**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 $\textbf{Sch} \rightarrow \textbf{Grpd}$.