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

Any comments and suggesting references are welcome !

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

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(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 ?

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 introduction of this paper)

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 (i.e, $[L'_1/G']$ 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 ?

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

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.

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

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.