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I’ve added a definition to locally cartesian closed model category, although I’m open to debate about whether this is the right definition. This definition is more or less exactly what one needs to interpret dependent products in type theory with function extensionality (I plan to add a proof of this). But it’s certainly less obviously correct from a pure model-categorical viewpoint. For one thing, it doesn’t imply that we have a cartesian closed model category, which one would naively expect a notion of “locally cartesian closed model category” to do.
after “Quillen adjunction” I have added the words “between the corresponding slice model structures”.
I have started a Properties with a remark that, dually, for $Q \to B$ a fibration between fibrant objects, we have that $[Q, -]_{\mathcal{C}/_B}$ is right Quillen.
I found myself thinking again about examples of right proper local simplicial (pre)sheaf model categories.
For $C$ a site, the Joyal model structure on simplicial sheaves over $C$ is not in general right proper. But what if $C$ has enough points?
Let’s see. Then then weak equivalences are the stalkwise weak equivalences. And since forming stalks preserves finite limits, we can deduce some extra properties.
Let $g : A \to B$ be a Joyal fibration of simplicial sheaves. This is in particular a Jardine fibration, hence in particular an objectwise fibration of simplicial sets.
Then for $f : X \to B$ a weak equivalence of simplicial sheaves, its pullback $g^* f$ along $g$ is a weak equivalence iff all its stalks $x^* (g^* (X \stackrel{f}{\to} B))$ are. But since forming stalks preserves pullbacks, we have pullback diagrams of simplicial sets
$\array{ x^*(g^* X) &\to& x^* X \\ \downarrow^{\mathrlap{x^* (g^* f)}} && \downarrow^{ \mathrlap{x^* f}} \\ x^* A &\stackrel{x^* g}{\to}& x^* B } \,.$Since here the right morphism is a weak equivalence of simplicial sets, it would follow that $g^* f$ is a weak equivalence if $x^* g$ is a fibration of simplicial sets, by right properness of $sSet_{Quillen}$.
So we need that
$(x^* A)^{\Delta[k]} \to (x^* A)^{\Lambda[k]^i} \times_{(x^* B)^{\Lambda[k]^i} } (x^* B)^{\Delta[k]}$is an epimorphism. Since stalks commute with finite limits, this is equivalent to
$x^* \left( A^{\Delta[k]} \to A^{\Lambda[k]^i} \times_{ B^{\Lambda[k]^i} } B^{\Delta[k]} \right)$being an epimorphism. But the morphism in parenthesis is epi since $f$ is in particular an epimorphism of presheaves. So because left adjoints preserve epimorphisms, the statement follows.
This argument seems to imply that the Joyal model structure over a site with enough points is right proper.
That model structure presents only hypercomplete $(\infty,1)$-toposes, right?
Yes.
By the way, I found the statement meanwhile mentioned in a book by Olsson. I was a bit hesitant because I see Jardine always ever mention that the “motivic” simplicial presheaves (over the Nisnevich site) are right proper.
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