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I have now typed out full proof of the adjoint equivalence.
It follows (not in the entry yet) with K. Brown’s lemma that for simplicial presheaves this adjoint equivalence is a Quillen adjunction for the projective model structure and the slice of that.
Still need to think about full proof that this is a Quillen equivalence. But need to interrupt now.
Added mentioning (here) of the application to the proof of cohesion of global over Gequivariant homotopy theory.
Following discussion in another thread (here)…
…I have added (here) a section which proves the equivalence of categories in the generality where the base object need not be representable.
Strictly speaking, the statement proven in this case is (currently) a little weaker than that given over a representable, (which proves an adjoint equivalence with identification of the right adjoint as the functor of sections). But the proof is general abstract and applies verbatim also in $\infty$category theory (where all the required properties, such as the interplay of homfunctors with colimits, still hold as natural equivalences of $\infty$groupoids).
I have added a brief remark on this here.
Added:
{#InSimplicialPresheafTheory}
We now promote the Quillen equivalence in the previous section to the case of Čech[[local model structures on simplicial presheaves].
Recall that these are obtained as a left Bousfield localization of the (say) projective model structure on simplicial presheaves with respect to Čech nerves of covering families.
We reuse the notation of the previous section.
\begin{proposition}
The Quillen equivalence of \ref{SimplicialLocalSectionsIsRightQuillen}
descends to a Quillen equivalence of the corresponding Čechlocal projective model structures.
\begin{tikzcd}
\big(
\mathrm{sPSh}(\mathcal{C}){/y{\mathcal{C}}(X)}
\big){\mathrm{\v Cech,proj}}
\ar[
rrr,
shift right=8pt,
“{
\mathrm{PSh}(\mathcal{C}){/y(X)}
\big(
(y_{\mathcal{C}}){/X}()
,,

\big)
}”{below}
]
&&&
\mathrm{sPSh}
\big(
\mathcal{C}{/X}
\big)_{\mathrm{\v Cech,proj}}
\ar[
lll,
shift right=8pt
]
\ar[
lll,
phantom,
“{ \scalebox{.6}{$\simeq_{\mathrlap{\mathrm{Qu}}}$} }”
]
\end{tikzcd}
\end{proposition}
\begin{proof} Both model categories are left proper and combinatorial. Therefore we can take left Bousfield localizations with respect to arbitrary sets of morphisms.
We localize both sides with respect to Čech nerves of respective covering families. Observe that Čech nerves of covering families in $\mathcal{C}_{/X}$ are mapped to Čech nerves of covering families in $sPSh(\mathcal{C})$ and therefore also in the slice category $sPSh(\mathcal{C})_{y_{\mathcal{C}}(X)}$. Thus, we have an induced Quillen adjunction between localized model categories.
It remains to show that this Quillen adjunction is a Quillen equivalence.
It suffices to show that the right adjoint reflects weak equivalences between fibrant objects. Here fibrant objects are objectwise Kan complexes that satisfy the appropriate variant of the homotopy descent property. Local weak equivalences between locally fibrant objects coincide with objectwise weak equivalences. As established in the previous section, the right adjoint functor reflects objectwise weak equivalences between objectwise fibrant presheaves, which completes the proof. \end{proof}
I have added (here), as a simple but fun example of the “fundamental theorem of $\infty$presheaf $\infty$topos theory”, that it gives the identification of the slice over delooping $\infty$groupoids with $\infty$group actions:
$\array{ \big(Grpd_\infty\big)_{/\mathbf{B}\mathcal{G}} \,\simeq\, \big( PSh_\infty(\ast) \big)_{/\mathbf{B}\mathcal{G}} & \underoverset {\phantom{}\sim\phantom{}} {\Gamma} {\longrightarrow} & PSh_\infty \big( \ast_{/\mathbf{B}\mathcal{G}} \big) \;\simeq\; PSh_\infty( \mathbf{B}\mathcal{G} \big) \;\simeq\; \mathcal{G} Act \big( Grpd_\infty \big) \,. }$This example shows that there is a gap in the previous argument that the proof in the entry generalizes immediately to slicing over nonrepresentables.
Namely, take, for instance, the simplicial site to be the terminal category $\mathcal{C} = \ast$; and take the simplicial preasheaf over this site to be $\overline{W}G \,\in\, sSet \,=\, sPSh(\ast)$ for a nontrivial group $G$.
