I appreciate that some people try to make others' questions better by fixing typos and LaTeX, correcting language mistakes, suggesting changes and so on, but, in this case, I would have liked to know why Andrej Bauer thought the title of my question would be better the way he has put it. In the past, I have edited the title of a question of mine after someone suggested I do so in a comment. If such a choice have been made here (suggesting instead of modifying arbitrarily), I could have replied (before the title was changed) that, for one thing, (continued)

Thanks for your comments. Perhaps I'm missing something (it's quite late here), but don't they both assume that the 2-category has a "terminal object" (for Eric Wofsey) or an object called "1" (Michal R. Przybylek) the definition of which is not clear for me right now? I'll try to understand tomorrow if I'm missing something indeed. But, yes, I agree that such an object need not be unique.

This is the name Joyal calls it in his notes on quasi-categories, see for instance p. 26 of math.uchicago.edu/~may/IMA/JOYAL/JoyalDec08.pdf. There are some other notes by Joyal which deal with that construction in slightly more details, if I am not mistaken, but I cannot recall whether he states a universal property or not. Anyway, he uses it to define slices of quasi-categories.

Thanks, Mike. I was not aware of that, yet I was planning to study that work anyway!

Many thanks to Peter May for blowing such an interesting horn. I am eagerly waiting for this book to appear. I have been wondering for a while why neither Dwyer-Kan nor Thomason mention the fact that the subdivision is merely that composite. I would accept this comment as an answer if I could. (P.S. I had to look for the meaning of "REU". It means "Research Experience for Undergraduates".)

As for the Quillen equivalence between the category of small categories and the category of simplicial sets, the standard references include Fritsch and Latch "Homotopy inverses for nerve" and Thomason's "Cat as a closed model category". Another reference is Cisinski's Astérisque "Les préfaisceaux comme modèles des types d'homotopie". However, while the Quillen adjunction is beautifully explained by Thomason, the Quillen equivalence part is somewhat less clear. Cisinski's Master's thesis might (I do not have it handy) explain the details, but it is not publicly available as far as I know.

I have (prompted by Peter May's answer) posted a question related to this construction, see mathoverflow.net/questions/103281/….

Thanks for the comment, Theo. I have not asked the Yale librarian yet. While I am very familiar with the French system, I do not have any idea as to how to get a copy of documentation held by a US university. (I know it may sound silly... Perhaps it is the easiest thing in the world... In France, this is not.) That is one of the reasons why I am asking this question: understanding the procedure probably will take me some time, identifiying first the right institution to contact would therefore help me a lot.

Yes, indeed. Thanks for pointing that out. I don't recall the details (I don't have copies of the books to hand) but the Ladd-Lipset-related documentation in "Challenges" is extracted from "The File".

Thanks for this answer! (And sorry for my tardiness: I was abroad for a few weeks.)