The n-Category Café

Now here’s the result stated more precisely: first in category-theoretic terms, then purely group-theoretically.

Let $I$ and $J$ be small categories, and let

$$D: I \times J \to \mathbf{Set}$$

be a functor. There’s a canonical map of sets

$$\lambda: colim_{j \in J} lim_{i \in I} D(i, j) \to lim_{i \in I} colim_{j \in J} D(i, j),$$

and the question is whether $\lambda$ is a bijection. If the answer is yes for all $D$, we say that limits over $I$ commute with colimits over $J$ in the category of sets. The statement is that when $I$ and $J$ are the one-object categories corresponding to finite groups with coprime orders, they do commute.

Here it is again, purely group-theoretically. To translate, we’re going to need the facts that when a group is viewed as a one-object category, a functor from that category into $\mathbf{Set}$ is the same thing as a left action of the group, the limit of such a functor is the set of fixed points of the action, and the colimit is the set of orbits.

Let $H$ and $G$ be groups, and let $X$ be a set equipped with both a left $H$-action and a left $G$-action in such a way that the actions commute: $g h x = h g x$ for all $h$, $g$ and $x$. Equivalently, $X$ has a left action by $G \times H$.

The set $Fix_H(X)$ of $H$-fixed points has a $G$-action, and we can take the set $Fix_H(X)/G$ of orbits. On the other hand, the set $X/G$ of $G$-orbits on $X$ has an $H$-action, and we can take the set $Fix_H(X/G)$ of fixed points. There’s a canonical map of sets

$$\lambda: Fix_H(X)/G \to Fix_H(X/G).$$

It’s straightforward to show that $\lambda$ is always injective. It’s not always surjective. But the fact is that it’s surjective (and therefore bijective) if $G$ and $H$ are finite with coprime orders. So then,

$$Fix_H(X)/G \cong Fix_H(X/G).$$

The proof is so short that I might as well include it. We have to show that $\lambda$ is surjective. Let $\xi \in Fix_H(X/G)$. Then $\xi = G x$ for some $x \in X$, and we know that $G x$ is a fixed point of $H$. It’s enough to show that $x$ itself is a fixed point of $H$: for then the element of $Fix_H(X)/G$ represented by $x$ is mapped by $\lambda$ to the element of $Fix_H(X/G)$ represented by $x$, which is $\xi$.

So, let $h \in H$. We must show that $h x = x$. Since $G x$ is a fixed point of $H$, we know that $h x = g x$ for some $g \in G$. Since the $G$- and $H$-actions on $X$ commute, $h^n x = g^n x$ for all integers $n$. But $\left|G\right|$ and $\left|H\right|$ are coprime, so we can choose an $n$ such that

$$n \equiv 1  (mod \left|H\right|), \qquad n \equiv 0  (mod \left|G\right|).$$

Then $h^n = h$ and $g^n = 1$, so $h x = x$, as required.

Although category theorists seem to have overlooked this result, I thought it might be well-known in group theory, so I asked on MathOverflow. No one has yet provided a reference, and the view of the expert group theorist Derek Holt was that “the proof is sufficiently straightforward as not to require a reference”. But what may not have been apparent to group theorists is its categorical significance.

You might ask whether this result can be generalized. In other words, for which pairs of groups do limits over one commute with colimits over the other in $\mathbf{Set}$? Although I didn’t ask for it, Will Sawin provided a complete answer — a necessary and sufficient condition. Here it is:

$H$-limits commute with $G$-colimits in $\mathbf{Set}$ if and only if no nontrivial quotient of $H$ is isomorphic to a subquotient of $G$.

Of course, this immediately implies the coprime-order case for finite groups. But it applies to other pairs of finite groups too (e.g. when $H$ is simple and $G$ has smaller order), as well as to some pairs where one or both groups are infinite.

Update (30 September 2014):

We’ve just arXived a short paper on this:

Marie Bjerrum, Peter Johnstone, Tom Leinster and Will Sawin, Notes on commutation of limits and colimits. arXiv:1409.7860, 5 pages, 2014.