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I have started an entry on shuffles. It is meant to be an ’elementary introduction’ so there will be room for deeper exploration of them in follow-on entries.
Thanks, Tim.
I have edited the entry a bit. Made your discussion of products of simplices a subsection of an Application-section. Have a look to see if you can live with it.
And I added a Definition! :-)
Tim, I am wondering if your nice discussion of the combinatorics of products of simplicial sets might be better visible if we moved it to an entry that deals with products of simplicial sets, such as the entry sSet in its section on the monoidal structure.
What do you think?
We could have a separate entry on Shuffles in the Combinatorics of products of simplices, just by renaming the entry and doing a minor edit. Then have a (for the moment shorter) shuffles with the definition. I paused for lunch and have some more that I can add. Let me add that and then we can see how best to put this. (The rest of the article that I sent you relates to Kan’s simplicial formula for Whitehead products. The formula is in Curtis’ paper but a proof has never been published.) Perhaps I can put something on Whitehead products as well, later on. We’ll see.
Your addition looks fine. My comment above is, I think, correct or almost. The problem of putting it within sSet is that it may get lost there as well or rather unbalance with the rest of that entry. Let me put a bit more on and we can evaluate what best to do. (and you can use some of it with your students if you want to.)
Okay, I gave it its own page products of simplices and linked to it from “shuffle” and from “sSet”.
you can use some of it with your students if you want to
Well, I need to tell them about the Eilenberg-Zilber map. It would be nice to have a direct relation from your discussion to that.
Let’s see. It’s nice that you are being expository, but maybe it would also be good to isolate the actual statements.
So I guess one proposition you want to state is that:
Proposition The non-degenerate simplices in are precisely those of the form for a -shuffle and non-degenerate simplices in in and , respectively.
Is that right?
Yes. except that we have two ins there!
Sorry, two ins??
Oh, haha. I get it. Okay, I’ll paste that into the entry, with one in removed.
It would be worth mentioning that it is exactly the link with products of simplices that makes the Eilenberg-Zilber map work. (approximations to the diagonal and all that sort of thing.)
Okay. You are currently locking the entry. I left the proposition in preliminary state. So I’ll wait for you to unlock it again.
Tim,
I have now logged in again and finished typing the proposition, including introducing the notation.
Not sure about the structure of the entry now. Do you think it is necessary to keep the “elementary introduction” separate from its statements? I’d think ideally it would be merged to a text that goes back and forth between motivating a statement and then making that statement. But you decide.
(For instance: is this really an introduction? Calling it an introduction suggests that later there will be some full-fledged theory of shuffles. But what more can there be said? I’d think this “introduction” probably is the theory, and could be written as such.)
Tim,
I tried to fix the codecogs image that you included at products of simplices, but not really successfully yet.
The problem was that you had “greater-than”-signs in the code. The instiki-compiler does not like these, as it confuses them with HTML tag delimiters.
So I tried replacing them with
\gt
but that apparently the codecogs compiler does not know how handle. So I am not sure what to do. Others here might know.
I left in phrases from the document so I am not fussed about the wording which was chosen for a different context.
You have a repeat in your system or in your fingers. Now there is a ’to to’ so two to’s! (and that does not make 4)! I will fix it.
It would be worth mentioning that it is exactly the link with products of simplices that makes the Eilenberg-Zilber map work. (approximations to the diagonal and all that sort of thing.)
I don’t know about that. I once did the brute-force computation that the EZ map works, not more. I’d be interested in hearing whatever insights you have to offer on this.
The point is that the free simplicial Abelian group on is the tensor of the two individual cases. In case this gives a diagonal map, but the diagonal is not there in the simplicial abelian group case, so you use a sum of bits that approximate the diagonal in the tensor. I will see if I can find it somewhere and send it to you as it make it clearer what is going on with the EZ and AW maps and why everything fits together.
The signs on the faces of the simplices pair up (and that pairing can be made explicit) so that you SEE how the EZ map really works. It is quite enlightening.
Sounds really good! Is there a chance that you might send this tonight?
It is in chapter 11 of the Menagerie. I will send it by e-mail. Pages 454-459. (This is still under construction so is not in the version on the Lab.)
Thanks a lot!
