## December 21, 2013

### Commuting Limits and Colimits over Groups

#### Posted by Tom Leinster

Limits commute with limits, and colimits commute with colimits, but limits and colimits don’t usually commute with each other — with some notable exceptions. The most famous of these is that in the category of sets, finite limits commute with filtered colimits.

Various other cases of limit-colimit commutation are known. There’s an nLab page listing some. But it seems that quite an easy case has been overlooked.

It came to light earlier this week, when I was visiting Cambridge. Peter Johnstone told me that he’d found a family of new limit-colimit commutations in the category of sets, I asked whether his result could be simplified in a certain way (to involve groups only), and we both realized that it could not only be simplified, but also generalized.

Here it is. Let $G$ and $H$ be finite groups whose orders are coprime. View them as one-object categories. Then $G$-colimits commute with $H$-limits in the category of sets.

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.

Posted at December 21, 2013 12:55 AM UTC

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### Re: Commuting Limits and Colimits over Groups

I wonder if reformulation in homotopy type theory might be revealing, e.g., as here, dependent sum is quotient and dependent product is the type of fixed points.

Posted by: David Corfield on December 21, 2013 3:03 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

Hi Tom (and David)

I may well have misunderstood the issue, but here is what I
think is a homotopy version of the result: it looks simpler
(indeed, just a question of dependent sums and products),
but I don't know how to derive the 0-type statement from it.
(Maybe it's the general phenomenon that homotopy colimits
are better behaved than 0-colimits.)

In the following, all slices are homotopy slices, and all
pullbacks and adjoints are homotopy.  Given groupoids G and H,
and the pullback diagram

q
GxH -> H
t |     |r
v     v
G --> 1
p

the question is whether (or when)

p\lowerstar t\lowershriek = r\lowershriek q\lowerstar.

(These are functors between homotopy slices of Grpd.)
Usually the left-hand side is computed by the distributive law:
put Q := p\lowerstar GxH, and P := p\upperstar Q.  Then there
is a counit e : P -> GxH, and a pullback diagram

a
P  -->  Q
e|       |
v       |
GxH      |b
t|       |
v       v
G --->  1
p

and the distributive law says

p\lowerstar  t\lowershriek = b\lowershriek  a\lowerstar  e\upperstar.

So the question is when we have Q = H, and hence e is an
equivalence, and hence the two right-hand sides agree.
In other words, the question is when

H = p\lowerstar p\upperstar H.

The right hand side is the mapping space Map(G,H).
If G and H are finite one-elements groupoids of coprime
order, then Map(G,H)=H, and that's it.

(Please correct me -- I often get even the most elementary
things wrong.)

Cheers,
Joachim.

Posted by: Joachim Kock on December 21, 2013 7:02 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

I think it works.

It is easier for me to understand it in old-fashioned terms. In spaces, I think you are saying this: if the product group $G\times H$ acts on $X$, then there is a general formula $(X_{hH})^{hG} \approx \coprod_{[\phi]} (X^{h\Gamma(\phi)})_{hC(\phi)}.$ The coproduct on the right is over conjugacy classes of homomorphisms $\phi\colon G\to H$. The group $G\approx \Gamma(\phi)\subseteq G\times H$ is the “graph” of $\phi$, and the group $C(\phi)\subseteq H$ is the centralizer of $\phi$. Note that $\Gamma(\phi)\times C(\phi)$ acts on $X$ in an evident way.

The left hand side is the space of sections of the composite of the maps $(EG\times EH) \times_{G\times H} X \to BG\times BH \to BG.$ The right-hand side is what you get if you compute sections in stages along the composite.

So I guess the claim is that if (for discrete groups $G$ and $H$) the only homomorphism $G\to H$ is the trivial one, then the right-hand side reduces to $(X^{hG})_{hH}$. Neat.

Posted by: Charles Rezk on December 22, 2013 3:55 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

Here’s a more consise proof:

Let $X$ be a space with a $G\times H$ action, and suppose $BH\to \mathrm{Map}(BG,BH)$ (inclusion of constant maps) is a weak equivalence. There is a homtopy fiber sequence of the form $X \to Y \to BH,$ where $Y=X\times_H EH \approx X_{hH}$. The group $G$ acts compatibly on this sequence, by the given action on $X$ and the trivial action on $EH$ and $BH$. Taking homotopy fixed points gives a fiber sequence $X^{hG} \to Y^{hG} \to (BH)^{hG}\approx \mathrm{Map}(BG,BH) \xleftarrow{\sim} BH.$ Such a fiber sequence exhibits the evident action of $H$ on $X^{hG}$, with the total space identified with the homotopy quotient of this action. Thus $(X_{hH})^{hG} \approx Y^{hG} \approx (X^{hG})_{hH},$ as desired.

