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nLab > Latest Changes: factorization lemma

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2010
    • CommentRowNumber2.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 18th 2014
    • (edited Mar 19th 2014)

    Hi,

    I happened to notice today that there are problems with the proof of Lemma 1. Unfortunately, I do not have time to fix this myself, but I thought I’d let you know.

    It can be fixed as follows.

    1) Define X^\hat{X} to be the pullback of f×id:X×Y→Y×Yf \times id : X \times Y \rightarrow Y \times Y and the map Y I→Y×YY^{I} \rightarrow Y \times Y which is currently denoted (I would not use this notation myself!) by (d 0,d 1)(d_0,d_1).

    2) The required map p:X^→Yp : \hat{X} \rightarrow Y is the composite of the map X^→X×Y\hat{X} \rightarrow X \times Y which is part of the pullback of 1), and the projection map X×Y→YX \times Y \rightarrow Y.

    3) The required fibration X^→X\hat{X} \rightarrow X arises in the same way as in 2), but composing with the the other projection map instead,

    4) The required map X→X^X \rightarrow \hat{X} arises via the universal property of the pullback by using the map f∘c:X→Y If \circ c : X \rightarrow Y^I, where cc is the map Y→Y IY \rightarrow Y^I appearing in the factorisation which defines Y IY^I, and the map id×f:X→X×Yid \times f : X \rightarrow X \times Y.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2014

    Thanks. I’ll look into it as soon as I find a calm minute. Probably tomorrow around noon CET.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2014
    • (edited Mar 19th 2014)

    I have looked at it now. What I saw was an typo in the indices of the projection maps – clearly a consequence of the bad idea of using indices in {0,1}\{0,1\} for the maps out of the path space objects and indices in {1,2}\{1,2\} for the projections out of a product. I have harmonized the indices now to the latter choice.

    Did you have more “problems with the proof” in mind than the nuisance with the labels?

  5. Yes, the principal problem is with the argument that pp is a fibration. Since f×idf \times id is not necessarily a fibration, I do not see how expressing pp as a composition of three maps as on the page helps. It seems to me that the definition of pp is incorrect, and should instead be as in the argument I gave above.

    A minor point is that it is the fact that XX is fibrant that ensures that the projection map X×Y→YX \times Y \rightarrow Y is a fibration, not the fact that YY is fibrant.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2014
    • (edited Mar 19th 2014)

    The diagram exhibits pp as the composite

    X^→X×Y⟶(f,id)→p 2Y \hat X \to X \times Y \stackrel{(f,id)}{\longrightarrow} \stackrel{p_2}{\to} Y

    The first map is a fibration because by construction it is the pullback of a fibration; and the composite of the second and third map is the projection which is also a fibration (right, strictly speaking because XX is fibrant, but both XX and YY are fibrant by assumption, but I’ll change that in the text).

    I am not in fact sure where your proposal differs from the given one. (And you seem to have typos in your item 4), no?) Also, I think the proof as on the nLab is just the one that K. Brown gives in his article, but I don’t have time to go back and check now.

    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 19th 2014
    • (edited Mar 19th 2014)

    Ah, thanks, it was the trivial observation that p 2∘(f,id)=p 2p_{2} \circ (f,id) = p_{2} which prevented me from joining up the dots!

    I do find the argument that I gave cleaner: just using p 2p_{2} directly, and using only one pullback. This is a matter of taste, though! The essence is definitely exactly the same as the argument on the page.

    Sorry to take up your time with this!

    Regarding the typos in my argument: thanks, I’ll fix them in a second! I used ZZ instead of X^\hat{X} when I was working out the argument on paper, and forgot to change everywhere!

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2014
    • (edited Mar 19th 2014)

    Okay, I see. I tried to clarify this better in the text now.

    And no need to apologise. I am glad you brought this up. There were indeed annoying typos in the indices of the proof and in parts it was indeed unclear, as we just saw. If you found it unclear, many (most) other readers would, too!

