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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?! :-]
Is the example correct? I totally believe that if you used the interval $\Delta^1$ that $\Lambda_{\Delta^1}(S, M)$ generates is the model structure for right fibrations… but it’s not obvious to me that $\Lambda_{\mathfrak{L}}(S, M)$ does too.
In particular, $\Lambda_{\Delta^1}(S, M)$ contains the maps $\partial \square^n \times \Delta^1 \cup \square^n \times \{1\} \hookrightarrow \square^n \times \Delta^1$ and IIRC the right horn inclusions are retracts of these.
However, $\Lambda_{\mathfrak{L}}(S, M)$ instead contains $\partial L^n \times \Delta^1 \cup L^n \times \{ 1 \} \subseteq L^n \times \Delta^1$ which makes things less obvious.
Here, by $\partial L^n$ I mean the union of the faces of the cube $L^n$. I.e. all the subobjects $L^i \times \{ e \} \times L^{n-i-1}$ with $e \in \{0,1\}$. The inclusion is the n-fold pushout product of $\{0, 1 \} \subseteq L$ with itself.
I know that the example would be correct if you included the right horn inclusions in $S$, but the example includes only $\{1\} \subseteq \Delta^1$.
That example is indeed incorrect. I also think that the “Proposition” at the end of §2 is not worth keeping; in its current state it is only implicitly connected to the subject of the page, and in any case is already treated in more detail on the page for Cisinski model structures. I think both example and proposition are fit for deletion.
I’m still waffling on the presentation, but wanted to get the related fact down while I”m writing.
I’m imagining that the construction of cofibration-trivial fibration factorizations should be a proposition itself. But I also want to give a proof $Opt_S(X) \to S$ is a trivial fibration – either separately using the characterization I just gave, or by generalizing the more explicit proof for $L \to 1$ that follows.
I’m saying things true of all toposes rather than just Grothendieck toposes (or just presheaf toposes), so I’ve been reluctant to actually use the term “trivial fibration”. At least, not until I get to a point where I can show everything is sufficiently model-structure-like. Am I being too pedantic and should use the phrase anyways?
I guess I should clarify what I mean about “sufficiently model-structure-like”. While the $\mathfrak{L}$-fibrant objects can be easily characterized as those objects $X$ where both endpoint evaluations $X^L \to X$ are trivial fibrations… what I’m missing is an elementary construction of fibrant replacements. So, in the case of presheaf toposes I can invoke Cisinski to say they exist and it all assembles into a model structure, but I don’t know about general toposes, especially non-Grothendieck ones. So, I’ve been wanting to avoid using that language outside of the presheaf topos case.
And I wanted to work out a better understanding of how it all works before I feel confident unilaterally declaring a definition of $\mathfrak{L}$-fibrant object for elementary toposes.
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