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Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$):

$Maps(S, LocSys_G(X)) := \text{Maps}(S \times X_{dR}, \text{pt}/G)$

Here $X_{dR}$ is defined in pg $12$ (see also the end of pg $3$) of this paper via: $\text{Maps}(S, X_{dR})= \text{Maps}((\pi_0(S)^{red}), X)$, where $\pi_0(S)$ is the reduced version of the classical scheme $\pi_0(S)$ corresponding to the $DG$ scheme $S$.

With this definition, how can we recover the "classical" stack $\text{LocSys}_{G}(X)$? The ``classical" stack is given by defining $\text{Hom}(S, \text{LocSys}_G(X))$ to be the groupoid of $G$-bundles on $S \times X$ with a flat connection along X. In the paper, it is stated on pg $103$ that we can recover the classical stack from Lemma $10.1.3$; but I don't understand how.

Q2: I was wondering what references are best to learn about DG (differential graded) schemes? Also, are there any good references for $\infty$-groupoids?

The reason I'm asking is that I've been trying to learn the definition given here of a pre-stack as a contravariant functor $\textbf{DGSch} \rightarrow \infty-\textbf{Grpd}$, and why it is a more suitable notion than the ordinary definition of a pre-stack as a contravariant functor $\textbf{Sch} \rightarrow \textbf{Grpd}$.

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    $\begingroup$ You applied $\pi_0$/red to the wrong object. It should be $Maps(\pi_0(S)^{red},X)$. You may be interested in the links here: ncatlab.org/nlab/show/de+Rham+space $\endgroup$
    – Ben Wieland
    Commented Jun 2, 2012 at 17:26
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    $\begingroup$ Are you sure your definition of the ``classical'' stack is correct? I feel like it should be something more like G-bundles on $S\times X$ with a flat connection along $X$. This would fit better with the Arinkin-Gaitsgory definition. $\endgroup$
    – Sam Gunningham
    Commented Jun 2, 2012 at 19:23
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    $\begingroup$ Sam is right. The correct definition is '$G(S)$-local systems on $X$' (which is then mapped to Sam's definition via Riemann-Hilbert). For question 1 it may be useful to know the description of $QCoh(S\times X_{dR})$ if $X$ is smooth: it is equivalent to the category of quasi-coherent sheaves on $S\times X$ with a $D_X$-action (for example, see Lurie's notes on crystals). $\endgroup$
    – Pavel Safronov
    Commented Jun 2, 2012 at 21:04
  • $\begingroup$ Thanks, Ben & Sam; I've edited it now. Ok; I'll have a look at the notes on crystals. $\endgroup$
    – Puraṭci Vinnani
    Commented Jun 3, 2012 at 0:55

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