DG Lie algebras and derived deformation theory

Question

As far as I understand it, in recent years there has been a lot of progress on generalizations of classical deformation theory in characteristic 0 using tools such as simplicial deformation functors or ∞-categories, generalizing from characteristic 0 to positive characteristic and from deformations over local Artinian $k$-algebras to "derived" deformation functors, defined on DG analogs of Artinian $k$-algebras.

I have the following naive question about these generalizations.

Given a classical deformation problem in characteristic 0 (such as deformations of associative algebras or complex manifolds) "controlled" by a DG Lie or L_∞ algebra $L$, in the sense that the deformation functor attached to this deformation problem (defined on the category of local commutative Artinian $k$-algebras) is isomorphic to the deformation functor attached to $L$, to what extent does $L$ encode the "derived" deformation theory of the original deformation problem?

What would be the steps and obstructions to turning the classical deformation functor associated to $L$ (from local commutative Artinian $k$-algebras to sets, say) into a "suped-up" version — for example as functor from the ∞-category of Artinian $\mathbb E_\infty$-algebras to the ∞-category of spaces as in Lurie's formulation of derived deformation theory?

Answer 1

Once you have your DGLA $L$, the simplest description is consider the functor from commutative dg Artinian local $k$-algebras $A= k \oplus \mathfrak{m}_A$ in non-positive cochain degrees to simplicial sets given by Hinich's simplicial nerve. Explicitly, in simplicial level $n$, you take Maurer-Cartan elements $\mathrm{MC}(L\otimes \mathfrak{m}_A \otimes \Omega^{\bullet}(\Delta^n))$. In particular, the DGLA completely determines a derived deformation functor.

Since the $\infty$-categories of connective Artinian local $E_{\infty}$-algebras and of cdgas as above are equivalent, this suffices to give a functor in Lurie's setup. Beware that the derived deformation functor is highly dependent on the choice of $L$, and is not determined by the classical deformation functor alone.

For the detailed description of this functor, see section 8 of Hinich https://arxiv.org/abs/math/9812034 . For the original proof that all simplicial derived deformation functors arise in this way, see https://arxiv.org/abs/0705.0344 or the notes on derived deformation theory here http://blog.poormansmath.net/lecture-notes/ ; another summary appears in https://ncatlab.org/nlab/show/model+structure+for+L-infinity+algebras .

If you're looking at Lurie's stuff, beware that it isn't as original as he first thought in his ICM address - see for instance Remark 0.0.14 of DAG X.