# nLab Formal Moduli Problems and DG-Lie Algebras

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

This entry provides some links concerning the notes

# Contents

## Idea

This text establishes a statement (theorem 5.3) that characterizes ∞-stacks over an (∞,1)-site of formal duals of Artin E-∞ algebras over a field (hence infinitesimally thickened points in derived geometry) such that

1. they have a contractible underlying ∞-groupoid

$\Gamma X \simeq *$

(in the terminology and notation for cohesive (∞,1)-toposes)

2. and are infinitesimally cohesive in that they respects homotopy fiber products of morphisms whose kernels are nilpotent ideals

with those associated to an ∞-Lie algebra $\mathfrak{g}$ in the sense of (Hinich). This is (Lurie, theorem 0.0.13, remark 0.0.15).

The construction is then generalized to noncommutative geometry.

## Keywords

Am earlier (2007) model category-presentation proof by (Pridham) exists, see the discussion at

A indication of how to use the main theorem here in order to implement Lie differentiation of smooth ∞-groups is at

The same result was proved earlier (theorem 2.30 and corollary 4.57, cf (Lurie, remark 0.0.14)), with further model category presentations and comparisons, in

See at model structure for L-infinity algebras for more along these lines.

The foundations of the theory appear in