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Given two categories $I$ and $J$ we say that colimits of shape $I$ commute with limits of shape $J$ in the category of sets, if for any functor $F : I \times J \to \text{Set}$ the canonical map $$\textrm{colim}_{i\in I} \text{lim}_{j\in J} F(i,j) \to \textrm{lim}_{j\in J} \text{colim}_{i\in I} F(i,j)$$ is an isomorphism.

The standard examples are a) filtered colimits commute with finite limits and b) sifted colimits commute with finite products. (Those statements can be regarded as definitions of which categories $I$ are filtered or sifted respectively, but both terms have independent definitions for which these commutation results are propositions.) A third, less known example is to take $I$ a finite group and $J$ a cofiltered category, in other words, if $G$ is a finite group and $X_j$ is an inverse system of $G$-sets, then the canonical map $$(\varprojlim_{j\in J} X_j)/G \to \varprojlim_{j \in J}(X_j/G)$$ is an isomorphism.

Now, all of these examples are easy to prove separately (here's a proof of the $G$-set result, for example) but I see no unifying pattern. Is there a simple criterion for when $I$-colimits and $J$-colimits commute in the category of sets?

[Note: It's true that $I$ is filtered (resp. sifted) if and only if for all finite (resp. finite discrete) $J$ the diagonal functor $I \to I^J$ is final; but I don't think that for arbitrary $I$ and $J$, if the diagonal $I \to I^J$ is final then $I$-colimits commute with $J$-limits. If I'm wrong and that condition on the diagonal actually is sufficient for commutation: why? and is it also necessary?]

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Also, if anyone sees this question and hasn't seen Steve Lack's request for a soft proof that filtered colimits commute with finite limits in sets, and knows of such a proof, please answer his question: mathoverflow.net/questions/57099/… –  Omar Antolín-Camarena Apr 5 '12 at 23:08
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@Martin: Joyal claims it's true in the appendix to his paper Foncteurs analytiques et especes de structures, (at least I think he does: does "limites projectives filtrantes" mean "filtered limits" or something else, like, say "directed limits"?) I wrote a proof for the case when J is a sequence, and of injectivity for arbitrary cofiltered J here: math.harvard.edu/~oantolin/notes/fingrpcomm.html I'll add the surjectivity as soon as I finish debugging my proof. –  Omar Antolín-Camarena Apr 6 '12 at 16:03
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This is not really an answer to your question, but have you read the paper "A classification of accessible categories" by Adamek, Borceaux, Lack, and Rosicky? They study general notions of accessibility relative to such notions of commutation. –  Mike Shulman Apr 12 '12 at 23:56
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Here are some further examples, all easy: (d) coproducts commute with connected limits; (e) initial objects commute with nonempty limits; (f) connected colimits commute with terminal objects. –  Tom Leinster Aug 10 '12 at 14:03
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Here are more examples still, in the setting where both $I$ and $J$ are one-object categories corresponding to groups: mathoverflow.net/questions/152193/… –  Tom Leinster Dec 21 '13 at 2:10

2 Answers 2

up vote 13 down vote accepted

Some very involved necessary and sufficient conditions are found in an obscure paper of Foltz (in French). I can't vouch for the accuracy of all his results, although I have looked at bits of the paper which seem to work out. Some observations on his paper:

  1. An elementary observation (Proposition 3, section 1, p. F 12): $I$-colimits commute in $\mathrm{Set}$ with $P$-limits iff $I$-limits commute with discrete $\pi_0(P)$-colimits and also with $P'$-colimits for each connected component $P'$ of $P$. Foltz then analyzes separately the cases of $P$ discrete and of $P$ connected.

  2. He separately analyzes the conditions that the canonical comparison map be always injective and that it be always surjective.

  3. He treats some examples of interest at the end, including the colimits that commute in $\mathrm{Set}$ with pullbacks and those that commute in $\mathrm{Set}$ with equalizers. But it doesn't appear that he discusses how to recover characterizations of filtered or sifted limits.

  4. Foltz's criteria are expressed in terms of certain certain subdivision categories, and a lot of zig-zags. Unfortunately, he doesn't discuss how to relate his criteria to other more familiar ones, such as the finality of certain diagonal functors. But it might be possible to convert his criteria into such forms.

Some things are known about the general phenomenon of limits commuting with colimits:

  • Albert and Kelly's "The Closure of a Class of Colimits" discusses which limit-weights commute in $\mathrm{Set}$ with all the colimit-weights that a given class commutes with -- which is sort of the "square" of the commutation relation you're interested in. This is what Albert and Kelly call the "closure" of a class of colimits, and nowadays is typically referred to as the saturation.
  • There are also some good notes by Kelly and Schmitt which discuss the formal aspects of the situation, which is enough to gain some meaningful insight into the important case of absolute colimits -- those which commute with every limit.

