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Just a note for later when the page can be edited again:
A model-category version of the “fundamental theorem of $\infty$-topos theory” is also discussed in:
Finally coming back to this.
I am looking for the generalization of the equivalence (of $\infty$-categories)
$PSh(\mathcal{S})_{/y(X)} \underoverset {\sim} { \Gamma_{(-)}(-) } {\longrightarrow} PSh \big( \mathcal{S}_{/X} \big)$to the case that $X$ is not necessarily an object of $\mathcal{S}$ but is an object of $PSh(\mathcal{S})$ (hence omitting the “$y(-)$” in the above formula), and interpreting $\mathcal{S}_{/X}$ as the evident sub-category of $PSh(\mathcal{S})_{/X}$.
I vaguley thought it was proven in this generality in HTT, but now I don’t find it there, and maybe it isn’t. (Corollary 6.3.5.1 in HTT does generalize the above statement, but in another direction).
In the special case that $X \,\in\, PSh(\mathcal{S})$ is 1-truncated, the statement is essentially Thm. 4.4 of Hollander’s article above (after identifying $\mathcal{S}_{/X}$ with the Grothendieck construction that she refers to).
For a while this was all I needed. But now I need the statement for $3$-truncated $X$, and it shouldn’t depend on the truncation of $X$ at all.
Is there any reference?
The equivalence of these presheaf categories is corollary 5.1.6.12 of HTT; the last thing proved in the section on complete compactness. However, the equivalence constructed there is the colimit-preserving functor extending the map $C_{/p} \to P(C)_{/yp}$.
(I’m confused. I’m pretty sure I’ve made an error in the following, but I can’t find it. It’s probably just reversing an orientation so I’m leaving it up)
For completeness, let $p : J \to C$ be such that $X = \lim(yp)$. $C_{/p} \to C$ is a pullback of $(C^J)^{[1]} \to C^J \times C^J$, and so it is contravariant Grothendieck construction $\overline{el}_C( C^J(\Delta-, p))$. However,
$C^J(\Delta-, p) \simeq PSh(C)^J(y \Delta -, yp) \simeq PSh(C)^J(\Delta y -, yp) \simeq PSh(C)(y-, X) \simeq X$so $C_{/p} \simeq \overline{el}_C(X)$, and the isomorphism is $PSh(\overline{el}_C(X)) \simeq PSh(C)_{/X}$.
I guess the part that threw me off is that 5.1.6.12 isn’t actually full generality like I had remembered; I guess it only applies when $X$ is a limit of representable functors.
Thanks for looking into this! Would be a great issue to settle.
Other people went through thinking that HTT 5.1.6.12 (right, that’s what I had meant to point to) gives the answer – e.g. MO:a/86209.
Myself, I don’t see, either, how HTT 5.1.6.12 would give the answer (which is what I meant by saying that it “generalizes in a different direction”) – but I’d be happy to be educated if it does.
Alternatively one could try to figure out if Hollander really needs her assumption that $X$ be 1-truncated. Possibly the only reason for this condition is that it allowed her to think of the slice in classical terms. But I haven’t dug through her proof yet.
Okay, here’s a way to establish the equivalence. The isomorphism $PSh(C_{/c}) \to PSh(C)_{/yc}$ when applied to an arrow $f : a \to b$ is the dependent sum $PSh(C)_{/ya} \to PSh(C)_{/yb}$ and the left Kan extension $PSh(C_{/a}) \to PSh(C_{/b})$. (when viewing $PSh$ as $Fun(-, \infty Gpd)$)
Taking local right adjoints, an arrow $a \to b$ acts on $PSh(C)_{/yb} \to PSh(C)_{/ya}$ by taking pullbacks and on $PSh(C_{/b}) \to PSh(C_{/a})$ by the restriction map (along the composition map $C_{/a} \to C_{/b}$).
Let $p : J \to C$ be such that $X = colim(yp)$. Then we have an isomorphism $\lim_j PSh(C_{/p(j)}) \simeq \lim_j PSh(C)_{/yp(j)}$. On the left side, we have
$\lim_j PSh(C_{/p(j)}) \simeq PSh(\colim_j C_{/p(j)}) \simeq PSh(\overline{el}_C(X))$where the latter isomorphism is given by computing the colimit in right fibrations over $C$ via the Grothendieck construction. On the right side, we have
$\lim_j PSh(C)_{/yp(j)} \simeq PSh(C)_{/\colim_j yp(j)} \simeq PSh(C)_{/X}$by applying HTT proposition 6.3.5.14 which implies slicing gives a limit-preserving functor $PSh(C)^{op} \to LTop$ and HTT 6.3.2.3 which says limits in $LTop$ are computed in $\widehat{\infty Cat}$.
I was hoping there was a simpler proof in terms of right fibrations; the proposition we’re asking for is that given a right fibration $X \to C$, we have
$RFib(C)_{/X} \simeq RFib(X)$These are full subcategories of $(\infty,1)Cat_{/X}$ spanned by, respectively:
It’s not true that these are the same subcategory is it? This would imply that if $g$ and $gf$ are right fibrations, then $f$ is a right fibration, but I didn’t see anything like that flipping through HTT or the nLab pages.
So, if these are isomorphic $\infty$-categories, the isomorphism $RFib(X) \to RFib(C)_{/X}$ would have to be more complicated than just composing with $X \to C$.
Actually, I think that cancellation property is true. Characterizing right fibrations by the homotopy lifting property, given $f : A \to B$ and $g:B \to C$, we have
(for some reason, tikzcd wasn’t working so I couldn’t draw the diagram)
This equivalence is very easy to implement as a Quillen equivalence of simplicial presheaves and their slice categories.