Hi Tim,
could you send me an update of the menagerie too, please? ”O_O”
Just popping my head round the door to thank Tim for writing this up – during my PhD I had to grapple with shuffles since they arise in Gerstenhaber and Schack’s “Hodge-type decomposition” of Hochschild cohomology, and since I then had negligible intuition for algebraic or geometric topology it was pretty hard going for me. So I look forward to seeing the evolution of this page.
Also: Tim, could you please send me a copy of the relevant pages of the Menagerie?
@David and Yemon Will do! I will send the current version. I have not done much since early this year as I got into another project in Lyon. The relevant section is in chapter 11. there is a lot still to be done….. but since the plan has always been open ended there always will be!
As usual suggestions for additions to both the n-Lab entries and the Menagerie are welcome… I don’t promise to do them :-)
Tim, I have looked at it, but maybe not in sufficient detail. Could you maybe highlight again for me where exactly you invoke some depper principle for understanding the Eilenberg-MacLane map. So far it looks to me as if you go and spell out the usual computations. You spell them out nicely, but I am not sure yet I see that you have a more abstract argument that would safe one from doing these tedious computations.
As usual suggestions for additions to both the n-Lab entries and the Menagerie are welcome.
My general suggestion: put everything from the Menagerie into the Lab! :-)
Deeper principles versus tedious computations.
One problem I have is that ’deeper’ and ’tedious’ are subjective judgements. I would say that the combinatorial principle (deep or otherwise) of explicitly pairing up the faces (and seeing why they do pair) is a useful insight, particularly the ’why’. Once that is seen then there is no need for tedious calculations at all! (at least in this instance:-)). What is interesting is then that the idea behind Kan’s formula for the Whitehead and Samelson products indicates that this works on a non-commutative level as well. That formula is hidden away in Curtis, but I sort of hope that it is IMPORTANT. e.g. that it sheds light on the higher homotopy structure in a lot of different contexts. I will have to look at that stuff again, but seem to be occupied 120% of my time… and I’m officially retired.
(PS. I think there will always be some ’tedious’ calculations.)
My general suggestion: put everything from the Menagerie into the Lab!
That sort of is my intention but there are only 48 hours in the day and 9 days in a week! I need to finish more of what I have started in both places.
I mean to add stuff by Aldrovandi and Noohi to the Menagerie and the n-Lab as that seems to have passed by without comment.
Once that is seen then there is no need for tedious calculations at all!
Just so that I understand you correctly, could you just say this in precise terms: which proposition are you saying is an immediate consequence of which lemma?
Ah that is a problem. I do not have an easily accessible version so have to work from memory and specific lemmas etc are hard to pin down at this moment. The analysis of the shuffle posets for the anti-lex order (in the shuffles and Whitehead product notes this is, I think, clearer.) gives you exactly the pairing up of shuffles that is the reason for the cancellation of terms. I do not think the version so far written in the Menagerie does things as well.
I will attempt to sort this out and to put it on the n-Lab page. The intuition is fairly clear, the details are a lot harder. The point is that in any p+q simplex of the product a (p+q-1) face is either in the boundary of the product and occurs once in the list of such or it occurs twice (so is inside). The shuffles that specify it then have to be checked to be of opposite sign. (There are various ways to explain that fact by an analysis of the combinatorics, but I do not really like any of them. The calculation is clearer.) This is the universal example i.e. it works at the level of models or ’shapes’ and is universal as such.
As I said I do not have a printed version to hand so it will take me some time to see what I can give as to ’chapter and verse’.(By the way, I used MacLane’s Homology and Andy Tonks thesis as sources for some of this.)
invoke some deeper principle for understanding the Eilenberg-MacLane map.
I am not sure exactly what you need. Here are some thoughts.
I checked back in MacLane’s Homology and he makes the point about the approximation to the diagonal. In the Eilenberg-Zilber theorem the shuffle map yields an approximation to the diagonal. He makes the point that there are two such and that these are not equal. (They are linked by Steenrod squares.)
There is an analogy possible with the stuff that Phil Ehlers and I did based on the ordinal subdivision. The point is that the ordinal subdivision goes into the subdiagonal of the product . The formula is very like the Alexander Whitney map but is induced by the ordinal sum (I forget the details at the moment but I think the map is somewhere in the Lab relating to ordinal sum.) The subdiagonal retracts to the image of the subdivision (the algorithm for constructing the retraction was in our paper.) I have never understood the link between our reasoning and the Eilenberg-Zilber map defined using shuffles. There clearly should be a close link. The combinatorics seem to be very close but I am missing one or two insights I think.
