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created diagram chasing lemmas - contents (with some information on what implies what) and included it as a floating TOC into the relevant entries.
If lemma A implies lemma B, is it because B is very easy to prove using A, or because every diagram to which B applies is a diagram to which A applies (and then applying A gives B’s result), or because these lemmas are true only in some categories and lemma B holds in every category in which lemma A holds, or what?
The first.
The salamander lemma makes the mechanism behind all these lemmas clear, they are simple corollaries once the salamander lemma is there.
And that the five lemma is the four lemma applied twice is standard and almost by definition. In fact the nLab entry on five lemma originally just listed three cases, the first two are what is called the four lemma, the third is “1 and 2 combined”. I made four lemma a separate entry, four I think it deserves one.
four I think it deserves one
(^_^)
Could somebody comment on terminology “4-lemma” ? I learned the five lemma from Cartan-Eilenberg when I was 20 year old, so this is one of my romantic early memories on advanced mathematics. By five lemma there in my memory they meant the statement listing the two dual cases which are here quoted as 4-lemma. Bourbaki also (Homological algebra, 1980) says five lemma for precisely what is here 4-lemma. The iso case is just a trivial special case of the combination of the two, and Bourbaki singles out a remark of even more special case that in particular if f1,f2,f4,f5 are all isos then f3 is so. Of course CE and here have the intermediate case of generality. Newer books often just do that iso special case and say 5-lemma what always made me uneasy as it is weaker.
Is 4-lemma terminology also old ? I admit that the terminology 4-lemma is completely new to me. Anybody ?
Lemmas ch I 3.2, 3.3 and ch XII 3.1 in Mac Lane’s Homology.
Thanks Urs for the reference, though it does not resolve the terminology issue yet. I have a Russian edition of MacLane’s Homology, and he calls it Lemma on five homomorphisms there 3.3, for precisely in (i),(ii) what is here called 4-lemma. He indeed has also something what he calls “strong lemma on 4 homomorphisms”, 3.2 but which is different from the statement called 4-lemma in $n$Lab in five lemma entry. He has a diagram with 4 homomorphisms, and makes an assertition about the comparison of sizes of kernel and image of the central two vertical arrows and assigns that one to J. Leight (Leicht ?) or so (I do not know the spelling as it is russified).
For information on the terminology: page 364 of Mac Lane’s Homology, lemma 3.1 is called the weak Four Lemma, 3.2. is the 3x3 lemma and 3.3 is the sharp 3x3 lemma. Lemma I.3.2 (page 14) is called the Strong Four Lemma.
I found it interesting that Mac Lane refers to Lubkin’s representation theorem from 1960. Can anyone shed light on the history of the Lubkin-Freyd-Michell theorem, as I should know that history but seem not to! I had not looked at Mac Lane recently.
Yes, the strong four lemma compares the sizes of the kernels and cokernels. This implies the weak four lemma.
I still want to put this weak/strong distinction into the entry, but I am too busy with other things. Maybe somebody feels like helping.
You can just copy it from MacLane. Or you copy it from Salamander lemma – Implications – Four lemma where in fact the strong four lemma is derived in the proof.
I still want to put this weak/strong distinction into the entry,
Now I did.
This is helpful. Now we have at four lemma both the weak and strong versions, in the terminology of MacLane. On the other hand, MacLane, Bourbaki and Cartan-Eilenberg still call 5-lemma Lemma 1, both parts in five lemma entry. Thus the four lemma in entry four lemma, neither weak nor strong version agrees with what is called four lemma in entry five lemma. Is there still a reason in literature to say four lemma for Lemma 1 (i) in entry five lemma ?
the four lemma in entry four lemma, neither weak nor strong version agrees with what is called four lemma in entry five lemma.
They do agree. The five lemma is the four lemma applied to the left four columns and then to the right four columns of a five-column diagram.
Good, you are right, I misread the weak 4-lemma under entry four lemma (of course I know the obvious which part one applies at). But still even MacLane calls in Lemma 3.3 page 14 as five lemma what is here an in his later chapter called the weak four lemma together with iso corollary (he says the weaker form of iso statement where all are iso and then says, more precisely and lists what you call 4-lemma). That means that MacLane is hesitant in terminology.
That means that MacLane is hesitant in terminology.
I am not sure what you are after. This is entirely standard terminology. It’s also on Wikipedia and on Mathworld, people use it on the SE forums and on blogs.
Well, the original terminology (Cartan-Eilenberg etc. and still used when quoting the argument in practice in algebraic topology/geometry) is that if we assume 5 vertical arrows and suppose a version with two epis and one mono or two monos and one epi, then they call it 5-lemma. Now I see that new secondary sources use also 4-lemma (after inessentially throwing out one vertical morphism in assumptions, not used in proof). It seems the clarification by naming the intermediate step as 4-lemma may be due MacLane, for pedagogical purposes (hence popularity in wikipedia and alike). Still if $n$Lab does not allow the original usage as well (that is that the 5-lemma includes the case with two epis, not only the iso corollary) we do not support the usage in many many references. Both terminologies seem to have place and one should write the historical remarks, including all variants in terminology.
You are now saying that there is a statement missing at five lemma? Then please add it.
