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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeNov 27th 2018

Correct the characterization of nerves of groupoids.

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 25th 2019

• CommentRowNumber3.
• CommentAuthorGuest
• CommentTimeDec 16th 2019
I don't believe the bar construction of a group is exactly the same, on-the-nose, as the nerve of the one-object group of a small category. In particular, the indexing is not the same - they are off by one. There are also a different number of maps in between G^n and G^n+1 at each n. If we are trying to understand these objects deeply we should be concerned with the details and not just identify objects when they look the same at a high level! Some nontrivial quotienting or downshifting or other fiddling has to be done to get them to agree.

To be precise, if you let G be a discrete group, and let G x - : Sets -> Sets be the monad that takes a set X and returns the "free G-set" G x X, then it's possible to carry out the bar resolution with respect to this monad described in the link in that section. This is clearly what is implied by the link. You would then apply the resulting series of functors to say, the singleton {*} equipped with the trivial group action, and get an augmented simplicial G-set. The empty set maps to *. The singleton [0] maps to the comonad iterated once, i.e. G x {*} \cong G. [n] maps to G^{n+1}.

But there is some confusion here. The nerve of a category is not augmented, unless you include the empty category; in this case there is an augmentation (but that's not the easy fix to this problem it might appear to be at first glance). In the diagram I wrote above, the ordinal [0] would map to G, and in general [n] to G^{n+1}. This is not what happens when you take the nerve of a one-object category. You get that [0] maps to the set of objects, which in the case of a one object of a category is a singleton, {*}. And [n] maps to G^n, the set of composable strings of n-morphisms.

A better insight into what's going on is provided by looking at for example Peter May's book "A Concise Course in Algebraic Topology." What the bar construction gives you, if I understand his writing correctly, is the simplicial G-set EG. The quotienting of this by the action of G is the desired BG.

Would love someone to write up a full exposition of the connection between these that takes this subtlety into account.
• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2019

Guest, you’re right. What the section gives you is $E G$, not $B G$. I usually keep this straight in my head by using two-sided bar constructions, so $N B G = B(1, G, 1)$, whereas $E G = B(G, G, 1)$ gives you the contractible total space. I will fix in a moment.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2019

Added a correction to the section on bar construction (h/t to Guest), linking instead to two-sided bar construction.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeDec 17th 2019

I’m confused; is the point that “the usual bar construction of $G$” would be interpreted by some people as referring to $E G$ rather than $B G$?

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2019

I think the real point Guest was making is that the article had misidentified the nerve of $B G$ (or $B A$), by writing down instead the simplicial object for $E G$. The article also linked to bar construction, largely written by me, where the discussion is about various acyclic resolutions of structures, for example the machine which produces a standard acyclic free $\mathbb{Z}G$-resolution of $\mathbb{Z}$ for the purpose of defining group cohomology. Formally that would be more like $E G = B(G, G, 1)$ which “resolves a point”.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeDec 17th 2019

Got it, thanks.

1. Off by one error

Béranger Seguin

2. Added 2 pictures: one of a $2$-simplex of $N(\mathcal{C})$ and the other of a $3$-simplex.

3. Added a pointer to the page on the Street nerve of a tricategory.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 18th 2021

I have added publication data to the reference to Segal’s “Classifying spaces and spectral sequences”, and moved it from the very bottom of the list of references to the very top.

(Not only is it possibly the first reference that explicitly states the notion of the nerve a acategory, but it is also more pertinent than several of the other references given here. For instance Dwyer-Kan’s “Singular functors…” and Isbell’s “Adequate subcategories” seem to be only rather vaguely relevant here. Maybe they need pointers to which page and verse the reader is meant to take note of.)

Together with Segal’s article there was the claim that:

The notion of the nerve of a category may be due to Grothendieck, based on the nerve of a covering from 1926 work of Pavel Sergeevič Aleksandrov.

Do we have any references confirming this?

Finally, I re-ordered the references into “For 1-categories” and “For higher catgegories” and added more publication data (doi-s) here and there.