    You should feel invited to either further streamline the proof on the nLab page as is or else just add your alternative proof right next to it. If it seems to improve the exposition from some point of view, then I am all in favor of it.

    I remember back when I created that entry I mostly wanted to have a place that I could keep pointing to when I referred to the factorization lemma. I wasn’t primarily motivated by writing a good new exposition and proof, I just wanted there to be something. So let’s improve further if there is room. It’s a beautiful and powerful lemma that deserves to receive all the love we have for it :-)

  9. That was kind of you to write, thank you!

    I will try to find time to make an edit at the weekend.

  10. I have now rewritten quite a lot of the factorization lemma page. I presented the argument of Comment 2, along with proofs of a couple of preliminary facts. I also rewrote the proof of Ken Brown’s lemma. I made a tiny edit at category of fibrant objects. The definition was missing the requirement that pullbacks of fibrations exist.

    I had a lot of trouble with the typesetting, especially with the commutative diagrams. Usually I like to argue in a very diagrammatic way, using diagrams for almost everything. For this, I usually use Tikz.

    I investigated the possibility of creating the diagrams in SVG, but SVG diagrams do not display correctly in my browser. It is an obsure webkit browser (dwb), but I think the problem is the same in all webkit browsers, such as Chromium. Thus I decided to go with the ’hack’ using arrays.

    I think that the diagrams are readable, but they do not look very polished. Unfortunately, I do not have time to experiment with further hacks to ensure that they look better! I would have liked to have displayed one of the diagrams as a ’universal arrow to a pullback’, but I guess that this is not really possible with the ’hack’ using arrays.

    I also had problems with creating lists. I have resorted to a hack here too, throughout the page.

    The numbering of the environments has also become rather bewildering. I would prefer everything to be increasing order from the top of the page to the bottom. Unfortunately, I don’t think this is possible without using the same environment all the way through, which means that the propositions, etc, cannot be typeset in italics, as they are elsewhere on the nLab.

    Finally, is there any way to ensure that text is not indented after a diagram, in such a way that the source code is readable?

    Back to the mathematics, feel free to make any edits/deletions that you wish. For example, if you wish to also have the proof as it was before on the page, feel free to just copy it from the earlier source, etc. I also cannot bring myself to use the American spelling of factorisation, but feel free to change it!

    I think that it is not usual elsewhere on the nLab to present the two facts before the proposition in the way that I did. They may well be proven elsewhere on the nLab. Perhaps it is not a problem to duplicate minor observations such as these, which probably would not have their own page, but feel free to reorganise/delete.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2014
    • (edited Mar 23rd 2014)

    Thanks!

    Regarding your struggle with the software: yes, it is one of the glaring issues that, of all places, the nnLab has no real support for typestting commuting diagrams. There are hacks such as including codecogs diagrams (explained somewhere here). I am hoping in the future it will be able to enter xy or tikz code into the nLab source and have it rendered properly.

    (Not sure what you have in mind with the text being indented after a diagram, though(?))

    Regarding duplication: no problem at all, in fact that’s probably better.

  12. I thought that the diagrams were causing the following text to be created as a new paragraph, as would happen in LaTeX if there were a blank line between the code for the diagram and the following text. I now see that in fact all lines in the statement of a proposition, etc, are indented after the first! This looks strange to my eyes, but that’s the way it is, I guess!

    Regarding commutative diagrams, perhaps a possibility would be to create by hand html (or whatever) code for common shapes of commutative diagram: triangles, squares, squares with a diagonal, pushouts, pullbacks, etc. There could then be some kind of script underneath the nLab which would allow one to write say \triangle{X,Y,Z,f,g,h}, and output a html triangle with these objects and arrows.

    In LaTeX, I write macros which do exactly this: I create ’generic’ commutative diagram shapes in tikz, with a few optional arguments, and then just write commands like the one in the previous paragraph.