BTW Andy Tonks gave a proof of the E-Z theorem for crossed complexes which may be useful. How this relates to higher order (less linear) models is a mystery. (Pilar and I are hoping to attack that soon.)
I am not sure exactly what you need.
Tim, I am still just trying to make you say more explicitly what you were alluding to in #11, where you said:
It would be worth mentioning that it is exactly the link with products of simplices that makes the Eilenberg-Zilber map work. (approximations to the diagonal and all that sort of thing.)
I didn’t feel the “need” for anything else until you said this. It seemed to me you were hinting at some deeper level of understanding of the EZ map.
I added a note to the article with a couple of simple (perhaps folklore?) equivalent characterizations of -shuffles that I learned about from a nice talk by Eric Hoffbeck (describing joint work with Ieke Moerdijk) on shuffles of trees. More sadly, while navigating this article I noticed that product of simplices has a bunch of “invalid equation”s, which seems to be the result of a problem with codecogs previously discussed on the nForum.
The reference should maybe go to shuffle of trees.
Would you have time to add a reference for ordinary shuffles?
They do discuss classical -shuffles in this article, though, including these two equivalent dual characterizations. I don’t know another reference for that, but if I find one I will add it.
What is the recommended solution for the codegogs / xymatrix problem?
(If anyone wants the derivation of Kan’s non-Abelian shuffle description of the Whitehead or Samelson rpoduct which was given in the Curtis survey and never published, I have one which is based on what is on our shuffles page.)
They do discuss classical (p,q)-shuffles
Sorry, I moved the reference back. But could you then add pointer to the page/proposition number in that article where the entry recalls these statements? Thanks!
What is the recommended solution for the codegogs / xymatrix problem?
It looks like the only solution is to re-do the typesetting by other means. If the diagrammatics is not too sophistcated with its arrows, I prefer the usual hack: Simply use an
\array{...}
environment and use
\downarrow
etc to get a diagram.
I fixed the first few diagrams in product of simplices, but I was rusty so it took a long time (much longer than It should have … mostly due to typos!) I will try to do a few more later on but any help would be very useful.
Urs wrote:
Sorry, I moved the reference back. But could you then add pointer to the page/proposition number in that article where the entry recalls these statements? Thanks!
I added a pointer to the section in Hoffbeck and Moerdijk’s article where these dual formulations are discussed, as well as a pointer to a place in Aguiar and Mahajan’s book where the permutation formulation is discussed.
Tim wrote:
I fixed the first few diagrams in product of simplices, but I was rusty so it took a long time (much longer than It should have … mostly due to typos!) I will try to do a few more later on but any help would be very useful.
Thanks! I tried fixing a couple more, but it also took some time and I’m worried about introducing mistakes… I might have time to do more later today or tomorrow though.
Noam, We have done the easy ones now the next lot need diagonal lines… oh dear, I will need to search for those to see how they are done! Thanks for the help.
@Noam: Thanks!
@Tim: maybe you need things like “” and “”? These are given by
\searrow
and
\swarrow
etc. (But maybe you mean something more sophisticated.)
Urs, Thanks. I knew of those but I need a line not an arrow. It is probably simple, but if someone knows the solution off hand it would save me a bit to searching. I will look at some of the Hasse diagrams as the diagrams that are not showing are just some simple Hasse diagrams.
(Edit later:I have used arrows as headless slanted lines do not seem to be available. That was tedious!)
I see. I suppose the intended way to deal with this is to create SVG graphics. The built-in SVG editor here never worked for me, so I can’t help with that (and the button to it seems defunct now anyway).
Another option is to simply do the diagrams in LaTeX on your personal machine, and then copy them as pictures into the nLab entry.
If you send me a LaTeX or pdf file with the required diagrams, I can take care of including them as pictures into the nLab entry.
Thanks for the offer. I will go on using the arrows for the moment. There may be a fix known to the iTex people and then a simple search and replace will fix it. It does not look too bad and is much better than the invalid equation signs!
I have updated the products of simplices page correcting some typos pointed out by Noam. Thanks. And I have added a bit more that I had on a draft document on shuffles and Whitehead products.
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