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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 9th 2021

starting something, meaning to record a transparent proof of an adjoint equivalence that becomes a Quillen equivalence for simplicial presheaves (haven’t seen this discussed anywhere).

But not done yet, just need to save…

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 9th 2021

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 11th 2021

Have typed out the argument (here) that the simplicial version of the adjoint equivalence is a Quillen adjunction.

Still need to show that it’s a Quillen equivalence.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 11th 2021

Finally I have typed out also the proof of the Quillen equivalence property (here)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 11th 2021

Have now added a concluding section (here) with the statement seen in $\infty$-category theory.

Also adjusted the Idea-section, to reflect this completion of the proofs.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 11th 2021

Added mentioning (here) of the application to the proof of cohesion of global- over G-equivariant homotopy theory.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 9th 2022
• (edited Jul 9th 2022)

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 hom-functors with colimits, still hold as natural equivalences of $\infty$-groupoids).

I have added a brief remark on this here.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 10th 2022

I have slightly adjusted and re-arranged throughout the entry, for better readability (hopefully).

• CommentRowNumber9.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 10th 2022

### In simplicial presheaf theory

{#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 Čech-local 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}