    Doing something like this for the nLab would, of course, give a uniformity to all the diagrams of a similar shape, and allow the typesetting of all to be configured in one go.

    Perhaps I am wildly optimistic, but I cannot believe that it is very hard to create, for instance, a commutative triangle in html. I do not have the time to look into this, and probably most of the contributors to the nLab do not, but maybe it is something which someone eventually would be interested in trying. I think it is important not to be too ambitious: not to try to create a new ’standard’ for commutative diagrams, just find a robust and simple method for creating the basic ones. Maybe then one can see how to generalise it.

    I can certainly say for myself that I am very willing to contribute to the nLab, but the difficulty in creating commutative diagrams (I tried out Instiki some years ago), and therefore the time it takes to do so, means that I am not able to do so in my day to day work. Thus I am reluctant to embark upon more ambitious projects.

    For instance, amongst other things I have worked a lot on the foundations of cubical homotopy theory and higher category theory over the last few years, and have a number of results which I am writing up when time permits (which is not very much during term time). I would be very willing to share these with people, for example by discussing this work on the nLab, writing up various entries which tie in to it, but at the moment I do not really feel that it is practical to do so.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2014
    • (edited Mar 23rd 2014)

    Yeah, I know.

    While I absolutely don’t have the time to do so, with some people we are experimenting with a new setup on a working wiki platform. There is a page here: Schreiber WorkingWiki for experimenting. (It will probably ask you for a password, that password is “schreiber”).

    As you can see there, this has support for all your LaTeX needs. As long as you are not concerned with moving old nLab material, you might start editing there right away. If you want your own wiki like this, you should ask Lee Worden.

    With a little luck Lee and Adeel Khan Yusufzai (one of the contributors here) might be producing a script that tranports present nLab content to the working wiki installation. But if you know people who might lend a hand, that would be most useful.

  14. That sounds interesting, thank you for pointing it out! It looks as though I would need to log in with a username as well as a password to see anything, but I had a very quick look at some other ’working wiki’ pages, and it does look very promising. In particular, there is an example of a commutative diagram in tikz which seemed to display fine. It also appears to be possible to write scripts in a straightforward manner to customise things.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2014

    Ah, the username is the wiki name, i.e. “Schreiber”.

    And yes, it is possible there to do a lot of things with some scripts. We still need some more people however to write such. Are you into such things? Maybe we should then discuss this further.

    • CommentRowNumber16.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 24th 2014
    • (edited Mar 24th 2014)

    I can’t find a way to log in with this username either, I’m afraid! The page that you linked to gives ’Permissions errors’. Trying to use the ’Log in’ link at the top right with the user name and password that you gave gives: ’there is no user by the name of Schreiber’!

    It’s no problem, though. I will not have time to contribute in any significant way for several months at least, on the nLab or elsewhere. I just felt duty bound to edit the page on the factorisation lemma, because I had mentioned it when teaching!

    With regard to the scripts, it was more a general observation that it would seem very desirable to be able to write simple scripts and incorporate them in the wiki, without these scripts being buried deep in the source code of the wiki. For instance, my feeling is that commutative diagrams are so central to a wiki on category theory, that one should be determining the typesetting of the most common shapes ’only once’ (of course, people would still have the option to create the diagrams differently if they wished to). This allows one to change the typesetting throughout the wiki if the need arises in the future, and would speed up creating entries greatly.

    I mentioned that SVG diagrams do not display correctly in my browser a few posts ago. I realise that it in fact seems to be the combination of SVG and MathML that appears not to display: SVG alone appears to display fine.

    Edit: Andrew pointed me towards Tom Leinster’s article on ’Publications of the nLab’, and in fact most of the SVG + MathML diagrams there do appear to display more or less correctly on my browser. In particular, the commutative squares and triangles. There is a little too much distance between the arrows and the labels, but they are still perfectly readable. It was the diagram at comma object that I was looking at, which is completely mangled in my browser.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2015
    • (edited Aug 12th 2015)

    Adeel,

    after Richard’s substantive edits to the entry factorization lemma you made another edit without announcing it here.