First, the functor Γ:PSh(S)/X⟶PSh(S/X) is a right Quillen functor between projective model structures. This follows from the fact that fibrations and acyclic fibrations are stable under base changes in any model category.
Next, the left adjoint functor PSh(S/X)→PSh(S)/X sends Čech nerves of open covers in S/X to Čech nerves of open covers in PSh(S). Thus, the Quillen adjunction descends to a Quillen adjunction between local projective model structures.
Finally, we have to show that the derived unit and derived counit of this Quillen adjunction are weak equivalences.
To this end we observe first that the right adjoint functor preserves weak equivalences and homotopy colimits. This implies that the derived unit and derived counit maps depend homotopy cocontinuously on the object. Thus, to show they are weak equivalences, it suffices to do it for the appropriate sets of (homotopy) generators.
Given a cofibrant object $A\in PSh(S/X)$, we can take it to be representable, in which case the unit map is an isomorphism.
Given a fibrant object $B\in PSh(S)/X$, we can take its underlying object in $PSh(S)$ to be representable, in which case the counit map is an isomorphism.
Thanks for all this. I’ll think about it.
Meanwhile, I was wondering about a general abstract argument. The following should be the easy proof in every context in which “category theory works”:
Using
$\big( X, \, f_X \big) \;\simeq\; \Big( \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \, s_X(i) ,\, (f_{s_X(i)})_{i \in \mathcal{I}} \big) \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \big( s_X(i) ,\, f_{s_X(i)} \big)$we check fully faithfulness by the following sequence of natural equivalences:
$\begin{array}{l} PSh(\mathcal{S})_{/B} \Big( \big( X, \, f_X \big) \,, \big( X ,\, f_{X'} \big) \Big) \\ \;\simeq\; PSh(\mathcal{S}) \big( X \,, X' \big) \underset{ PSh(\mathcal{S}) \big( X \,, B \big) }{\times} \{f_X\} \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, PSh(\mathcal{S}) \big( s_X(i) \,, X' \big) \underset{ \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, PSh(\mathcal{S}) \big( s_X(i) \,, B \big) }{\times} \Big\{ \big( f_{s_X(i)} \big)_{i \in \mathcal{I}} \Big\} \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \bigg( PSh(\mathcal{S}) \big( s_X(i) \,, X' \big) \underset{ PSh(\mathcal{S}) \big( s_X(i) \,, B \big) }{\times} \big\{ f_{s_X(i)} \big\} \bigg) \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \bigg( \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} PSh(\mathcal{S}) \big( s_X(i) \,, s_{X'}(i') \big) \underset{ PSh(\mathcal{S}) \big( s_X(i) \,, B \big) }{\times} \big\{ f_{s_X(i)} \big\} \bigg) \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} \, \bigg( PSh(\mathcal{S}_{/B}) \Big( \big( s_X(i) ,\, f_{s_X(i)} \big) \,, \big( s_{X'}(i') ,\, f_{s_{X'}(i')} \big) \Big) \bigg) \\ \;\simeq\; PSh(\mathcal{S}_{/B}) \Big( \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \big( s_X(i) ,\, f_{s_X(i)} \big) \,, \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} \big( s_{X'}(i') ,\, f_{s_{X'}(i')} \big) \Big) \bigg) \\ \;\simeq\; PSh(\mathcal{S}_{/B}) \Big( \big( X ,\, f_x \big) \,, \big( X' ,\, f_{X'} \big) \Big) \bigg) \end{array}$On the other hand, the previous proof over a representable which I had typed up In October 2021 (here at slice of presheaves is presheaves on slice) works verbatim also for non-representable base objects, and in this case gives the adjoint equivalence whose right adjoint is given by pullback.
This is what you (Dmitri) are appealing to in the first statement of #11:
First, the functor Γ:PSh(S)/X⟶PSh(S/X) is a right Quillen functor between projective model structures. This follows from the fact that fibrations and acyclic fibrations are stable under base changes in any model category.
and that is the kind of argument that has been being spelled out in the above entry here. There I had found it necessary to add a few more observations to really exhibit a simplicial Quillen equivalence for enriched simplicial presheaves over an $sSet$-site, but it’s all straightforward, of course.
What I had not done before is type up any generalization of this argument to non-trivial Grothendieck topologies, which is of course the bulk of the argument in #11 above. Would be nice to type this up in the $n$Lab entry here on the “fundamental theorem”.
I added the Čech-local argument to slice of presheaves is presheaves on slice. At some point the article switches from arbitrary presheaves to representable presheaves, for no particular reason, perhaps it should be generalized?
There must be some gap in the argument in #11.
Namely, consider the example (here) where the simplicial site is trivial, $S = \ast$, and where the simplicial presheaf $X$ being sliced over is not 0-skeletal, e.g. $X = \overline{W}G$ for a non-trivial group $G$. Then the slice site is still trivial, $\ast/\overline{W}G \,=\,\ast$, since its hom-objects are computed by plain pullbacks which don’t pick up the higher cells in $X$.
So it won’t be true that $sPSh(S)/X$ is (Quillen) equivalent to $sPSh(S/X)$ in this case. Some fibrancy condition on the site slicing is missing.
The problem is the appeal to the enriched slice category. It needs to be replaced by the “discrete enriched Grothendieck construction”. I’ll try to make the necessary edits at slice of presheaves is presheaves on slice.
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