added pointer to:

- Saunders MacLane, §XII.2 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

Added the reference:

- Stephen Lack and Simona Paoli,
*2-nerves for bicategories*, K-Theory,**38**, 2008. (arXiv:math/0607271, doi:10.1007/s10977-007-9013-2)

added also pointer to

- Charles Rezk, Part 1 of:
*Introduction to quasicategories*(2022) [pdf]

Added pointer to

for a proof of the claim (here) that the nerve $N \colon Cat^{smll} \to sSet$ is fully faithful.

But it would be good to add more classical references for these classical facts…

]]>added statement and proof (here) that the nerve functor on small strict categories preserves mapping objects

]]>I have added the statement (here) that the nerve functor $Cat \to sSet$ preserves finite products

]]>I have copied the paragraph on coskeletality+ characterizing nerves of categories/groupoids which yesterday I had added to *coskeleton* (as announced there) also to *nerve*, now here.

This says nothing that wasn’t already said on this page here, but it means to wrap it up more recognizably, so that reader’s confusion such as on MO:q/107951 is avoided.

]]>typo

julian rohrhuber

]]>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).

]]>added pointer to:

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

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

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

]]>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

- Graeme Segal, Section 2 of:
*Classifying spaces and spectral sequences*, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 105-112 (numdam:PMIHES_1968__34__105_0)

]]>

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).

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

This 1961 paper definitely precedes Segal’s.

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

]]>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?

]]>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 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)

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.

]]>Added:

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.

]]>

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.

]]>[deleted]

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