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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 4th 2010

edited homotopy coherent nerve a bit

I tried to bring out the structure more by adding more subsections. Have a look at the new table of contents. Then I did a bunch of trivial edits like indenting some equations etc. Have a look at "See changes" if you want to see it precisely.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 4th 2010
• (edited Mar 4th 2010)

added a remark on the relation to the Boardman-Vogt W-construction but this needs more attention

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 4th 2010

started a section Properties

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 4th 2010

added to the end of the section on Illustration of the cosimplicial sSet-category a discussion of how that spurious sixth face of $|N(P_{0,4})|$ encodes the exchange law.

That's pretty neat how the exchange law and its higher cousins can be seen here. Very nice.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

I thought that as well. There was a thesis by a student at Montpellier who encoded the higher cousins neatly. I do not have a copy. Her name was, I think, Florence Marty. She gave a talk at one of the Como Cat theory meetings.

The way I like to state the interchange law is $5\neq 6$. The extra face is not really 'spurious'. Sometimes I have thought that this is really the key to understanding 'everything' about homotopy coherence, and even the structure of homotopy types, but each time I try to grasp it it slips through my fingers.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

There was a thesis by a student at Montpellier who encoded the higher cousins neatly. I do not have a copy. Her name was, I think, Florence Marty. She gave a talk at one of the Como Cat theory meetings.

Ah, thanks. After googling a bit: on this page here -- which lists preprint from Montpellier university in 1998 -- here are the lines

98-02 MARTY Florence : Sur une généralisation des 3-catégories augmentées d’Olivier Leroy

I suppose that's the thesis you are thinking of? Unfortunately this doesn't come with a link to a file.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

That looks right. I have a copy of the earlier work of Olivier Leroy which Clemens Berger sent me. (Again not a file).

The point I noted was that the interchange law is the variant of this with the 4 simplex, but already with the 3-simplex, the actions (whiskering) are examples of the same sort of thing.

Some of the things that Ali Mutlu and I did within simplicial groups were motivated by this same combinatorics, and I seem to recall Pilar told me that she had had a similar reaction when doing her hypercrossed complexes.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMar 5th 2010

Interesting. Whatever electronic copies you may have, i'd be interested in.

Meanwhile, i slightly expanded the entry exchange law, just to link back to the discussion on the homotopy coherent nerve page.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

I have nothing really in the way of electronic copies.

I like the fact that in S[5] there are two new laws, since $6\neq 8$!

• CommentRowNumber10.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

I do have my S-cat notes from the IMA conference, but you should have seen them??? If not they have been put on my personal n-lab page: (http://ncatlab.org/timporter/show/HomePage)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

I like the fact that in S[5] there are two new laws, since $6\neq 8$!

Yeah, that's nice! And in $S[n]$ there are $2(n-1) - (n+1) = n-3$ exchange laws .

• CommentRowNumber12.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

These laws can be seen in a simplicial group and its Moore complex, by taking a p-simplex and an q-simplex and forming the commutator. That is the cycle around the p+q face in the relevant dimension. Now reflect that back to the Moore complex. Ali and I called this the Peiffer pairing and it is fun to calculate in low dimensions. It is related to Whitehead products in simplicial groups and the old formula of Kan that appears in Curtis' survey article.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

(By the way, I can't count. So I had to fix my above message. Now it should be right. :-)

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMar 5th 2010

It would be nice if we could indicate the relation to Whitehead products on the page exchange law eventually...

• CommentRowNumber15.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

That is more complicated to do as although Ronnie has a formula for some of this sort of thing it is not 100% clear how to give an exposition.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeMar 5th 2010

I do have my S-cat notes from the IMA conference, but you should have seen them???

This is not about me having seen something, this is about the entry giving the information to other people!

I am currently using the entry for a seminar. A student is preparing a talk with the help of this entry. I want that entry to provide as much helpful information as possible!

In general, if I may say this, I think you should adopt the habit of giving links, links, links to your work. When asked for a reference, you tend to say thinks like "In the Menagerie there is a reference given." Probably not many people solve these little detective stories. They have to google your name and "Menagerie" then be lucky and find the right page, then be lucky to find the link to the pdf on it, then be lucky to correctly identify the chapter you have in mind, and so on.

Whenever possible. let's provide direct links on the nLab. One can even link directly to a specific page in a pdf. I do this often when referencing particular proofs or definitions.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeMar 5th 2010

That is more complicated to do as although Ronnie has a formula for some of this sort of thing it is not 100% clear how to give an exposition.

Hm, too bad. But didn't you just say that there is a nice way to see this in a simplicial group?

• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

Mea culpa!

I have partially hidden the Menagerie with a link from my n-Lab page to my personal page and then a pdf linked to that. (The version there is 10 chapters long.) The presently editing version is longer although some sections are as yet very short.

As the notes grow (and I have just been rewriting stuff on simplicial fibre bundles that will change the page numbering yet again) I have not tended to give page numbers in links because of that. The topics are hyperlinked within the notes and the index is intended to be quite complete. References are hyperlinked both from the bibliography and to it.

When you say that a student is preparing a seminar on that entry, do you mean on the homotopy coherent nerve?

The S-cat notes are on my personal page WITH A LINK that works!!!!

When I have time, I will put the Whitehead product formula on the lab (I though I already had!!!! I don't quite understand that.)

An interesting point is that the reflection into the Moore complex is some sort of multi-derivation. There is the shuffle product of two elements in the Moore complex with Kan's cunning order and this is in the Moore complex, yet if you do the simplest cases, it is not really clear why. Some of this is unpublished, so I should put that on the Lab when I can.

• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

On my Lab page, I have added a list of the 'goodies' that can be downloaded from my personal page. I have added references and links in several places and also a n-Lab entry for the Menagerie notes.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeMar 5th 2010

I have partially hidden the Menagerie with a link from my n-Lab page to my personal page and then a pdf linked to that.

Right, so why is that? Why is the link being kept so secret?

I have added references and links in several places and also a n-Lab entry for the Menagerie notes.

Okay, so I go to Menagerie .

There it says a link to the Menagerie is available on Tim Porter's nLab page .

Okay, so I follow the link to your nLab page. And look for the word "Menagerie". The first I see is a link under "List of Goodies." It says "The Menagerie: a set of notes...". I follow that link and arrive at... the page I just came from! Still no pdf.

So I go back again to your nLab page and scan it more closely. Ah, now I see that there is a clause saying "See also my private nLab area.

All right, so I follow that link. On your private page then I see a link called crossed menagerie. Clicking on that leads me to a new page... and there is finally a link to a pdf!

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeMar 5th 2010
• (edited Mar 5th 2010)

When you say that a student is preparing a seminar on that entry, do you mean on the homotopy coherent nerve?

Yes, today in the seminar a student gave a talk on the homotopy coherent nerve and how it relates $sSet_{Joyal}$ with $sCat_{Bergner}$ .

• CommentRowNumber22.
• CommentAuthorTim_Porter
• CommentTimeMar 5th 2010

There is now. I had my reasons for the tortuous trail, but your points are valid so here is a 10 chapter version produced today. (Some edits are 'en cours'.)

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeMar 5th 2010

Thanks, Tim. Sorry for pestering you with this.