# nLab formal moduli problem

Contents

### Context

#### Small objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## 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)

###### 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.

## References

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

Last revised on April 15, 2023 at 17:00:38. See the history of this page for a list of all contributions to it.