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Added to commutativity of limits and colimits the case of coproducts commuting with connected limits in a topos, and the generalization to higher topoi. This particular instance of commutativity is not mentioned very often, probably because it’s not very impressive in Set, but its generalization to higher topoi (for which I couldn’t find a reference) is more interesting. For instance, cofiltered limits commute with taking quotients by an ∞-group in an ∞-topos.
I can believe the claim for finite connected limits – that property is really just inherited from $Set$. But why is true in the general case?
I meant Grothendieck topos, was that the issue?
Here’s a proof of the general case. If $A$ is an ∞-groupoid, the colimit functor $\mathbf{H}^A\to\mathbf{H}$ can be identified with the forgetful functor $\mathbf{H}_{/Disc(A)}\to\mathbf{H}$, by the ∞-Giraud axioms. If $C$ is an ∞-category such that $\lim_C Disc(A)\simeq Disc(A)$ (e.g. if A is n-truncated and C is n-connected), then that forgetful functor creates $C$-limits.
Not being expert in $\infty$-categories, I will take your word for it. In the ordinary case, you are saying that the forgetful functor $\mathcal{E}_{/ \Delta X} \to \mathcal{E}$ creates connected limits (which is well-known) and identifying $\mathcal{E}_{/ \Delta X}$ with $\mathcal{E}^X$. This is true so long as $\mathcal{E}$ is an infinitary extensive category.
At https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits#ColimitsStableByBaseChange surely one could also list regular extensive categories? Just checking I’m not being stupid (could be the case…)
Yes… but why is that section there at all? It’s not an example of commutativity of limits and colimits. And we already have universal colimit.
It’s not an example of commutativity of limits and colimits.
Are we talking about
$\underset{d \in D}{colim} (F(d) \times_Z Y) \stackrel{\simeq}{\longrightarrow} (\underset{d \in D}{colim} F(d)) \times_Z Y$?
I thought so.
Yes, that’s what we’re talking about.
So how is it not an example?
How is it an example? Commutativity of limits and colimits means $colim^D \circ lim^E = lim^E \circ colim^D$ as functors on a diagram category $C^{D\times E}$.
Ah, I see your point. Not immediately obvious, if at all true.
Surely, if $E$ is $\bullet \rightarrow \bullet \leftarrow \bullet$, and $J : D \times E \rightarrow C$ sends $(d, \bullet \rightarrow \bullet \leftarrow \bullet)$ to $F(d) \rightarrow Z \leftarrow Y$, and does the obvious levelwise thing on arrows in $D$ (i.e. constant on $Z$ and $Y$), then the equation $colim^{D} \circ lim^{E}(J) = lim^{E} \circ \colim^{D}(J)$ expresses exactly that $\underset{d \in D}{colim} (F(d) \times_Z Y) \stackrel{\simeq}{\longrightarrow} (\underset{d \in D}{colim} F(d)) \times_Z Y$?
@Richard: only if the colimit of a constant $D$-diagram is equal to the object that it’s constant at. For 1-colimits this is true if $D$ is connected; for $\infty$-colimits it’s only true if $D$ has a contractible nerve.
I’m only interested in the case of 1-categories, which is not treated at universal colimit.
Well, it should be. Just copy and paste the definition given and remove all the $\infty$s.
While it’s not generally an example of the functorial commutaticity
$\underset{\underset{D}{\longrightarrow}}{\lim} \circ \underset{\underset{C}{\longleftarrow}}{\lim} \simeq \underset{\underset{C}{\longleftarrow}}{\lim} \circ \underset{\underset{D}{\longrightarrow}}{\lim} \;\colon\; \mathcal{C}^{C \times D} \to \mathcal{C}$the formula
$\underset{d \in D}{colim} (F(d) \times_Z Y) \stackrel{\simeq}{\longrightarrow} (\underset{d \in D}{colim} F(d)) \times_Z Y$is still syntactically an exchange of the order of taking limits and colimits and as such is something that a reader may appreciate finding discussed on this page.
