nLab
formal moduli problem

Context

Small objects

Higher geometry

Contents

Definition

Definition

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 (are infinitesimally cohesive)

(Lurie, def. 1.1.14)

Remark

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.

Properties

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”.

Definition

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). See at model structure for L-∞ algebras for various other incarnations of this equivalence.

Relation to Lie differentiation

Proposition

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}} \,.
Remark

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.

References

Revised on October 26, 2016 14:25:49 by Urs Schreiber (89.204.155.11)