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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.
Added a correction to the section on bar construction (h/t to Guest), linking instead to two-sided bar construction.
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 $B G$?
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”.
Got it, thanks.
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.
Added:
The original definition was given in 1926 by Paul Alexandroff:
Thanks!
And I see that Segal points to
but I haven’t yet found the exact document to link to.
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:
- Alexander Grothendieck, Théeorie de la descente, etc., Seminaire Bourbaki, 195 (1959-1960)
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).
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?
Yes, it appears that Segal really meant to reference Part III, not Part II.
This 1961 paper definitely precedes Segal’s.
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).
Corrected the references:
The notion of the nerve of a general category already appears in Proposition 4.1 of
Another early appearance in print is
By the way, Corollary 4.2 in Grothendieck’s paper talks about nerves of internal categories, internal groupoids, etc. May be relevant for internalization.
BTW in the document I linked to p.27 was a retyping of p. 350 of the original SGA4.
Tim, Dmitri, thanks. That’s an excellent reference. I’ll be adding this also at Segal conditions and elsewhere.
I have added a new section (here)
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).
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