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1. I would like to rearrange Kan complexes as ∞-groupoids to something like

1. general description

2. 2-dimensional example

In particular I think the word oriental should occur more prominently in the beginning of this section.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 26th 2012

You are right, the entry is not really in good shape.

For your second item, I guess you should simply add a new subsection under Examples titled 2-groupoids or the like.

For the first: this should really go in the sections Definition and Properties, I think.

I have done a quick rearranging, where I merged the old "Remarks"-section (I never like that, as it is so unspecific and undecided between "Properties" and "Related concepts") with the "How to think of Kan complexes as $\infty$-groupoids".

Still not really good. But maybe a step in the right direction.

2. I edited the introductory sentence of the section on Kan complexes as models for $\infty$-groupoids and added a reference for this section.

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

Stephan, that sentence is accurate but slightly gives the impression that that was the order in which the ideas came out historically. I am not sure but I think Boardman and Vogt probably had the idea in about 1973that Kan complexes were behaving like infinity groupoids,and hence thought of weak Kan complexes as infinity categories. Certainly by the mid 1980s I was asking why people were searching for models for infinity groupoids when there were Kan complexes around which clearly fitted the bill and Ronnie Brown was advocating the related simplicial T-complexes as strict infinity groupoids (i.e. crossed complexes). Perhaps some historical ancestry might be put somewhere. (It may be in some other entry and I have not checked across.)

• CommentRowNumber5.
• CommentAuthorStephan A Spahn
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

For your second item, I guess you should simply add a new subsection under Examples titled 2-groupoids or the like.

My point was that in this section there was a thread which always has been preluded by a phrase like ”in low dimension…” then something was said on general orientals then the explanation in low dimension was continued. I have changed this now to illustrate what I meant. But I won’t insist on this. Some people might prefer to have the examples in front position.

• CommentRowNumber6.
• CommentAuthorStephan A Spahn
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

that sentence is accurate but slightly gives the impression that that was the order in which the ideas came out historically.

Yes, when I read the ”Idea”-section I also had qualms if the idea of Kan complexes ”is” to play a role in ($\infty$,1)-category theory. I admit what I wrote was somehow biased. Maybe we should change the order of the mentioned sentences into some historical order - or at least to no ahistorical order.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

I will try something to see if we can get that looking a bit more to our mutual liking. (We can always go back to this version if it does not work ;-))

(Edit: done that and got rid of a lovely 22-simplex!)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

Stephan,

thanks for working on the entry!

You and Tim would help me if you could indicate (reproduce) here which sentences you are debating about. I am at a slow connection and have trouble chasing through the history-files.

I have edited the formatting of the Reference-item to the "guidebook" that you edited. Have a look to see what I did.

I also added some question marks "section (???)". Which section of the guidebook would you like the reader to be aware of in relation to Kan complexes? We should add that.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeApr 26th 2012

Urs: I rearranged the ideas section at the top of the entry. Mostly shifting sentences around but also adding some new linking ones. I do not know if it is yet satisfactory but some of the sentences did seem out of place.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

I see. Thanks. Maybe with a little more work we can bring this entry to very-nice-form™. That would suit the nLab well.

• CommentRowNumber11.
• CommentAuthorStephan A Spahn
• CommentTimeApr 26th 2012
• (edited Apr 26th 2012)

I have edited the formatting of the Reference-item to the “guidebook” that you edited. Have a look to see what I did.

Yes, I forgot that we are doing it this way.

I also added some question marks “section (???)”. Which section of the guidebook would you like the reader to be aware of in relation to Kan complexes? We should add that.

I transferred the reference to the guidebook to oriental and mentioned the references there in Kan complex since the chapter 6 I had in mind from the guidebook is rather on the more general weak $\omega$-categories.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeApr 26th 2012

Okay, but on which page does the "guidebook" discuss Kan complexes? (I can't see it right now.)

3. Kan complexes are not discussed there, that’s why this reference is ill placed at Kan complex. On the other hand what is said in the subsection Kan complexes as ∞-groupoids on interpreting simplices as n-cells (in an $\omega$-category) is explained in section 6.2.3 p.111

In fact the subsection Kan complexes as ∞-groupoids is (or should be) partly a transclusion of material to be expected at oriental.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeApr 26th 2012

Okay, I see you have edited all the pointers accordingly. Good.

is (or should be) partly a transclusion of material to be expected at oriental.

True. As a first-order fix (don't have time for more) I added a pointer to the very first paragraph at oriental.