Then the slice enriched category is still trivial: $\ast\big/\overline{W}G \,=\, \ast$, since the hompullback does not pick up the higher cells in $\overline{W}G$.
Therefore $sPSh(\ast)/{\overline{W}G}$ is not equivalent to $sPSh\big( \ast/\overline{W}G \big)$, contrary to what the above discussion is assuming.
[edit: 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 edits. ]
I have accordingly adjusted the material on simplicial presheaves:
The previous material I have given a section header “Over 0truncated objects” (now here).
The pointers to Hollander’s article I have (slightly expanded and) put under “Over 1truncated objects” (now here).
In another direction, I guess over 0connected simplicial presheaves, one could regard the coefficient sSets as oneobject sSetcategories and then model the proper slice sSetsite by the sSetenriched Grothendieck construction from Beardsley & Wong 2018, Def. 4.2 .
But in full generality, without any truncation of connectedness assumptions, I am afraid one does need to invoke some straightening yoga.
Random aside: did the meaning of “slice” change over time? I.e. given a functor $p : X \to C$, I’m reasonably certain I’ve seen the comma category $(C \downarrow p)$ called the slice category in the past, including here at nLab, rather than as the category of cones over $p$ which seems the standard today.
But maybe I just got mixed up by seeing the notation $C/p$ used for both notions and there was never the term “slice” attached to it when used for commas.
Re #12: There is an easy way to define the slice category of a simplicial presheaf without any straightening.
Suppose $F$ is a simplicial presheaf on a small category $C$.
We define the slice site $C/F$ as follows.
Objects are morphisms $\sigma\colon\Delta^n\times y(c)\to F$, i.e., elements of the set $F(c)_n$.
Morphisms $([n],c,\sigma)\to([n'],c',\sigma')$ are given by pairs $(f\colon[n]\to[n'],g\colon c\to c')$ such that $F(g)(c')_f=c$.
The Grothendieck topology on $C/F$ is induced by the forgetful functor to $C$.
Now consider the category of simplicial presheaves on $C/F$. Equip it with the projective (or injective) model structure, then left Bousfield localize at Čech covers (or hypercovers), then further left Bousfield localize at morphisms of representable presheaves induced by morphisms $(f,g)$ in $C$ such that $g$ is the identity map.
It is not difficult to show that the resulting model category $M$ is Quillen equivalent to the slice model category of $F$.
The left Quillen equivalence $M\to sPSh(C)/F$ sends the representable presheaf of $([n],c,\sigma)$ to the object in $sPSh(C)/F$ given by the morphism $\sigma\colon\Delta^n\times y(c)\to F$.
I see, thanks. What I’d need is is proof of the fact that the equivalence is induced by forming homotopy sections. This becomes a little opaque with the category of simplices.
Re #15: The right adjoint functor R to the left adjoint functor I described above indeed sends an object G in the slice category sPSh(C)/F to its sections over $\Delta^n\times y(c)$ (which become homotopy sections once you pass to right derived functors):
$R(G)(\sigma\colon\Delta^n\times y(c))=Map(\sigma,G).$In particular, after right deriving,
$R(G)(\sigma\colon\Delta^n\times y(c))$no longer depends on $n$ in the following precise sense: taking any vertex of $\Delta^n$ produces a weakly equivalent fiber
$R(G)(\sigma\colon\Delta^0\times y(c)).$While I haven’t yet checked the Quillen equivalence that you are claiming now, what puzzles me at the moment is that it would seem to imply that all slices of 1localic $\infty$toposes (i.e. with a 1site of definition) are again 1localic (have again a 1categorical site of definition).
But a slice $\infty$topos over an object that is not $n$truncated should be at least $n+1$localic (e.g. here). What am I missing here?
all slices of 1localic ∞\inftytoposes (i.e. with a 1site of definition) are again 1localic (have again a 1categorical site of definition).
No, it does not. Observe that I also left Bousfield localized at morphisms of representable presheaves induced by morphisms (f,g) in C such that g is the identity map.
This produces a relative site, i.e., a site equipped with a notion of a weak equivalence, equivalently, a relative category equipped with a coverage.
Of course, you can take its Dwyer–Kan hammock localization and convert it to a simplicial site. However, the resulting site is not 1categorical.
Additionally, the ∞topos of a 1site need not be 1localic, if that site does not have finite limits.
I see, thanks!
Yeah, I had missed that you have a nontrivial Bousfield localization in #14 even in the case that the original topology was trivial. That explains it.
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