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Formal Moduli Problems and DG-Lie Algebras

Contents

Context

Higher algebra

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

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

    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.

Keywords

A model category-presentation of much of the theory 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

A similar result in less generality was discussed earlier in

And further model category presentations of much of the structure involved here are constructed and compared with each other in

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

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 October 26, 2016 at 14:15:33. See the history of this page for a list of all contributions to it.