3 edited body; added 25 characters in body

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: $\text{Maps}(S, X_{dR})= \text{Maps}(S, (\pi_0(X))^{red})$, text{Maps}((\pi_0(S)^{red}), X)$, where$\pi_0(X)$\pi_0(S)$ is the reduced version of the classical scheme $\pi_0(X)$ \pi_0(S)$corresponding to the$DG$scheme$X$.S$.

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 $G$-local systems 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$\textbf{Sch} \rightarrow \textbf{Grpd}$. 2 added 33 characters in body; deleted 1 characters in body 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:$\text{Maps}(S, X_{dR})= \text{Maps}(S, (\pi_0(X))^{red})$, where$\pi_0(X)$is the reduced version of the classical scheme$\pi_0(X)$corresponding to the$DG$scheme$X$. With this definition, how can we recover the classical" "classical" stack$\text{LocSys}_{G}(X)$? It The classical" stack is given by defining$\text{Hom}(S, \text{LocSys}_G(X))$to be the groupoid of$G$-local systems on$S \times X$. In the paper, it is stated that on pg$103$that it should follow we can recover the classical stack from Lemma$10.1.3$, 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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# The De Rham Stack and $\text{LocSys}$

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: $\text{Maps}(S, X_{dR})= \text{Maps}(S, (\pi_0(X))^{red})$, where $\pi_0(X)$ is the reduced version of the classical scheme $\pi_0(X)$ corresponding to the $DG$ scheme $X$.

With this definition, how can we recover the classical" stack $\text{LocSys}_{G}(X)$? It is given by defining $\text{Hom}(S, \text{LocSys}_G(X))$ to be the groupoid of $G$-local systems on $S \times X$. In the paper, it is stated that on pg $103$ that it should follow 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 $\textbf{Sch} \rightarrow \textbf{Grpd}$.