nLab formal moduli problem

Definition

Local definition as functors on Artinian objects

Definition

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 $\mathcal{Y}^{inf}$

$$Moduli^\mathcal{Y} \hookrightarrow [\mathcal{Y}^{inf}, \infty Grpd]$$

on those (∞,1)-functors $X \colon \mathcal{Y}^{inf} \to \infty Grpd$ such that

1. over the terminal object they are contractible: $X(*) \simeq *$ (hence they are anti-reduced);

2. they preserve (∞,1)-pullbacks of small morphisms (are infinitesimally cohesive)

(Lurie, Def. 1.1.14 with Def. 1.1.8)

Remark

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.

Remark

The clause about pullbacks is what makes the behaviour at arbitrary infinitesimal order be all controlled by that at first order, see Calaque-Grivaux 18, top of p. 8.

This ability to understand deformations order-by-order is related to the existence of a good obstruction theory. Indeed, evaluating a formal moduli problem $X$ on a pullback diagram defining an elementary morphism exhibits $X(\Omega^{\infty - n}E)$ as an obstruction space.

Properties

Relation to $L_\infty$-algebras

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”.

Theorem

There is an equivalence of (∞,1)-categories

$$L_\infty Alg_k \stackrel{\simeq}{\to} Moduli^{CAlg^{sm}_k}$$

with that of L-∞ algebras.

In this form this is (Lurie, theorem 0.0.13), originally proved as (Pridham, theorem 2.30 and proposition 4.42). See at model structure for L-∞ algebras for various other incarnations of this equivalence.

Relation to Lie differentiation

Proposition

Given a deformation context $\mathcal{Y}$, the restricted (∞,1)-Yoneda embedding gives an (∞,1)-functor

$$Lie \colon \mathcal{Y} \to Moduli^{\mathcal{Y}} \,.$$

Remark

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.

Related concepts

• moduli problem

• model structure for L-∞ algebras

• sheaf of L-∞ algebras

• synthetic differential geometry

• synthetic differential ∞-groupoid

References

• Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)

• Jacob Lurie, Formal moduli problems

• Jacob Lurie, Moduli problems for ring spectra ICM 2010 proceedings contribution pdf

• Mauro Porta, Derived formal moduli problems, master thesis 2013, pdf.

• Damien Calaque, Julien Grivaux, Formal moduli problems and formal derived stacks (arXiv:1802.09556)

The correspondence between formal moduli problems and dg-Lie algebras is extended to positive characteristic in

• Lukas Brantner, Akhil Mathew, Deformation Theory and Partition Lie Algebras, arXiv:1904.07352