I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.

If we fix a derived $k$-Artin stack, and denote $(Sch^{aff})_{/X, sm}$ the full subcategory of $(Sch^{aff})_{/X}$ consisting of of those $ S \rightarrow X$ which are smooth as an $(k-1)$-representable map. Let's also denote $ QCoh(X)$ to be the evaluation at $X$ of the right Kan extension along the yoneda embedding of the functor that sends an an affine scheme $ Spec(A)$ to $ D(A)$ (or if you prefer $Mod_A$).

Then the claim that I would like to understand is that there is an equivalence of categories:

$ QCoh(X) \simeq lim_{(S \rightarrow X) \in ((Sch^{aff})_{/X, sm})^{op}} QCoh(S) $.

I'm having a hard time following the proof in loc. cit. Here are some specific problems I am having:

1) The first point I don't really see is why this is obvious for $k = 0$. Im guessing the point is that for $X$ an affine scheme the fully-faithful embedding

$(Sch^{aff})_{/X, sm} \hookrightarrow (Sch^{aff})_{/X}$

is cofinal because for any $ f: X' \rightarrow X \in (Sch^{aff}) $ the category

$ (Sch^{aff})_{/X, sm} \times_{(Sch^{aff})_{/X})} ((Sch^{aff})_{/X})_{f/}$

is contractible since it has a final object namely $X'\overset{f}{\rightarrow} X \overset{id}{\rightarrow}X$. Is this reasoning correct?

2) If what I wrote in 1) isn't bullshit, it seems this works more generally for $X $ an $n$-Artin stack for abtitrary $n$ and we'd be done? I must be missing something(s).

3) I really get lost after displayed equation (1.6). In particular:

(i) I don't understand the equation that follows the line "The validity of point (b) for $k -1$ is equivalent...."

(ii) I dont understand how to construct the "restriction" map in the last displayed equation in the proof.

(iii) Finally and most embarrassingly Im not totally clear on how the induction hypothesis was used.

Any comments or help on any of these points would be greatly appreciated :).


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