    Now the “diff”-functionality of the nLab is such as to be essentially useless in this case for seeing what it is, by and large, that you changed. But now I see that over here Richard feels that his “substantial edit on the nLab, to the factorization lemma page, it was mostly written over”.

    Glancing over your edits, I suspect you did not intend to overwrite it as much as making some polishing. But since it is tedious for me to check what exactly changed, maybe you could say here what you did, also for Richard to have a chance to react. Thanks.

    • CommentRowNumber18.
    • CommentAuthorRichard Williamson
    • CommentTimeAug 12th 2015
    • (edited Aug 12th 2015)

    Thank you for taking it up, Urs, I appreciate the sentiment, but there is no need for Adeel to respond to this. I am happy to leave the page as it is.

    If I recall correctly, the reason that Adeel made the changes was because he had asked a question about something on another page (I cannot remember which), a detail of which had relied upon a specific aspect of the proof of the factorisation lemma as it was before I changed it, which was no longer visible after my changes. Hence Adeel reverted (and I think may have been asked to revert) what I had changed sufficiently far to allow this specific aspect of the proof to be visible again. There were also a few stylistic reversions, and so on.

    As I discussed on the other page that you link to, the style in which I wrote the proof does seem somewhat at odds with much of the rest of the nLab, and therefore the way the page is now no doubt fits more harmoniously in with the rest of the nLab. Thus I am entirely content for the page to be left as it is; there is no need for further discussion, or for further comparison of what I wrote, what was written before that, and what is written now. I myself prefer the way I had it, but that is only natural, and I can access what I wrote in the revision history, so there is no problem there.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2016

    Coming back to #17: In fact it seems to me that the lemma 3 introduced in that revision 14 is not correct. The morphism ss shown there does not make the triangle commute.

  20. Yes, certainly the diagram in Lemma 3 in Revision 14 does not commute for the stated gg! I have not looked to see whether the proof can be repaired in a simple way, but I remember that I wrote the proof in Revision 13 rather carefully, so that one at least should be correct.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2016

    I’ll go now and revert back to before revision 14.

    I remember that I wrote the proof in Revision 13 rather carefully, so that one at least should be correct.

    For what it’s worth, I allow myself to say that the proof I had written (present until Revision 10) was also careful and correct. :-)

    • CommentRowNumber22.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 6th 2016
    • (edited Apr 6th 2016)

    It might perhaps be best to revert all the way back to your proof, because I think something in it is referred to elsewhere in the nLab, which is not, explicitly at least, present in the proof I gave.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2016
    • (edited Apr 6th 2016)

    I have reverted to rev 13 and just re-instantiated a few later formatting edits I made (expanded the Idea-section, made the redirects “Ken Brown’s lemma” properly capitalized, added labels to the props and cross-pointers to them).

    I am happy that you have contributed here, I hope to motivate you to contribute more, so I’ll keep your revision.

    I have re-extracted my original version as part of the Introduction to Stable homotopy theory…

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2020
    • (edited Aug 22nd 2020)

    added pointer to Hovey’s book for the terminology “Ken Brown’s lemma” (for that corollary of the factorization lemma)

    diff, v26, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2020
    • (edited Aug 22nd 2020)

    [ duplicate removed ]

    • CommentRowNumber26.
    • CommentAuthorJem Lord
    • CommentTimeNov 26th 2020

    replaced the proof with one which refers to a single tikzcd diagram, which contains all of the morphisms involved. the proof method is unchanged, but hopefully presented more clearly

    diff, v27, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020

    Thanks for this!

    I haven’t looked in detail at your addition yet. But this reminds me that I had another version of the proof in the entry earlier (as witnessed by the discussion above) which used more comprehensive diagrams. My version of the proof is still at Introduction to Homotopy Theory, here.

1 to 27 of 27

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