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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 11th 2010

added to homotopy coherent nerve two diagrams in the section Examples and illustrations that are supposed to illustrate the hom-SSets of the simplicial category on $\Delta^n$

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeFeb 11th 2010

I have added several 'old' references to this. As one of the originators of that theory I get a bit 'browned off' by papers that do not make any mention of Vogt, Cordier (and of course myself). As I feel a bit like moaning, I would say that some of the papers I have seen show a lack of 'scholarship' in this regard. (end of moan.. promise!)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 11th 2010

I think it is good that you make these points! I am certainly interested in getting this information from you.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 11th 2010

I have now made some more of the author names link to their respectivbe nLab entries: Boardman, Vogt and you. Maybe you could create a page for Cordier?

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeFeb 12th 2010

WIll do (in time!)

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeJan 17th 2011
• (edited Jan 17th 2011)

I added the following text in homotopy coherent nerve (I hope correct; I learned this roughly from Joyal’s texts, but blame me if I misinterpreted) and reorganized the definition part:

Recall that a reflexive graph is a simplicial set of dimension $1$, i.e. 1-coskeletal; they form a full subcategory $\mathrm{reflGraph}↪\mathrm{Cat}$. The forgetful functor $U:\mathrm{Cat}\to \mathrm{reflGraph}$ has a left adjoint $F$ hence $G=\mathrm{FU}:\mathrm{Cat}\to \mathrm{Cat}$ is a comonad. By the definition its cobar construction is an augmented simplicial endofunctor $S\to \mathrm{Id}$ featuring $S:\Delta \to \mathrm{sSet}\mathrm{Cat}$ and whose augmentation is a cofibrant replacement of a 1-category in the Bergner model structure on $\mathrm{sSet}\mathrm{Cat}$ (“model structure for simplicially enriched categories”).

(By the way if David Roberts is around, he wondered about usual nerves on the history of the distinction between looking it as simplicial sety and as a topological space – of course for simplicial complexes the passage back and forth was elementarily well known since 1920s (about the same time when Aleksandrov introduced a nerve of a covering) but the full fledged treatment and the definition of a geometric realization is due John Milnor: The geometric realization of a semi-simplicial complex. Ann. of Math. (2) 65 (1957), 357–362 (of course at the time simplicial sets were “complete semi-simplicial complexes”). This is about the same time as Grothendieck started looking at more general nerves of categories.)

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeJan 17th 2011

Here I archive two discussions from query boxes at homotopy coherent nerve

QUERY 1

Todd: I am learning this for the first time, and I had some difficulty with how the definition of ${P}_{i,j}$ reads. From the ensuing discussion, it seems you want the elements of the poset to be $I\subseteq \left[i,j\right]:i,j\in I$, ordered by inclusion. But in the definition, it’s not clear what sort of thing $J$ is supposed to be, and it looks like the elements of the poset are instances of inclusions. (Another minor thing I don’t understand is why $\subset$ is being used instead of $\subseteq$, since for many mathematicians $\subset$ means strict inclusion. I see this preference for $\subset$ all over the nLab in fact; has this been discussed somewhere?)

Tim: I did not originate the poset based description as I always think of things as being paths through the $N$-simplex from $i$ to $j$ and then use a rewrite idea for the link. I will try to clean up the poset definition a bit and see if it helps, otherwise we can switch to the path based description and use the poset as a second way. I’m not bothered either way.

Some minutes later! Does this read well now?

Todd: Thank you! Yes, me happy now.

QUERY 2 – on terminology (at the time, it was at the entry then called simplicial nerve of simplicial categories)

Is there a simplicial nerve that's not of simplicial categories? If not, I'd put the article here instead of there. —Toby

Urs: yes, it seems to be called just “simplicial nerve” in the literature, but I found that a bit undescriptive, since every nerve is “simplicial” and here the point is really that we take the nerve of a simplicial category. I also seem to recall that Tim said he doesn’t like the term “simplicial nerve”. Maybe Tim should decide, he is probably the one among us who has thought about this notion the most.

Toby: Ah, I see how ’simplicial nerve’ is confusing; so how about just nerve of a simplicial category?

Urs: right, that might be the best option – I have to run now, maybe you can implement that?

Toby: I'll wait to hear from Tim.

Mike: Not all nerves are simplicial; it depends on what you are taking the nerve of. The nerve of a multicategory is a dendroidal set (a presheaf on the category of trees). The nerve of a compact symmetric multicategory is a presheaf on the category of Feynman graphs. And an $n$-category has a nerve that is a simplicial set, but also one that is a ${\Theta }_{n}$-set and one that is an $n$-fold simplicial set.

FWIW, I have sometimes seen the “simplicial nerve of simplicial categories” called the “homotopy coherent nerve,” which to me captures the intuition better.

Urs: true, I actually know that not every notion of nerve is simplicial, should have thought before typing.

Now that you mention it, maybe Tim Porter also said he favored “homotopy coherent nerve”? I’ll send him an email.

Tim: Back from a short absence: the term ’homotopy coherent nerve’ is probably a good one only when it fairly directly relates to homotopy coherence.

Any 2-category can be thought of as a simplicially enriched category and the Duskin nerve of a bicategory specialises to the same construction on 2-categories. Various people use the term ’geometric nerve’ for this. (It is interesting to compare the ’simplicial nerve’ of a simplicial group (as SSet-enriched category) with Wbar of the same thing.) I am trying to write something for the Menagerie that looks at the h.c. nerve with these aspects accentuated and also with links with Behrang Noohi’s weak maps of crossed modules. I may put some of this on nLab when I see more clearly how it all fits together.

In the meantime, I suggest we keep the term h.c. nerve although it is probably not ’best possible’. I agree that ’simplicial nerve’ is probably not a good term.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeJan 17th 2011

Hi Zoran, thanks for the pointer to the history. I was thinking more along the lines of how some people these days say ’simplicial space’ and mean bisimplicial set, and when talking about spaces actually mean simplicial set. But once Milnor pointed out the link between sSet and Top, and Kan (I think) showed that sSet modelled all homotopy types, then I suppose those in the know would have been comfortable with switching back and forth. But as you say, simplicial sets came from simplicial complexes, which were definitely used interchangeably with (nice) spaces.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJan 18th 2011

they form a full subcategory reflGraph↪Cat

Do you mean reflGraph↪sSet?

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeJan 18th 2011

Yes, that is what I meant, but I was writing and erasing various things and somehow the final version got incorrect.