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At the very beginning of chapter 5 of Joyal's lectures on Quasi-Categories (http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), he uses a notation which I think he has never explained/used (though I might be wrong, and this is due to the fact that this "book" developes a lot of tools in the appendix, and I might have lost something).

To be precise, given a simplicial set $A$, he consider a subobject denoted by $A_0 \subset A$, but none of the interpretations which I tried to give seem to work. He wants to prove some facts about the induced projection $X^{A} \to X^{A_0}$ for a quasi-category $X$, hence I need to understand what he means.

Any help will be highly appreciated, thanks in advance.

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    $\begingroup$ I would guess, based on the statement of Theorem C, that $A_0$ is simply the set of vertices of $A$. If you prefer, you can think of it as a discrete simplicial set. $\endgroup$
    – Zhen Lin
    Commented Dec 6, 2014 at 16:09
  • $\begingroup$ Yes, I have just figured it out by myself by looking at his notes on simplicial homotopy theory. I am pretty sure he means the 0-skeleton of $A$, i.e. the discrete simplicial set associated to the 0-simplexes of $A$. Should I erase the question or leave it for others with the same doubt? $\endgroup$
    – Edoardo Lanari
    Commented Dec 6, 2014 at 16:17
  • $\begingroup$ Have you tried e-mailing him? $\endgroup$
    – Ryan Budney
    Commented Dec 10, 2014 at 18:36
  • $\begingroup$ I was going to, but I solved my doubt (which was quite silly: I immediately thought of the discrete simplicial set given by the 0-cells, but at a first glance I couldn't see why it was a subobject of the starting simplicial set) $\endgroup$
    – Edoardo Lanari
    Commented Dec 10, 2014 at 20:29
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    $\begingroup$ I suggest closing as no longer relevant. $\endgroup$
    – Daniel Moskovich
    Commented Dec 11, 2014 at 10:28

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Yeah, it's the discrete simplicial set associated to the 0-simplexes of A, see page 19 of his lectures on Quasi-Categories, where he says the vertex map (a map here) takes its values in a discrete simplicial set.

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