By the way, I find this suggestion that the word “four lemma” is used only for “pedagogical purposes” wrong. The four lemma is the stronger, more general and sharper result. It has four vertical maps instead of five. It is by all accounts most reasonable not to conflate it with the five lemma.
But I guess we have discussed that sufficiently now. Let me know when you add more content to five lemma.
Yes, Urs, the thing is that it is conflated in the present version of five lemma (unlike in MacLane’s book where lemma 3.3 on page 14 has it clear) while I am happy with the entry four lemma which does not conflate. I will deconflate it when i can. I have the feeling that you still did not understand my point. I learned some things about terminology and strong version of four lemma thanks to this, but still I am not clear about the history here and true level of appearance of new terminology.
I have made the careful work, please check now if you agree.
The present version in five lemma is what I will be happy with as it agrees with the literature which I have seen so far. I have also added MacLane 1963, Bourbaki 1980, Cartan-Eilenberg 1956 and Borceux-Bourn to the references.
Lemma (the lemma on five homomorphisms or the five lemma)
sharp five lemma (essentially the weak four lemma)
(weak) five lemma (conjunction of the two statements above)
If $f_2$ and $f_4$ are isos, $f_1$ is epi, and $f_5$ is mono, then $f_3$ is iso.
The weak four lemma is another terminology (cf. MacLane, Homology) for the same as 1.1 and 1.2 except that in 1.1 $f_1$ is not required to exist, and in 1.2 $f_5$ is not required to exist (see four lemma), where the dropped requirements are inessential as not used in the proof. The four lemma follows directly also with the salamander lemma, as discussed at salamander lemma - impliciations - four lemma. The following is a direct proof.
+++++++++++++++++++++++
P.S. By the way, I said pedagogical because anybody who ever read the proof of (strong version of) five lemma knows that the existence of fifth vertical arrow is not important for strong version of the half of the statement. Formulating the well known as a separate fact is for pedagogical (clarification of exposition and memorizing) purposes I think, as it is anyway practically always used when the ladder can be prolonged. It is obvious of course that it is formally stronger.
I should also note that Cartan-Eilenberg 1956 (what is the most influential historical reference on classical homological algebra) has, as I just checked, only 1.1 and 1.2 as five lemma (proposition 1.1, page 5), while not even mentioning second, weaker, combined, statement. Massey’s simple book has only the weaker statement. The author of what MacLane calls strong 4-lemma is in English spelling J. LEICHT. The name is in MathReviews with three listed references
MR0266949 Lorenz, F.; Leicht, J. Die Primideale des Wittschen Ringes. (German) Invent. Math. 10 1970 82–88. (Reviewer: H. Gross) 15.70 (12.00)
MR0080695 Leicht, J. Über ZPE-Ringe in der algebraischen Geometrie. (German) Monatsh. Math. 60 (1956), 214–222. (Reviewer: P. Samuel) 10.0X
It is not clear if any of the three has the lemma, or his contribution is a folklore fact known more precisely to MacLane…
Looks good, thanks.
Thanks for taking care with this, Zoran.
Added a section to diagram chasing lemmas - contents, since this seemed the most fitting place for it, for the time being. Explanations in standout-box there should be sufficient.
I suppose you want the square(s) on the right in lemmas about equalizers to be serially commutative.
I don’t understand (in the lemma about pushouts) why you need $\mathcal{C}$ on top of $C$ (in whatever font it was). Seems needlessly redundant; you could just say “diagram in $C$”.
I’m a little conflicted as well about grafting these results onto a page where the original intention seems clearly to be about diagram-chasing in abelian categories. Undoubtedly these types of lemmas do deserve a place somewhere on the nLab.
Actually, coming back to the lemma on equalizers, it seems worthwhile to note the converse as well, that if the top row is an equalizer, then the left square is a pullback.
Fair enough. Can’t think right now of a better thing to do, so it’s okay as a holding pattern.
Wait. The entries with titles of the form
topic - contents
are being used as “floating tables of contents” in pulldown menus in the top right of the entries being linked to.
For instance if you go do salamander lemma you see a pulldown menu “Context – Diagram chasing lemmas” in the top right. Clicking on that shows the content of the file “diagram chasing lemmas - contents”.
Therefore this file must contain a vertically-ordered list of hyperlinked keywords .. and nothing else.
If you have a further diagram-chasing-like lemma to add you need to
create a new page, and add your content there
add the page’s name (and nothing else) to the list at “diagram chasing lemmas - contents”.
Therefore I needed to undo your edit in the file, but the content is still kept here. Just copy-and-paste the relevant bits into a new entry.
And maybe the best entry to include the new material is at pasting law.
Okay. I have added proposition-environments at pasting law.
And the statement about pushouts of strong monomorphisms I moved, since this is is not an example of a pasting law, as far as I see. I copied it to pushout here and to strong monomorphism here.
Thanks. More or less in the same spirit, added to pullback some elementary preservation statements. The intent is to add to the nLab some collection of statements which is more systematic and reference-work-like than statements “pullbacks of monos are monos” used within other contexts.
Peter, the pasting statement is already in the entry, in the section above the one you added. For all the other statements which you added “the right square is irrelevant”, as you note, so I suggest that you remove the right square and just talk about pullback of monomorphisms.
I have edited accordingly, here.
Thanks for the edits.
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