• CommentRowNumber13.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2021

### For covers

The original definition was given in 1926 by Paul Alexandroff:

• Paul Alexandroff, Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, Mathematische Annalen 98 (1928), 617–635. doi:10.1007/BF01451612.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJun 18th 2021
• (edited Jun 18th 2021)

Thanks!

And I see that Segal points to

• A. Grothendieck, Théorie de la descente, etc., Seminaire Bourbaki, 195 (1959-1960)

but I haven’t yet found the exact document to link to.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJun 18th 2021

I found where the text refers to Isbell, and turned that into a hyoerlink.

I give up searching Numdam for where Grothendieck may have defined nerves, but I added Segal’s way of referencing Grothendieck:

• CommentRowNumber16.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2021
• (edited Jun 18th 2021)

Segal refers to the following paper by Grothendieck

Séminaire BOURBAKI

12e année, 1959/60, n° 195

Février I960

TECHNIQUE DE DESCENTE ET THÉORÈMES D’EXISTENCE EN GEOMETRIE ALGEBRIQUES

II. LE THÉORÈME D’EXISTENCE EN THEORIE FORMELLE DES MODULES

par Alexander GROTHENDIECK

It is available in electronic form here: http://libgen.rs/book/index.php?md5=90E371DADC483E85157578F2506D0E26

See page 369 (referring to page numbers appearing in the book itself).

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJun 18th 2021

I was looking at that p. 369 on Numdam earlier. But it states not the nerve construction but the Yoneda embedding (in fact we might cite it there). Am I missing something?

• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeJun 18th 2021

Perhaps here Proposition 4.1. on page 107. There is a reference to it in SGA4 p 27.

• CommentRowNumber19.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2021

Yes, it appears that Segal really meant to reference Part III, not Part II.

This 1961 paper definitely precedes Segal’s.

• CommentRowNumber20.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2021

I could not find anything on page 27 of SGA4, but page 350 of SGA4 refers back to Part III, Proposition 4.1, and refers to the resulting object as Nerf(C).

• CommentRowNumber21.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2021

Corrected the references:

The notion of the nerve of a general category already appears in Proposition 4.1 of

• Alexander Grothendieck, TECHNIQUES DE CONSTRUCTION ET THÉORÈMES D’EXISTENCE EN GÉOMÉTRIE ALGÉBRIQUE. III : PRÉSCHEMAS QUOTIENTS, Séminaire BOURBAKI, 13e année, 1960/61, no. 212, February 1961. PDF.

Another early appearance in print is

• CommentRowNumber22.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2021

By the way, Corollary 4.2 in Grothendieck’s paper talks about nerves of internal categories, internal groupoids, etc. May be relevant for internalization.

• CommentRowNumber23.
• CommentAuthorTim_Porter
• CommentTimeJun 18th 2021

BTW in the document I linked to p.27 was a retyping of p. 350 of the original SGA4.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJun 18th 2021
• (edited Jun 18th 2021)

Tim, Dmitri, thanks. That’s an excellent reference. I’ll be adding this also at Segal conditions and elsewhere.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeJul 27th 2021
• (edited Jul 27th 2021)

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeAug 25th 2021
• (edited Aug 25th 2021)

I have added a new section (here)

• Properties – (Non-)Preservation of colimits

with

• one example of the nerve not respecting a class of colimits

• one key example where it does respect the colimits.

I was prodded to this from reading Guillou, May & Merling 2017, who emphasize the relevance of these elementary but crucial points for the theory of universal bundles – but their corresponding Exp. 2.9 seems a little broken (unless I am missing something?) and their corresponding Lem. 2.10 seems to go only half-way along the argument for which it is later on quoted.

In any case, it’s an elementary but important point, worth recording. I have included as a final example the observation that, more generally, the nerve preserves left group actions on right action groupoids of sets eqipped with commuting left and right actions (as maybe suggested by the notation around that Lem. 2.10).

4. typo

julian rohrhuber