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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.