Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?


Let me take an example from notes of Bertrand Toen, page 41 of <https://arxiv.org/pdf/math/0604504.pdf>.


Let $C$ be a smooth projective curve and consider the functor $i: \mathrm{St}(k) \to \mathrm{dSt}(k)$. Take $X$ to be the mapping stack
$$
X = \mathrm{Map}(C,BG) = \mathrm{Bun}\_G(C).
$$
The derived enhancement is
$$
\mathbb{R}X = \mathbb{R}\operatorname{Map}(i(C), i(BG)),
$$
and $X$ is called a *truncation* of $\mathbb{R}X$. I am therefore requesting examples where this truncation procedure is trivial!


My naive hope is that this is the case since $\dim(C) = 1$ and so there are no higher cohomologies/homotopies above degree 1.



Examples when algebraic 1-stack = derived enhancement?
If $S$ is a smooth projective variety of dimension $d$, then the derived stack $X=Bun\_G(S)$ has cotangent complex perfect of amplitude $[-(d-1), 1]$. If $S=C$ is a curve then it is in $[0,1]$, in particular $H^\*(\mathbb{L}\_X)=0$ for $\*<0$. For stacks, this condition (+ finite presentation) is exactly smoothness.


Smooth morphisms $X \to Y$ are flat, which in the derived context means flatness on classical truncations but also $\pi\_i(O\_Y) \otimes\_{\pi\_0(O\_Y)} \pi\_0(O\_X) \simeq \pi\_i(O\_X)$ on higher homotopy. In particular, saying that a derived stack is smooth over the base field $k$ automatically implies that its derived structure is trivial. I would refer you to the original references on DAG, such as [HAG II, 2.2.2].


For $X = Higgs\_G(S)$ we typically have one extra obstruction $H^{-d}(\mathbb{L}\_X) \ne 0$ compared to $Bun\_G(S)$. For $S=C$ a curve this means that $X$ is only quasi-smooth in general. The extra obstruction $H^{-1}$ does vanish in certain special cases, e.g. if $G$ is semisimple, see Beilinson-Drinfeld "Quantization of Hitchin..." 2.1.2 and also Halpern-Leistner's paper on Verlinde for Higgs, Prop 3.14.


