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I created a new page distributivity of products and colimits, where I recorded what I learned after asking this question: http://nforum.mathforge.org/discussion/6255/commutativity-of-homotopy-sifted-colimits-and-products-in-categories-other-than-sets-or-spaces/
Thanks! I have cross-linked with sifted colimit and with commutativity of limits and colimits. Also, I have made your pointers to the references come out hyperlinked.
I added a new example: sifted colimits distribute over limits in an algebraically exact category, in particular in a variety.
I also renamed the page to “distributivity of colimits over limits” in order to described its content more accurately.
Shouldn’t it be “distributivity of limits over colimits”?
Also, note that \rm doesn’t work on the nLab.
note that \rm doesn’t work on the nLab.
Dmitri, regarding this: for better or worse every consecutive string of characters in math mode is automatically romanized by Instiki
@MikeShulman: I checked all 4 references, and 3 of them say “colimits distribute over limits”.
Only one reference (Continuous Categories Revisited) reverses the order.
What is funny is that exactly the same three authors wrote another paper (How algebraic is algebra) with the opposite conventions!
Also, I’m sorry about \rm. It is really annoying to try to keep in mind all these proprietary extensions to TeX made for no good reason…
Well, I still think it’s wrong and we should change it. It’s 100% standard that multiplication distributes over addition. Thus, products distribute over coproducts. Thus, limits distribute over colimits.
I don’t think I agree with the name change that replaces “commutativity” with “distributivity”. Ordinarily I would say “sifted colimits commute with products”, and I don’t think there is any real danger of misunderstanding what this means. If anyone feels that “commutative” is a slightly loose use of language, then I would recommend “interchange of sifted colimits and products” in its place, and I think the latter expression is pretty much unimpeachable.
In my opinion, “distributivity” in this context is no improvement over “commutativity”. It’s hard to parse what it even means to say that “filtered colimits distribute over equalizers” or “equalizers distribute over filtered colimits” by extrapolating from “products distribute over coproducts” (which we interpret as saying that a one-sided product , an unary operation, preserves binary coproducts). Interchange, the term used in Categories for the Working Mathematician, captures more accurately the phenomenon we are after (“distributivity” just seems wrong to me here).
I think the meaning of “distributes” here is actually different from “commutes with”. If I read it correctly, saying that “binary products distribute over binary coproducts” means that we have the expected
whereas “binary products commute with binary coproducts” would mean instead that
Dmitri, the lack of \rm is not an extension to TeX, and there’s certainly nothing proprietary about it (iTeX is open source). Rather, iTeX is (basically) a simplified subset of TeX. (I’m pretty sure this is also the case for other web math engines like MathJax; I doubt that any of them implements the full Turing-complete TeX language.)
Re #9: of course, but this only strengthens my feeling that “distributivity” really isn’t the correct term for the types of things that the nLab article is trying to describe.
A special case is where all the domain categories in the Idea section are the same category , and is a discrete category. Here we have a canonical 2-cell
and if is sifted then the 2-cell is an iso. This is what I would call commutativity or interchange of limits and colimits, and naturally so. I’d think it should still be referred to as interchange even if the aren’t all the same.
Distributivity refers to something else, as you said, and lacks the symmetry exhibited in the square above. While I’m at it, remark that for commutative rings, binary coproducts distribute over binary products and not the other way around. So I wouldn’t insist that it should be “distributivity of limits over colimits” or the other way around; rather I’d say (again) that I don’t think “distributive” is the right word in the first place for that article, unless I’m missing something.
Dmitri that’s true, but the question was about \rm, not about that. Note that iTeX does support the standard LaTeX version of the command, \mathrm. Also, it’s not as if you have to memorize all of this, since you can just see what it looks like, and when you notice that \rm isn’t working, get rid of it.
Todd, that 2-cell being an iso is indeed what should be called commutativity or interchange. But unless I’m very much mistaken, that is not the condition that the article distributivity of limits over colimits is considering; the latter does in fact specialize to the standard notion of distributivity of finite products over finite coproducts.
Mike, I’ll try to think about it. But isn’t it true that that condition also specializes to what I mentioned in #12? If so, why is distributivity then the preferred term?
No, I don’t think it does specialize to your #12; that’s commutativity, not distributivity.
Oh, I see now where I was going wrong. Sorry for the noise.
I agree that it would be better if iTeX supported \rm too, I was just saying I don’t think the current situation is as bad as all that. In the grand scheme of things, it would probably be better if the nLab used MathJax just because it’s more common now (even though I personally prefer iTeX to MathJax — I find it a real pain to have to type \mathrm or \rm every time I want a multi-character identifier in MathJax). We have the classic “early adopter” problem of having implemented something before a standard existed (iTeX and the nLab predated MathJax), and now people have chosen a different standard and it would be substantial effort to migrate ourselves to it. There’ve been discussions of moving the nLab to MediaWiki+MathJax+etc., but I don’t know the current status of them.