The result thus applies to some pairs of non-discrete groups; for instance, it holds when $G$ is finite and $BH$ is homotopy equivalent to a finite dimensional CW-complex, by Miller’s theorem. I do not recall seeing statements like this anywhere, though I would suppose it is well known to the sorts of people who know such things; i.e., people who study classifying spaces of Lie groups and p-compact groups.

Posted by: Charles Rezk on December 23, 2013 6:29 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

That looks right to me! Although I think you switched the roles of $G$ and $H$ relative to the original post.

More generally, for groups $G$ and $H$ (with corresponding one-object groupoids $\mathbf{B}G$ and $\mathbf{B}H$) we have $Map(\mathbf{B}H,\mathbf{B}G) = \mathbf{B}H$ just when every homomorphism from $H$ to $G$ is trivial, which is to say that no nontrivial quotient of $H$ is isomorphic to a subgroup of $G$. Thus, this yields a stronger version of Will Sawin’s condition, in which “subquotient” is replaced by “subgroup”.

I also don’t see how to relate this directly to the statement for sets, however.

Posted by: Mike Shulman on December 22, 2013 4:32 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

To go from the homotopy theoretic statement to the set one, the obstruction seems to be to show that, if $X$ is a set, then $\pi_0((X_{hH})^{hG}) \approx (X_H)^G$. (I.e., the left-hand side of my equation; the right-hand side is ok, since $\pi_0[(X^{h\Gamma})_{hC}] \approx (X^\Gamma)_C$ when $X$ is a set always.)

If $X$ is a set, then under the $H$-action it is a disjoint union of $H$-orbits. If $G$ preserves an $H$-orbit $O\approx H/H'$, then it must act through the $H$-set automorphisms of $O$ (since $G$ and $H$ actions commute), which is the Weyl group $W=N_H(H')/H'$. Since $W$ is a subquotient of $H$, Sawin’s condition implies that $G\to W=\mathrm{Aut}(O)$ is trivial.

The orbit $O$ therefore contributes $(O_{hH})^{hG}\approx \mathrm{Map}(BG,BH')\approx BH'$ to $(X_{hH})^{hG}$, and we have $\pi_0[(O_{hH})^{hG}] \approx * \approx (O_H)^G$.

Posted by: Charles Rezk on December 23, 2013 6:00 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

Hi Joachim,

Where does this distributive law come from? I’ve never seen it before and got stuck when looking for a proof.

Posted by: Emily Riehl on December 27, 2013 11:01 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

The distributive law is a basic ingredient in the theory of polynomial functors; personally I recommend the reference [Gambino-Kock, “Polynomial functors and polynomial monads”, Math.Proc.Cambridge 2013], where a more detailed version of the following explanations can be found.

The category of finite polynomial functors (over finite sets) is the Lawvere theory for commutative semirings: lowershriek along 2 –> 1 gives addition, lowerstar gives multiplication, and similarly 0 –> 1 gives their neutral elements. The general distributive law stated then specialises to the usual distributive law of elementary arithmetic, a(x+y)=ax+ay. The role of pullback along e is to copy (or discard) variables as needed, as illustrated by the elementary formula where there are two occurrences of the variable a on the right-hand side. In type theory, the distributive law is also called ‘dependent choice’, for reasons I don’t quite understand.

As to the proof of the distributive law, the 2-cell appears as the mate of the Beck-Chevalley of the pullback square, under the lowershriek-upperstar adjunction; since this is a cartesian adjunction, to check that the whole 2-cell is invertible it is enough to check its component at the terminal object, which is easily seen to be invertible since lowerstar preserves the terminal object.

‘Distributivity pentagon diagrams’ like this were first studied by Tambara [On multiplicative transfer, 1993], who essentially proved the Lawvere-theory result mentioned. They have been analysed to depth by Mark Weber [Polynomials in categories with pullbacks].

All this has an infinity version which is essentially straightforward, once things have been set up properly. This is in a forthcoming(?) paper by Gepner and Kock, which unfortunately has been ‘nearly finished’ for the several years :-(

Posted by: Joachim Kock on December 28, 2013 9:55 PM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

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.

Posted by: Tom Leinster on September 30, 2014 1:57 AM | Permalink | Reply to this

### Re: Commuting Limits and Colimits over Groups

I wonder if any of the contributors to the thread above could tell us what the story would be in the homotopy case.

Is it just a matter of replacing ‘subquotient’ with ‘subgroup’, as Mike remarked? Could other higher groupoids provide commuting limits-colimits?

Posted by: David Corfield on September 30, 2014 9:41 AM | Permalink | Reply to this

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