nLab
Formal Moduli Problems and DG-Lie Algebras

This entry provides some links concerning the notes

• Jacob Lurie, Formal moduli problems (pdf)

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

• deformation context

• formal moduli problem

• model structure for L-infinity algebras

Related entries

A model category-presentation of much of the theory exists, see the discussion at

• model structure for L-infinity algebras .

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

• synthetic differential infinity-groupoid – Lie differentiation

Related references

A similar result in less generality was discussed earlier in

• Vladimir Hinich, DG coalgebras as formal stacks (arXiv:math/9812034)

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

• Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)

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

For more related discussion see also

• Jacob Lurie, Higher Algebra

A survey is in

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

category: reference