formal moduli problem


Small objects

Higher geometry




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)


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.


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). 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.


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