So I vote for leaving the material there. But I added the warning: here
I agree that a reader may appreciate finding something about it, but I think what they should find is a link to the place where it is properly discussed.
Sounds hardly objectionable. Maybe I am missing what is under debate.
I just wanted to expand the list of examples, wherever the list is given. I was trying to find alternative hypotheses for a result of Todd’s (at this point, Lemma 1.5 here) and double checking before adding to the page on colimits/limits.
Re #14: Thanks, Mike!
To me, though, it seems more like the definition at commutativity of limits and colimits should not necessarily be taken as the default one in the non-connected case. Take a coproduct, for instance. Using the fold arrows out of the coproduct, one recovers the canonical arrow we are discussing, and it seems to me that the latter one is actually the one that more ’correctly’ expresses a useful notion of commutativity of limits and colimits. As Urs put it, syntactically it seems equally justifiable as well.
Great that the warning has been added to the page, that is the most important thing I think.
If anyone can formulate a general notion of “commutativity of limits and colimits” that includes pullback-stability of colimits and also all the other examples on the page, I’d love to hear it. Otherwise, I’d like to move the former to a different page. Having precise definitions and categorizing things appropriately helps to alleviate just this sort of confusion.
As long as $C$ has an initial object, maybe we can just formally add an initial object $\emptyset$ to $D$ and send $(\emptyset, \bullet \rightarrow \bullet \leftarrow \bullet)$ to $\emptyset \rightarrow Z \leftarrow Y$? Haven’t checked carefully…
I think universal colimits is an example of distributivity of limits over colimits, for the functor $(a\to b\leftarrow c) \to Cat$ sending $a$ to the given indexing category and $b,c$ to the point.
FWIW I added the example I originally asked about.
Re #24, I think is indeed the construction I was getting at in #21. Seems more relevant than plain commutativity in the non-connected case. But maybe #23 provides an alternative way to see it as an actual instance of plain commutativity if one insists on that (one can always add an initial object to $C$ as well, of course, and then just use the forgetful functor back to $C$ at the end). If I have not made a mistake, then #23 should also work in the $\infty$-case.
Yes, I think #24 works. Good job Marc!
I think #23 works as long as the pullback of $\emptyset \to Z\leftarrow Y$ in $C$ is $\emptyset$, i.e. as long as we already know that initial objects are stable under pullback (which is equivalent to the initial object being strict). But if we don’t yet know whether the initial object of $C$ is strict, and we want to formulate the condition for its being pullback-stable, then #23 doesn’t work to write that as a commutativity condition. I don’t think it helps to add a new initial object to $C$ either, since then the colimit of the modified diagram would be the new initial object rather than the “real” one in $C$ that we want to be talking about.
Given that pullback-stability of colimits is an instance of distributivity, but is also sometimes an instance of commutativity, and the technicalities of regarding it as the latter are a little subtle, I still think we should move all substantial discussion and examples of pullback-stability to universal colimit. We can include explanations at commutativity of limits and colimits and distributivity of limits over colimits of how (and when) they specialize to pullback-stability, and links to these discussions from universal colimit. Objections?
I went ahead and added #24 as an example to distributivity of limits over colimits.
Re #27 good idea, thanks.
I implemented my suggestion at #27, and also moved the definition into a “Definition” section and added a section on preservation by functor categories and localizations (explaining in particular why filtered colimits commute with finite limits in all toposes).
tocuched the section Sifted colimits commute with finite products. Added hyperlinks and pointer to the reference given.
One thing I’m not sure about is that Eckmann and Hilton state:
The transformation w, together with the question when w is an equivalence, have already been considered by Roos [8], under the heading of the distributivity of lim with respect to lim
but on distributivity of limits over colimits, Roos is cited as defining distributivity rather than commutativity. So either Eckmann and Hilton are misinformed, or the nLab page is incorrect.
It strikes me that commutation of limits and colimits can be seen as a kind of limit–colimit coincidence, which is essentially an absolute colimit phenomenon (see Mike’s answer to this MO question, for instance). Does this suggest that the enriched perspective of absolute weighted (co)limits could shed some light on commutation of limits and colimits?
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