nLab formal moduli problem




Local definition as functors on Artinian objects


Given a deformation context (𝒴,{E α} α)(\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 𝒴 inf\mathcal{Y}^{inf}

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

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

  1. over the terminal object they are contractible: X(*)*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)


This means that a “formal deformation problem” is a space in higher geometry whose geometric structure is detected by the “test spaces” in 𝒴 op\mathcal{Y}^{op} in a way that respects gluing (descent) in 𝒴 op\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.


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 XX on a pullback diagram defining an elementary morphism exhibits X(Ω nE)X(\Omega^{\infty - n}E) as an obstruction space.


Relation to L L_\infty-algebras

For kk a field of characteristic 0, write write CAlg k smCAlg kCAlg_k^{sm} \hookrightarrow CAlg_k for the (∞,1)-category of Artinian connective E-∞ algebras over kk, or equivalently that of “small” commutative dg-algebras over kk.

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 E_\infty/dg-algebras are to be thought of as function algebras on “derived infinitesimally thickened points”.


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

L Alg kModuli CAlg k sm 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


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

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

For Y𝒴 opY \in \mathcal{Y}^{op}, the object Lie(Y)Lie(Y) represents the formal neighbourhood of the basepoint of YY as seen by the infinitesimally thickened points dual to the {E α}\{E_\alpha\}.

Hence we may call this the operaton of Lie differentiation of spaces in 𝒴 op\mathcal{Y}^{op} around their given base point.


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

Last revised on March 30, 2024 at 20:01:26. See the history of this page for a list of all contributions to it.