higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Given a deformation context $(\mathcal{Y}, \{E_\alpha\}_\alpha)$, the (∞,1)-category of formal moduli problems over it is the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-presheaves over $\mathcal{Y}^{inf}$
on those (∞,1)-functors $X \colon \mathcal{Y}^{inf} \to \infty Grpd$ such that
over the terminal object they are contractible: $X(*) \simeq *$ (hence they are anti-reduced);
they preserve (∞,1)-pullbacks (are infinitesimally cohesive)
This means that a “formal deformation problem” is a space in higher geometry whose geometric structure is detected by the “test spaces” in $\mathcal{Y}^{op}$ in a way that respects gluing (descent) in $\mathcal{Y}^{op}$ as given by (∞,1)-pullbacks there. The first condition requires that there is an essentially unique such probe by the point, hence that these higher geometric space has essentially a single global point. This is the condition that reflects the infinitesimal nature of the deformation problem.
For $k$ a field of characteristic 0, write write $CAlg_k^{sm} \hookrightarrow CAlg_k$ for the (∞,1)-category of Artinian connective E-∞ algebras over $k$, or equivalently that of “small” commutative dg-algebras over $k$.
The smallness condition implies connectivity (Lurie, prop. 1.1.11 (1)), hence that the homotopy group of these E-∞ algebras vanish in negative degree. Notice that for the dg-algebras this means that the chain homology vanishes in negative degree if the differential is taken to have degree -1 (see Porta 13, def. 3.1.14 for emphasis). This is the natural condition for the function algebra in derived geometry. Here these small $E_\infty$/dg-algebras are to be thought of as function algebras on “derived infinitesimally thickened points”.
There is an equivalence of (∞,1)-categories
with that of L-∞ algebras.
In this form this is (Lurie, theorem 0.0.13). See at model structure for L-∞ algebras for various other incarnations of this equivalence.
Given a deformation context $\mathcal{Y}$, the restricted (∞,1)-Yoneda embedding gives an (∞,1)-functor
For $Y \in \mathcal{Y}^{op}$, the object $Lie(Y)$ represents the formal neighbourhood of the basepoint of $Y$ as seen by the infinitesimally thickened points dual to the $\{E_\alpha\}$.
Hence we may call this the operaton of Lie differentiation of spaces in $\mathcal{Y}^{op}$ around their given base point.
Jacob Lurie Moduli problems for ring spectra ICM 2010 proceedings contribution pdf
Mauro Porta, Derived formal moduli problems, master thesis 2013, pdf.