Both of these papers are written in the context of enriched categories, which means they don't provide terribly specific information about the case of $\mathrm{Set}$-enrichment, but at least clarify the formal situation.

More specifically, as Mike Shulman notes, you might want to take a look at the

  • ABLR paper, available from Steve Lack's website. They use a condition on a class of limit weights $\mathbb{D}$ that they call "soundness." In fact, soundness is explicitly a simplifying assumption about which colimits commute with $\mathbb{D}$-limits in $\mathrm{Set}$. All the examples which are well-known (like finite/filtered and finite-discrete/sifted) satisfy soundness; it seems to account for why they're so nice to work with.

  • Some further work has been done on developing the theory of these "sound doctrines", especially by Claudia Centazzo; Lack and Rosicky's "On the notion of Lawvere Theory" also starts to consider what the enriched case might look like.

But very little seems to be known about which "doctrines" (classes of limit-weights) are sound in general. In fact, the only examples given by ABLR of non-sound doctrines are the doctrine of pullbacks, and the doctrine of pullbacks + terminal objects -- neither of which is saturated! The saturation of the latter is, of course, all finite limits, which is sound. The conical saturation of pullbacks is the class of simply-connected and finitely-presentable categories, as discovered by Paré, which is not sound -- this can be seen by adapting ABLR's argument concerning pullbacks (Example 2.3.vii).

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Summary (in case your original question got lost in my long-winded response): -Foltz provides necessary and sufficient criteria in full generality for $I$-colimits to commute in $\mathrm{Set}$ with $J$-colimits. But he doesn't connect these general criteria to the more conceptual criteria familiar from special cases (e.g. finality of diagonal functors). -With the simplifying assumption of a sound doctrine, one can unify many interesting and useful conceptual criteria, but the nature of soundness doesn't seem to be well-understood--judging from the published literature, at any rate. –  Tim Campion Aug 24 '12 at 18:46
    
I said something false in the last paragraph: finite limits are not saturated as a I claimed. Their saturation is the class of L-finite categories -- those which have an initial finitely-generated subcategory. Similarly, the saturation of pullbacks is the class of categories which are simply connected and L-finite. –  Tim Campion Apr 25 at 14:10
    
Also, the Albert-Kelly paper doesn't actually talk about this. It talks about the closure condition $\Phi \mapsto$ {the set of $\psi$ such that $\Phi$-cocomplete categories have $\psi$-colimits and $\Phi$-cocontinuous functors preserve $\psi$-colimits}. As observed in the Kelly-Schmitt notes, this is a finer closure operation than the one arising from your Galois connection: your closed classes of colimits are closed in the Albert-Kelly sense, but not conversely. –  Tim Campion May 5 at 21:11

Hello, I'm sorry for having been so slow to react, I was only made aware of this discussion by a colleage of mine recently.. I have been working for several years now on almost exactly all of the above questions in the context of my thesis to be submitted in very near future. Concerning first question: That I-limits commute with J-colimits in set implies that the diagonal functor J-->J^I is final, is a necessary fact wich is only/exactly sufficient when I is a so-called sound class of "limit index", but not in general. I suggest in my thesis that we define J to be I-filtered when the diagonal functor J-->J^I is final in stead of when I-limits commute with J-colimits in set (to keep diagrammatic interpretations). In fact though sound doctrines are easy to work with they are included in a (nearly just as nice) slightly larger "type of doctrines" (including "pullbacks+terminal objects") namely those classes of index "essentially closed" in the galois correspondence between the classes of "limit index" and classess of "colimit index" that commute in set. I give in my thesis a complete picture of these classes and by this we "decrypt" the abstract conditions of F. Foltz. I'm sorry that I don't yet have nice and short versions of my work to put down here, nor an article ready to refere to, but I'm happy to discuss or answer any related questions in detail either off-line or by email with anyone interested! While finishing written versions.. Cheers, Marie Bjerrum.

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Thanks for mentioning this! I hope you plan to make your thesis public and available online; if and when you do, please add a link to your answer. I would like to make a link to it from the nLab as well, as this sort of technical material would be very useful for researchers. –  Todd Trimble Sep 10 '12 at 14:33
    
Fantastic - it's high time for a real, systematic study. I second Todd's call for online public availability! –  Tim Campion Sep 10 '12 at 15:38
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This is great news, Marie. I'm looking forward to reading your treatment when it's ready. –  Omar Antolín-Camarena Sep 10 '12 at 16:05

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