nLab Formal Moduli Problems and DG-Lie Algebras



Higher algebra

\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids



Related topics


\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

This entry provides some links concerning the notes



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

    ΓX* \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 𝔤\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.


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

For more related discussion see also

A survey is in

  • Mauro Porta, Derived formal moduli problems, master thesis 2013, pdf.
category: reference

Last revised on November 5, 2022 at 09:10:05. See the history of this page for a list of all contributions to it.