In [Def 2.5.1 in " Derived Quot schemes"][1] 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.

Can we consider an analogous definition in the case $X^0$ is infinite dimensional or $Q^{-i}$ is infinite rank ?

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.
(I am also considering fixing constructions of derived quot schemes by them because that constructions are failed according to [this paper][2])


  [1]: https://arxiv.org/abs/math/9905174
  [2]: https://arxiv.org/abs/1908.03021