In Def 2.5.1 in " Derived Quot schemes" by Ciocan-Fontanine and Kapranov, we can find the notion of dg-manifolds.

In detail, let $X$ be a dg-scheme over $k$ ($k$ : an algebraic closed field of $ch(k) = 0$), then $X = (X^0, \mathcal{O}_X^{\bullet})$ is called a dg-manifold if $X^0$ is a smooth algebraic variety and $\mathcal{O}^{\bullet}_X$ is locally isomorphic (as graded sheaves ) to $\operatorname{Sym}_{\mathcal{O}_X^0}(\bigoplus_i Q^{-i})$ where the degree $-i$-part $Q^{-i}$ are locally free sheaf on $X^0$ of finite ranks.

(a) Can we consider an analogous definition in the case $X^0$ is infinite dimensional or $Q^{-i}$ is infinite rank ? (Edit: Any reference is also welcome.)

(b) Moreover, if we can define that notion, are there any conditions such that an infinite dg-manifold can be quasi-isomorphic to some dg-manifolds defined by Ciocan-Fontanine and Kapranov.

At least, I will treat the example whose classical part is finite dimensional and their tangent complexes have finite cohomology at each point. (Edit: I am also considering fixing constructions of derived quot schemes by them because that constructions are failed according to this paper)

Edit: Can we also extend (b) in dg-stacks ?

In particular, I'm interested the following case:

Let $A$ be a graded $k$-algebra and $V = \oplus_{i \geq 0} V_i$ be graded $k$-vector space (I assume that each $V_i$ is finite dimensional but $V$ is not necessarily. ).

We consider $L := \oplus_{i \geq 0} L^i := \oplus_{i \geq 0} {Hom}_{k-gr}(A^{\otimes n} \otimes_k V,V)$. Then, we have a dg-lie algebra structure on $L$. In detail, the. differential $d : L_n \rightarrow L_{n+1}$ is given by $d\mu(a_1 \otimes \cdots \otimes a_{n+1} \otimes v ) = \sum_i (-1)^{n-i}\mu(\cdots \otimes a_ia_{i+1} \otimes \cdots \otimes v)$. The bracket is given by commutators of compositions of linear maps. We also have the group action of $G = \prod GL(V_i)$ on $L$ by conjugates(Then, the Lie algebra of $G$ is $L_0$).

Next, we consider $L_1 := Spec(Sym(L_1^{\vee}))$. Then, we have a bundle of curved dg lie-algebra $\mathcal{L} = \oplus_{i>1} \mathcal{L}_i = \oplus_{i>1} L_i \times L_1$. The curvature map is given by $f(-) = d(-) + \frac{1}{2}[-,-]$. The differential on the fiber $\mu$ is given by $d(-) + [\mu,-]$. The bracket is equal to that of $L$.

Finally, $\mathcal{L}$ descends to $[L_1/G]$.

What I want to study is the dg-stack $(\mathcal{X}=[L_1/G], \mathcal{O}_{\mathcal{X}}^\bullet= Sym(\mathcal{L}[1])^\vee)$ (cf. Section 1 in this paper)

I am interested in the case of dg-stacks constructed from dg Lie algebras (i.e, for any given $L := \oplus_{i \geq 0 }L_i$, the underlying stack is $\mathcal{X}= [L_1/ \text{exp}(L_0)]$ and the corresponding sheaf $\mathcal{O}_{\mathcal{X}}^\bullet$ of dg-algebras on $\mathcal{X}$ is a descent of the sheaf of dg-algebras on $L_1$. From this construction, $(\mathcal{X}, \mathcal{O}^{\bullet}_{\mathcal{X}})$ can be called a "infinity dimensional smooth dg manifold" when $\mathcal{X}$ is infinite dimensional or $\mathcal{O}_{\mathcal{X}}^\bullet$ is generated by infinite rank bundles as above. See, also Section 1 in this paper).

My question is the following: When the the classical part of $(\mathcal{X}, \mathcal{O}^{\bullet}_{\mathcal{X}})$ is finite dimensional and has tangent complexes with finite dimensional cohomology at each point, is this quasi-isomorphic to a smooth dg-stack defined by Ciocan-Fontanine and Kapranov in page 31 in "Derived Hilbert schemes"?

Edit(5/9): In the above case, I especially interested in when both $L_1$ and $G$ are infinite dimensional. Then, we may be able to replace the structure dg-sheaves $\mathcal{O}^{\bullet}_{[L_1/G]}$ with $Sym_{\mathcal{O}_{[L_1/G]}}(Q^{'-i})$ with each $Q^{'-i}$ finite rank if we consider replacing the corresponding bundles on $L_1$. However, I have no idea how can $[L_1/G]$ be replaced with a finite dimensional model (for example, $[L'_1/G']$ with $L'_1,G'$ finite dimensional).

Edit(5/7): In this paper, dg-stacks of logarithmic flat connections is treated. By using bundles of curved dgla, the dg-moduli stack is constructed. But, it is infinite type (in detail, the gauge group is infinite dimensional). The author construct a replacement which is a finite dimensional model.

His approach seems to depend on the subject he is dealing with. Can we make such replacements in more general situations ?