About the category $\mathrm{Set}(G)$

Question

I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I have to give it a try. In particular, I need to understand the category of left $G$-sets for a group $G$ because of the classification theorem for covering maps.

Of course I know what a $G$-set is, and I know some examples. I know that any set can be made a $G$-set for any group $G$ by defining $$gx=x$$ for all $g\in G$ and $x\in X$. When $X$ and $Y$ are sets of different cardinalities, such $G$-sets are not isomorphic, which means that the isomorphism classes of objects in $\mathrm{Set}(G)$ don't form a set. So I'm guessing a classification could be impossible.

But I'm wondering if the important examples of $G$-sets are somehow special (maybe definable) in this category. By important examples I mean expecially these:

• the left coset spaces of subgroups of $G$ (in particular, the homomorphic images of $G$) acted on by multiplication;
• the normal subgroups of $G$ acted on by conjugation;
• the power set of $G$ acted on by multiplication.

If I haven't put something important on this list, I would like to learn about it too.

I would like to know if there's something that characterizes these examples up to isomorphism in $\mathrm{Set}(G)$ and whether perhaps there is some (maybe partial) classification or characterization of $G$-sets using these examples. For example, something like this would be good: every $G$-set is isomorphic to a left coset space of some subgroup of $G$. This is clearly false, but this is the kind of thing I have in mind. I'm not sure I can explain myself better.

I would love to learn some category theory from the answers, but please keep in mind that my understanding of it is very limited so I will probably need more explanation than others might.

Answer 1

This is usally answered in the context of Grothendieck's Galois Theory (where $G$ is a profinite group, for example the etale fundamental group of a scheme, and we only consider finite continuous $G$-sets). See Lenstra's notes on Galois theory for this. I only mention this as a background, the rest of this answer explains some of these ideas and hopefully answers your question.

In $\mathsf{Set}(G)$ coproducts are just disjoint unions endowed with the obvious $G$-action, and the initial object is the empty set endowed with the unique $G$-action. More generally, the forgetful functor $U : \mathsf{Set}(G) \to \mathsf{Set}$ creates all colimits. (By the way, $U$ is represented by $G$, so that in particular $\mathrm{End}(U) \cong \mathrm{End}(G) = G$ and we can reconstruct the group $G$ from $U$ - this is the discrete Tannakian reconstruction.)

An object of a category with coproducts is called connected if it is not initial and cannot be written as a coproduct of two non-initial objects.

One can easily check that the connected objects of $\mathsf{Set}(G)$ are precisely the transitive $G$-sets. Besides, every object is a unique(!) coproduct of connected objects: This is the usual decomposition of an action into its orbits. The connected objects can be classified (up to non-canonical isomorphism), they are precisely the coset sets $G/U$, where $U$ is a subgroup of $G$. Here, $G/U$ and $G/V$ are isomorphic iff $U$ and $V$ are conjugated to each other. How to characterize the case that $U$ is normal?

A connected object $X$ of a category with coproducts and quotients is called Galois if the quotient $X/\mathrm{Aut}(X)$ is a terminal object.

One should imagine this as "the automorphism group of $X$ acts transitively on $X$". For $G/U \in \mathsf{Set}(G)$, the condition means that for all $g \in G$ there is some automorphism $\sigma : G/U \to G/U$ with $\sigma([1])=[g]$. But then for $u \in U$ we have $[ug] = u [g] = u \sigma([1]) = \sigma([u])=\sigma([1])=[g]$, i.e. $g^{-1} u g \in U$. Hence, $U$ is normal. The converse is also easy to check. Hence, the Galois objects of $\mathsf{Set}(G)$ correspond to the normal subgroups of $G$.

Because you are learning topology, here is a reason why these category-theoretic notions are useful: If $X$ is a nice pointed topological space, then covering theory tells us that there is an equivalence of categories $\mathrm{Cov}(X) \cong \mathsf{Set}(\pi_1(X))$. Every equivalence of categories preserves objects and properties defined in the language of category theory. So for example we see that every covering space is a unique disjoint union of connected coverings, that normal subgroups of $\pi_1(X)$ correspond to Galois coverings of $X$, etc. We don't need extra arguments for this, the equivalence of categories is everything we need!

Answer 2

If $G$ acts transitively on $S$ and $s\in S$, we get a bijection $G/G_s\to S$, $gG_s\mapsto gs$, so the examples from your first bullit point are precisely the cases with transitive action.