[Paper Review] Deligne groupoid revisited

Paul Bressler, Alexander Gorokhovsky+2 more|arXiv (Cornell University)|Nov 28, 2012

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper establishes a homotopy equivalence between the simplicial nerve of the Deligne 2-groupoid associated to a nilpotent differential graded Lie algebra (DGLA) concentrated in degrees ≥ −1 and the simplicial set of g-valued differential forms introduced by Hinich. The result generalizes Hinich's earlier equivalence for DGLAs concentrated in non-negative degrees and provides a homotopy-theoretic bridge between deformation-theoretic structures and differential forms, with applications to formal deformation theory of algebras and gerbes via Kontsevich's formality theorem.

ABSTRACT

We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms introduced by V.Hinich.

Motivation & Objective

• To establish a homotopy equivalence between two simplicial sets associated with a nilpotent DGLA satisfying deg(𝔤) ≥ −1: the nerve of the Deligne 2-groupoid and Hinich’s simplicial set Σ(𝔤).
• To extend Hinich’s earlier result—valid for DGLAs concentrated in non-negative degrees—to the case where the DGLA has non-trivial components in degree −1.
• To demonstrate that the Deligne 2-groupoid of a DGLA 𝔤⊗𝔪 (for nilpotent 𝔪) can be reconstructed up to equivalence from the homotopy type of Σ(𝔤⊗𝔪), using L∞-quasi-isomorphisms.
• To provide a homotopy-theoretic framework for formal deformation theory of associative algebras and algebroid stacks, particularly in the context of Kontsevich’s formality theorem.

Proposed method

• The proof constructs canonical homotopy equivalences from both Σ(𝔤) and 𝔑MC²(𝔤) to a third, intermediate simplicial set, thereby establishing their mutual homotopy equivalence.
• It uses the simplicial nerve construction for 2-groupoids and the homotopy coherent nerve to model higher categorical structures as simplicial sets.
• The key technical step involves showing that the functor MC²(𝔤) → MC²(Ωₙ⊗𝔤) is an equivalence of 2-groupoids for each n, via quasi-isomorphisms of DGLAs induced by the inclusion [0] → [n] and evaluation at 0.
• It leverages the fact that the DGLA (Ωₙ⊗𝔤, d+δ) is quasi-isomorphic to (𝔤, δ) via the evaluation map ev₀, and that this induces isomorphisms on the relevant cohomology groups in degrees −1 and 0.
• The proof relies on the identification of the automorphism group of a Maurer–Cartan element μ in MC²(𝔤) with the action of exp(𝔨⁻¹) on exp(ker(δ_μ⁻¹)), and shows this action is preserved under the quasi-isomorphism.
• It applies results from Duskin’s theory of nerves and homotopy coherent nerves, and uses the fact that quasi-isomorphisms of DGLAs induce equivalences on their associated MC²-groupoids.

Experimental results

Research questions

• RQ1 Is the simplicial nerve of the Deligne 2-groupoid homotopy equivalent to Hinich’s simplicial set Σ(𝔤) for a DGLA 𝔤 with components vanishing in degrees below −1?
• RQ2 Does the homotopy equivalence between Σ(𝔤) and 𝔑MC²(𝔤) extend beyond the case of DGLAs concentrated in non-negative degrees?
• RQ3 Can the deformation 2-groupoid of an algebra A over a field k, twisted by a nilpotent k-algebra 𝔪, be recovered up to equivalence from the homotopy type of Σ(𝔤(A)⊗𝔪)?
• RQ4 How do L∞-quasi-isomorphisms interact with the construction of Σ(𝔤) and the nerve of MC²(𝔤)?
• RQ5 What is the role of the simplicial DGLA Ωₙ⊗𝔤 in relating the homotopy types of MC²(𝔤) and Σ(𝔤)?

Key findings

• The simplicial sets 𝔑MC²(𝔤) and Σ(𝔤) are homotopy equivalent for any nilpotent DGLA 𝔤 with 𝔤ⁱ = 0 for i < −1.
• The functor MC²(𝔤) → MC²(Ωₙ⊗𝔤) is an equivalence of 2-groupoids for each n, induced by the quasi-isomorphism (𝔤, δ) → (Ωₙ⊗𝔤, d+δ) via the inclusion [0] → [n].
• The evaluation map ev₀: Ωₙ⊗𝔤 → 𝔤 is a quasi-isomorphism of DGLAs with inverse given by the inclusion, and this induces isomorphisms on the relevant cohomology groups in degrees −1 and 0.
• The automorphism group of a Maurer–Cartan element μ in MC²(𝔤) is isomorphic to the action of exp(𝔨⁻¹) on exp(ker(δ_μ⁻¹)), and this structure is preserved under the quasi-isomorphism.
• The induced map on homotopy categories (sets of isomorphism classes) is a bijection, and the induced map on automorphism groupoids is an equivalence.
• The result implies that the deformation 2-groupoid of an algebra A over a field k, twisted by a nilpotent k-algebra 𝔪, is homotopy equivalent to Σ(𝔤⊗𝔪) whenever 𝔤 is an L∞-algebra quasi-isomorphic to 𝔤(A)⊗𝔪.