This is a crosspost of this MSE question.

In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is mentioned:

Theorem 1. $H$-limits commute with $G$-colimits in $\mathsf{Set}$ iff no nontrivial quotient of $H$ is isomorphic to a subquotient of $G$.

On the nlab page about commutativity of limits and colimits, it's also mentioned that taking orbits under the action of a finite group commutes with cofiltered limits, i.e:

Theorem 2. Let $G$ be a finite group and $\mathsf C$ be a small cofiltered category. Let $F:\mathsf C\longrightarrow G\mathsf{Set}$ be a functor. Then $(\varprojlim F)/G\cong \varprojlim _{j\in F}(F(f)/G)$.

I'm reasonably familiar with category theory, but I've only ever taken one course on groups - a first semester course about finite groups. I'm curious whether one could derive from these commutation results some basic facts in the theory of (finite) groups.

What kind of familiar elementary facts about (finite) groups can be derived from these results?

  • $\begingroup$ Although they do not seem to answer your question, the discussions at mathoverflow.net/questions/152193/… may be relevant $\endgroup$ – Yemon Choi Oct 4 '15 at 21:58
  • $\begingroup$ @YemonChoi they are certainly relevant - that question is what prompted this one. I just want to see some applications of these commutation results since I have no feel for them at all. $\endgroup$ – Arrow Oct 5 '15 at 17:42

Perhaps this is something you seek.

There is one type of groups that is closely related to limits. These groups are called profinite groups, i.e groups $G$ that can be expressed as $\varprojlim_{i\in I}G_i$, where $I$ is a directed set and each $G_i$ is a finite group endowed with the discrete topology. In the same manner, a topological space $X$ is called profinite, if $X=\varprojlim_{i\in I}X_i$, where this time $X_i$ is a finite set endowed with the discrete topology. There is the following statement:

Let $G$ be a profinite group and let $X$ be a profinite $G$-space. Then $X$ can be written as $X=\varprojlim_iX_i$, where each $X_i$ is a $G$-space such that $X_i$ is a finite $G$-space. Furthermore, $X/G$ is profinite.(See Lemma 5.6.3 and Lemma 5.6.4 in 'Ribes, Zalesskii, Profinite groups').

Suppose that $G$ is just a finite group endowed with the discrete topology and $X=\varprojlim_iX_i$ a $G$-space. Then the second assertion in the statement above follows directly from the first: Assume that each $X_i$ is a $G$-space. Now Theorem 2 $X/G\cong \varprojlim_iX_i/G$ implies that $X/G$ is profinite. Note, that the second assertion follows just from the fact that $X$ is a profinite $G$-space (no matter if $G$ is finite or not). The proof, however, is not that abvious as the one above.
There is a subtle application concerning $X/G\cong \varprojlim_iX_i/G$. As mentioned before, the identification $X/G\cong \varprojlim_iX_i/G$ shows that if $X$ is profinite so is $X/G$. This fact is used in several propositions in Chapter 5.6 of the book 'Ribes, Zalesskii, Profinite groups'. For instance, see Lemma 5.6.7, which basically tells us that under certain circumstances the quotient map $X\rightarrow X/G$ admits a continuous section $X/G\rightarrow X$.


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