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looking over this entry and polishing here and there, I noticed that the proof offered (since rev 1) for why the subobject classifier is Cisinski fibrant (this Prop.) was no proof. So I went ahead and spelled out a proof.
Also added pointer to Cisinski (2006), 1.3.9 for the terminology in the entry (previously it sounded as if “we” made this up). Incidentally, that paragraph points to MacLane and Moerdijk (1992), VI 10.1 for the proof of the proposition, but checking out what it says there, it seems only rather vaguely related.
[edit: as pointed out below, it must be “IV.10.1”]
More generally, I believe you can characterize the trivial fibrations $X \to Y$ as being those that induce a weak pullback square
$\begin{matrix} X \times \Omega &\to& X_\bot \\ \downarrow && \downarrow \\ Y \times \Omega &\to& Y_\bot \end{matrix}$where $\Omega$ is the subobject classifier, $X_\bot$ is the partial map classifier (which is itself trivially fibrant; the nLab page mentions it’s an injective object), and the horizontal maps are restriction, and might be even more simply described with the partial map classifier for $PSh(A)_{/Y}$.
If I’ve not made errors, this is mainly writing down, for some $p : X \to Y$, the (representable!) presheaf of lifting problems for monomorphisms against $p$, and expressing it in terms of partial maps, and the argument works in any topos (even elementary ones).
Also, I think the fibrations (in the minimal Cisinski model structure) are as being those such that the pullback powers $X^\Omega \to X \times_Y Y^\Omega$ are trivial fibrations for both endpoints, at least if you assume Cisinski’s characterization of the generating acyclic cofibrations as the pushout products of generating cofibrations with endpoints $1 \to \Omega$ (which I think he only proves for presheaf toposes?).
I was thinking about adding some of the stuff I’ve worked through about the minimal Cisinski model structure to this page, but I’m not sure if I’m going to be able to get around to it.
That proposition in MacLane and Moerdijk (on p. 210 of my copy) states that the subobject classifier is an injective object (i.e. has the RLP wrt all monos), which indeed implies immediately that it is fibrant.
p. 210 of my copy
Oh, I see what may have happened here: Cisinski’s thesis points the reader to VI.10.1 (which is what I was looking at) but you are saying it must be IV.10.1.
Right, I’ll edit the wording in the entry now, accordingly.
[edit: Ironically, I now see I made a typo in copying Cisinski’s typo to the nForum above. With an even number of $\mathbb{Z}/2$-valued typos this gave the correct reference, by chance, which may be how you discovered it?! :-]
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