group object structure given by $\cdot\text{!}$ At this point one might start to wonder, is $\cdot = \times\text{?}$ So let’s throw in some elements, what about
Ah hah! So the group operations were really the same!
So the answer is just totally boring then? Every group is a group object with the expected operation alone? Well not quite, so far we’ve just been talking about monoids really, i.e. we haven’t mentioned the inverse at all. In order to be a group object the “new” inversion map must be a group homomorphism for the underlying multiplication structure, which is really the group object one too, so by uniqueness of inverses must be the same map. So the cases where this all goes through are the groups for which inversion is a group homomorphism. This is precisely the abelian groups, no more no less!
Okay so in fact one can see the commutativity from the above discussion of the compatibility of multiplications as
so one obtains the corresponding result even just for monoids, but thinking about abelianness and inverses being homomorphisms is what sent me down this little diversion so it seemed rude to cut it out.
It turns out this is known as the Eckmann-Hilton argument/principle/theorem/show. It goes back a long way (1961) and one can read much more elsewhere, there is even a Catsters video or two.
To give a couple of more useful (though admittedly slightly less amusing) applications: This in fact shows that higher homotopy groups $\pi_n(X),\,n \ge 2$ are abelian. If you (like me) think you know a different proof, you don’t! (or maybe you do, who knows what you know).
Finally this also shows that $\pi_1(G,e)$ is abelian for a topological group $G$! No such luck for etale fundamental groups of algebraic groups though.
P.S. anyone who wishes to have someone to blame for the outpouring of uselessness seen here (for example; my advisor) need look no further than Sachi for inspiring me to write something here again.