There is something wrong with the page distributivity of limits over colimits: the “idea” section defines only distributivity of products over colimits (with the terminology in reversed order), but then the examples include distributivity of more general limits over colimits. We need a good general definition; is there one in one of the references?
Definition 5.11 in Continuous Categories Revisited can be a candidate: D-limits distribute over E-colimits if the functor Ind_E(A)→A that replaces a formal E-colimit with its actual colimit in A preserves D-limits.
Is that really a definition of distributivity rather than commutativity? Their Theorem 5.13 suggests that it incorporates both some commutativity and some distributivity.
I feel like there should be some general equipment-theoretic definition that would include all the examples on the page and also others such as that encoded by a distributivity pullback.
Re #22: It seems like there is no “standard” treatment of distributivity of limits over colimits, and the existing expositions are a bit sloppy on this matter.
However, the authors of that paper explicitly refer to this particular property as “distributivity” in Definition 5.11. I don’t see why it should be problematic that the distributivity of small limits over filtered colimits implies the commutativity of finite limits and filtered colimits (why not?).
Also, take a look at their other paper (How algebraic is algebra?). Definition 4.5 there contains a similar statement for sifted colimits and small limits. They also explicitly refer to this property as distributivity and prove in Theorem 5.1 that the distributivity of small limits over sifted colimits implies the commutativity of finite product and sifted colimits.
Also, in both papers they prove that the free completion monad Lim and the free filtered (respectively sifted) cocompletion monad Ind (respectively Sind) admit a unique distributive law in the sense of Beck. This is another argument in favor of the above definition.
I agree that the current article on distributivity is quite sloppy and should be improved. Originally, I wrote it out of frustration with the lack of sources on distributivity, so I simply recorded the facts that I learned from these papers.
I have slightly edited the Idea-section for readability, here (for instance I added the clause “assuming that these colimits exist” )
Let me rephrase that: their Theorem 5.13 seems to be saying that there is no real notion of “limits distributing over colimits” in general: all we can talk about is limits commuting with colimits, and products distributing over colimits. Saying “they call it distributivity” doesn’t convince me; I want an explicit formulation that looks like distributivity.
Re #25:
I think one can make the following observation.
Given small categories I and K, for any diagram D: I×K→C we have a morphism f: colim_K lim_I D → lim_I colim_K D that factors as a composition hg: colim_K lim_I D → colim_{K^I} lim_I D’ → lim_I colim_K D.
Here D’: I×K^I→I×K→C is obtained from D by precomposition.
I-limits commute with K-colimits if and only if f is an isomorphism. I-limits distribute over K-colimits if and only if h is an isomorphism.
This works for arbitrary categories K and I, in particular I does not have to be discrete.
Thus distributivity coincides with commutativity if and only if g is an isomorphism, i.e., the diagonal map K→K^I is cofinal.
Thus finite limits distribute over filtered colimits if and only if finite limits commute with filtered colimits. The same is true for finite products and sifted colimits. So in these two cases there is no difference.
However, I see no reason why this would hold for other cases. For instance, I see no reason why distributivity of finite limits over sifted colimits should be the same as commutativity.
As a justification for the new definition I cite Theorem 2.1 and its proof in the paper “On Algebraically Exact Categories and Essential Localizations of Varieties.”, which shows that products distribute over filtered colimits (respectively sifted colimits) if and only if the evaluation functor Ind(A)→A (respectively Sind(A)→A) preserves products.
Ah-hah! Now there’s a definition I can get behind. The connection to commutativity makes much more sense now too.
Here’s a generalization to Kan extensions. Let be a Grothendieck fibration, hence an exponentiable functor, let be any functor (or perhaps a Grothendieck opfibration), and form the distributivity pullback
So that and . Then in any sufficiently complete category there is a Beck-Chevalley isomorphism
(using the fact that is a fibration, so that any pullback of it is an exact square). The mate of this isomorphism is a transformation
and we say that distributes over (in some category) if this map is an isomorphism.
When generalized to internal categories and fibrations, this includes the distributivity encoded by a distributivity pullback: the latter says exactly that indexed products distribute over indexed coproducts in the self-indexing of a locally cartesian closed category. It also generalizes to derivators and other contexts for formal category theory.
It becomes your definition when specialized to and a product projection. Conversely, if is an opfibration, then the general statement follows from its pullback along all points , since such pullbacks preserve dependent products, left Kan extensions along opfibrations, and right Kan extensions along fibrations, and jointly reflect isomorphisms (one of the axioms of a derivator). But even in the case it’s not obvious to me that we can restrict further to the case when is a product projection without loss of generality.
Re #27: I completely rewrote the article and incorporated my comments above.
Perhaps you can add your observations about Kan extensions to it? Can we give an example of distributing Kan extensions apart from distributivity pullbacks?
Thanks! I’ll see what I can do. Can you add a proof or a citation that the “abstract formulation” agrees with the explicit one?
Okay, I added the Kan extensions definition, and reorganized the page a bit to put all the definitions together and all the examples together. Still missing a comparison of the definitions.
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