Abstract
We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of
finite type over an algebraically closed field of characteristic zero) is
locally equivalent to the derived intersection of two Lagrangian morphisms to a
$(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent
stack of a smooth classical scheme. This leads to the possibility of the
following viewpoint that is, at least to us, new: any $n$-shifted symplectic
derived scheme can be obtained, locally, by repeated derived Lagrangian
intersections in a smooth classical scheme.
We also give a separate proof of our main result in the case where the local
Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree
$(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is
in fact locally equivalent to the derived critical locus of a shifted function,
which we've been told is a folklore result in the field. We indicate the
implications of this for derived moduli stacks of sheaves on Calabi-Yau
fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$.