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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 GG and HH be finite groups whose orders are coprime. View them as one-object categories. Then GG-colimits commute with HH-limits in the category of sets.

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

Let II and JJ be small categories, and let

D:I×JSet D: I \times J \to \mathbf{Set}

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

λ:colim jJlim iID(i,j)lim iIcolim jJD(i,j), \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 λ\lambda is a bijection. If the answer is yes for all DD, we say that limits over II commute with colimits over JJ in the category of sets. The statement is that when II and JJ 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 Set\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 HH and GG be groups, and let XX be a set equipped with both a left HH-action and a left GG-action in such a way that the actions commute: ghx=hgxg h x = h g x for all hh, gg and xx. Equivalently, XX has a left action by G×HG \times H.

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

λ:Fix H(X)/GFix H(X/G). \lambda: Fix_H(X)/G \to Fix_H(X/G).

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

Fix H(X)/GFix H(X/G). 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 λ\lambda is surjective. Let ξFix H(X/G)\xi \in Fix_H(X/G). Then ξ=Gx\xi = G x for some xXx \in X, and we know that GxG x is a fixed point of HH. It’s enough to show that xx itself is a fixed point of HH: for then the element of Fix H(X)/GFix_H(X)/G represented by xx is mapped by λ\lambda to the element of Fix H(X/G)Fix_H(X/G) represented by xx, which is ξ\xi.

So, let hHh \in H. We must show that hx=xh x = x. Since GxG x is a fixed point of HH, we know that hx=gxh x = g x for some gGg \in G. Since the GG- and HH-actions on XX commute, h nx=g nxh^n x = g^n x for all integers nn. But |G|\left|G\right| and |H|\left|H\right| are coprime, so we can choose an nn such that

n1 (mod|H|),n0 (mod|G|). n \equiv 1  (mod \left|H\right|), \qquad n \equiv 0  (mod \left|G\right|).

Then h n=hh^n = h and g n=1g^n = 1, so hx=xh 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 Set\mathbf{Set}? Although I didn’t ask for it, Will Sawin provided a complete answer — a necessary and sufficient condition. Here it is:

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

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 HH is simple and GG has smaller order), as well as to some pairs where one or both groups are infinite.

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×HG\times H acts on XX, then there is a general formula (X hH) hG [ϕ](X hΓ(ϕ)) hC(ϕ). (X_{hH})^{hG} \approx \coprod_{[\phi]} (X^{h\Gamma(\phi)})_{hC(\phi)}. The coproduct on the right is over conjugacy classes of homomorphisms ϕ:GH\phi\colon G\to H. The group GΓ(ϕ)G×HG\approx \Gamma(\phi)\subseteq G\times H is the “graph” of ϕ\phi, and the group C(ϕ)HC(\phi)\subseteq H is the centralizer of ϕ\phi. Note that Γ(ϕ)×C(ϕ)\Gamma(\phi)\times C(\phi) acts on XX in an evident way.

The left hand side is the space of sections of the composite of the maps (EG×EH)× G×HXBG×BHBG. (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 GG and HH) the only homomorphism GHG\to H is the trivial one, then the right-hand side reduces to (X hG) hH(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 XX be a space with a G×HG\times H action, and suppose BHMap(BG,BH)BH\to \mathrm{Map}(BG,BH) (inclusion of constant maps) is a weak equivalence. There is a homtopy fiber sequence of the form XYBH, X \to Y \to BH, where Y=X× HEHX hHY=X\times_H EH \approx X_{hH}. The group GG acts compatibly on this sequence, by the given action on XX and the trivial action on EHEH and BHBH. Taking homotopy fixed points gives a fiber sequence X hGY hG(BH) hGMap(BG,BH)BH. 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 HH on X hGX^{hG}, with the total space identified with the homotopy quotient of this action. Thus (X hH) hGY hG(X hG) hH, (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 GG is finite and BHBH 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 GG and HH relative to the original post.

More generally, for groups GG and HH (with corresponding one-object groupoids BG\mathbf{B}G and BH\mathbf{B}H) we have Map(BH,BG)=BHMap(\mathbf{B}H,\mathbf{B}G) = \mathbf{B}H just when every homomorphism from HH to GG is trivial, which is to say that no nontrivial quotient of HH is isomorphic to a subgroup of GG. 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 XX is a set, then π 0((X hH) hG)(X H) G\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 π 0[(X hΓ) hC](X Γ) C\pi_0[(X^{h\Gamma})_{hC}] \approx (X^\Gamma)_C when XX is a set always.)

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

The orbit OO therefore contributes (O hH) hGMap(BG,BH)BH(O_{hH})^{hG}\approx \mathrm{Map}(BG,BH')\approx BH' to (X hH) hG(X_{hH})^{hG}, and we have π 0[(O hH) hG]*(O H) G\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

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