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formal moduli problem (changes) in nLab

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higher geometry / derived geometry

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Definition

Local definition as functors on Artinian objects

Definition

Given a deformation context (𝒴,{E α} α)(\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 𝒴 inf\mathcal{Y}^{inf}

Moduli 𝒴↪[𝒴 inf,∞Grpd] Moduli^\mathcal{Y} \hookrightarrow [\mathcal{Y}^{inf}, \infty Grpd]

on those (∞,1)-functors X:𝒴 inf→∞GrpdX \colon \mathcal{Y}^{inf} \to \infty Grpd such that

  1. over the terminal object they are contractible: X(*)≃*X(*) \simeq * (hence they are anti-reduced);

  2. they preserve (∞,1)-pullbacks of small morphisms (are infinitesimally cohesive)

(Lurie, Def. 1.1.14 with Def. 1.1.8)

Properties

Relation to L ∞L_\infty-algebras

For kk a field of characteristic 0, write write CAlg k sm↪CAlg kCAlg_k^{sm} \hookrightarrow CAlg_k for the (∞,1)-category of Artinian connective E-∞ algebras over kk, or equivalently that of “small” commutative dg-algebras over kk.

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 ∞E_\infty/dg-algebras are to be thought of as function algebras on “derived infinitesimally thickened points”.

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

References

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

  • Jacob Lurie, Formal moduli problems

  • Jacob Lurie, Moduli problems for ring spectra ICM 2010 proceedings contribution pdf

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

  • Damien Calaque, Julien Grivaux, Formal moduli problems and formal derived stacks (arXiv:1802.09556)

  • Arjen Baarsma: Deformation problems and L ∞L_\infty-algebras of Fréchet type, PhD thesis, Utrecht University (2019) [[dspace:1874/386311](https://dspace.library.uu.nl/handle/1874/386311), pdf]

  • Brice Le Grignou, Victor Roca i Lucio: A new approach to formal moduli problems [[arXiv:2306.07227](https://arxiv.org/abs/2306.07227)]

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

Last revised on January 19, 2025 at 11:09:10. See the history of this page for a list of all contributions to it.