TITLE: Deligne's letter to Millson
QUESTION [20 upvotes]: The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan formalism indeed produces a deformation problem and conversely for most deformation problems the dg-Lie algebra is well known. In the derived setting this principle has even become a theorem following Lurie-Pridham. However, I was wondering if anyone could explain the original intuition Deligne had behind this principle?

REPLY [8 votes]: I suspect, Deligne's intuition went along the following lines. Deformation theory describes the
tangent cone at a point $x$ of the moduli space $M$ of the
problem. The tangent cone is Spec Gr $\mathcal{O}_{M,x}$, where the
associated graded Gr is taken with respect to powers of the maximal
ideal of the local ring $\mathcal{O}_{M,x}$, the stalk of the structure sheaf
$\mathcal{O}_{M}$ at $x$. Usually, this cone is singular. You may resolve this
singularity within derived algebraic geometry, for instance, find a
free dg-commutative algebra whose cohomology is Gr $\mathcal{O}_{M,x}$. A free
dg-commutative algebra is equivalent to an $L_\infty$-algebra. Then you take a
quasi-isomorphic dg-Lie algebra, and you are done.