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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.
added (here) proof of the first example, slightly expanded and beautified the list of the following examples, and cross-linked with the discussion of right properness at local model structure on simplicial presheaves
With the definitions given, it is not clear that locally cartesian closed model categories are slice-wise cartesian closed model categories, as the terminology would suggest:
The axioms given here at locally cartesian closed model category readily imply the required pushout-produt axiom and pullback-power axiom only “in the second variable”.
Does the full pullback-power axiom follow with more work?
searching, I found about half a dozen publications that use the term “locally cartesian closed” for model categories, but I saw None that would state the definition or cite the $n$Lab page, or give any other indication of what is meant (though I gather they all took the definition from the $n$Lab or the discussion surrounding it)
[ removed duplicate ]
The requirement that the internal hom in the slice category is a Quillen bifunctor only for fibrant targets is particularly weird to me.
I would expect that the “correct” definition of a locally cartesian closed model category should postulate that the slice internal hom is a Quillen adjunction in three variables (including the base object as an additional variable).
As a special case, if the base object is fibrant, this will imply the current version.
Huh. I hadn’t realized until reading comment #11, but I think regarding LCCCs that I’m far more interested in the existence of a right adjoint to $f^*$ than I am in the existence of the internal hom in the slices. I’m not sure I can really opine on what is “correct”, but I think a definition oriented around that property is at least reasonable.
In most examples (in fact, all the nontrivial ones I can think of off the top of my head) it’s not true that the monoidal structure of the slice categories is a Quillen two-variable adjunction. For instance, if all objects are cofibrant, then this property would imply that acyclic cofibrations are stable under pullback along arbitrary morphisms, which is very rare.
Mike, thanks. So I have added a remark here warning that the LCC-model axioms do not imply the CC-model axioms.
I came to this from wondering whether an LCC model category equipped with a Quillen reflection+coreflection (hence with a model category version of $\flat \dashv \sharp$) would become enriched over its $\flat$-objects with enriched hom being $\sharp [-,-]$.
I have made explicit where in the text the assumption/condition plays a role that $g \colon A \to B$ be a fibration between fibrant objects (namely here and here).
On a different note: where it said:
Modulo questions of strictness and coherence (see identity type for details) …
I have changed it to:
… (see initiality conjecture for details) …
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