Theory and Applications of Categories, Vol. 30, No. 29, 2015, pp. 1001–1016.

DELIGNE GROUPOID REVISITED

PAUL BRESSLER, ALEXANDER GOROKHOVSKY, RYSZARD NEST AND BORIS TSYGAN

Abstract. We show that for a differential graded Lie algebra 𝔤 whose components vanish in degrees below −1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of 𝔤-valued differential forms introduced by V. Hinich [Hinich, 1997].

1. Introduction

The principal result of the present note compares two spaces (simplicial sets) naturally associated with a nilpotent differential graded Lie algebra (DGLA) subject to certain restrictions. Our interest in this problem has its origins in formal deformation theory of associative algebras and, more generally, algebroid stacks ([Bressler, Gorokhovsky, Nest & Tsygan, 2007]). The results of the present note are used in [Bressler, Gorokhovsky, Nest & Tsygan, 2015] to deduce a quasi-classical description of the deformation theory of a gerbe from the formality theorem of M. Kontsevich ([Kontsevich, 2003]).

To a nilpotent DGLA 𝔥 which satisfies the additional condition

𝔥^{i} = 0 for i < −1  (1)

P. Deligne [Deligne, 1994] and, independently, E. Getzler [Getzler, 2009] associated a (strict) 2-groupoid which we denote MC^{2}(𝔥) and refer to as the Deligne 2-groupoid.

Our principal result (Theorem 4.2) compares the simplicial nerve 𝔑 MC^{2}(𝔥) of the 2-groupoid MC^{2}(𝔥), 𝔥 a nilpotent DGLA satisfying (1), to another simplicial set, denoted Σ(𝔥), introduced by V. Hinich [Hinich, 1997]:

1.1. Theorem. (Main theorem) Suppose that 𝔥 is a nilpotent DGLA such that 𝔥^{i} = 0 for i < −1. Then, the simplicial sets 𝔑 MC^{2}(𝔥) and Σ(𝔥) are weakly homotopy equivalent.

In the case when the nilpotent DGLA 𝔥 satisfies 𝔥^{i} = 0 for i < 0 and, consequently, MC^{2}(𝔥) is an ordinary groupoid a homotopy equivalence between Σ(𝔥) and the nerve of MC^{2}(𝔥) was constructed by V. Hinich in [Hinich, 1997].

A. Gorokhovsky was partially supported by NSF grant DMS-0900968. B. Tsygan was partially supported by NSF grant DMS-0906391. R. Nest was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92)

Received by the editors 2015-05-23 and, in revised form, 2015-07-14.

Transmitted by James Stasheff. Published on 2015-07-16.

2010 Mathematics Subject Classification: 18G55, 55U10.

Key words and phrases: groupoid, L_{∞}-algebra, simplicial nerve.

\begin{matrix}c \\ ◯ \end{matrix} Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan, 2015. Permission to copy for private use granted.

Differential graded Lie algebras satisfying (1) arise in formal deformation theory of algebraic structures such as Lie algebras, commutative algebras, associative algebras to name a few. In what follows we shall concentrate on the latter example. Let k denote an algebraically closed field of characteristic zero. For an associative algebra A over k the shifted Hochschild cochain complex C^{•}(A)[1] has a canonical structure of a DGLA under the Gerstenhaber bracket; we denote this DGLA by 𝔤(A) for short. Suppose that 𝔪 is a nilpotent commutative k-algebra (without unit). Then, 𝔤(A) ⊗_{k} 𝔪 is a nilpotent DGLA which satisfies (1). Thus, the Deligne 2-groupoid MC^{2}(𝔤(A) ⊗_{k} 𝔪) is defined. For an Artin k-algebra R with maximal ideal 𝔪_{R} the 2-groupoid MC^{2}(𝔤(A) ⊗_{k} 𝔪_{R}) is naturally equivalent to the 2-groupoid of R-deformations of the algebra A. In this sense the DGLA 𝔤(A) controls the formal deformation theory of A.

The reason for considering the space Σ(𝔥) is that it is defined not just for a DGLA (V. Hinich, [Hinich, 1997]), but, more generally, for any nilpotent L_{∞} algebra (E. Getzler, [Getzler, 2009]). Homotopy invariance properties of the functor Σ (Proposition 3.9), the theory of J.W. Duskin ([Duskin, 2001/02]) and the theorem above yield the following result. If 𝔥 is a DGLA satisfying (1), 𝔤 is a L_{∞} algebra L_{∞}-quasi-isomorphic to 𝔥 and 𝔪 is a nilpotent commutative k-algebra, then 𝔑 MC^{2}(𝔥 ⊗_{k} 𝔪) is homotopy equivalent to Σ(𝔤 ⊗_{k} 𝔪). Thus, the 2-groupoid MC^{2}(𝔥 ⊗_{k} 𝔪) can be reconstructed, up to equivalence, from the space Σ(𝔤⊗_{k} 𝔪). The situation envisaged above arises naturally. Any DGLA 𝔥 is L_{∞}-quasi-isomorphic to an L_{∞} algebra with trivial univalent operation (the differential).

The paper is organized as follows. In Section 2 we review various constructions of nerves of 2-groupoids and their properties. In section 3 we recall the definitions of the functor Σ (3.4) and of the Deligne 2-groupoid (3.10) and prove basic properties thereof. The proof of the main theorem (Theorem 4.2) given in Section 4 proceeds by exhibiting canonical weak homotopy equivalences from Σ(𝔥) and 𝔑 MC^{2}(𝔥) to a third naturally defined simplicial set.

2. The homotopy type of a strict 2-groupoid

2.1. Nerves of simplicial groupoids.

2.1.1. Simplicial groupoids. In what follows a simplicial category is a category enriched over the category of simplicial sets. A small simplicial category consists of a set of objects and a simplicial set of morphisms for each pair of objects.

A simplicial category G is a particular case of a simplicial object [p] ↦ G_{p} in Cat whose simplicial set of objects [p] ↦ N_{0}G_{p} is constant.

A simplicial category is a simplicial groupoid if it is a groupoid in each (simplicial) degree.

2.1.2. The naïve nerve. Suppose that G is a simplicial category. Applying the nerve functor degree-wise we obtain the bi-simplicial set NG: ([p], [q]) ↦ N_{q}G_{p} whose diagonal we denote by N G and refer to as the naïve nerve of G.

2.1.3. The simplicial nerve. For a simplicial category G the simplicial nerve, also known as the homotopy coherent nerve, 𝔑G is represented by the cosimplicial object in [p] ↦ Δ_{𝔑}^{p} ∈ Cat_{Δ}, i.e

𝔑_{p}G = Hom_{Cat_{Δ}}(Δ_{𝔑}^{p}, G).

Here, Δ_{𝔑}^{p} is the canonical free simplicial resolution of [p] which admits the following explicit description ([Cordier, 1982]).

The set of objects of Δ_{𝔑}^{p} is {0, 1, …, p}. For 0 ≤ i ≤ j ≤ p the simplicial set of morphisms is given by Hom_{Δ_{𝔑}^{p}}(i, j) = NP(i, j). The category P(i, j) is a sub-poset of 2^{{0,...,p}} (with the induced partial ordering whereby viewed as a category) given by

P(i, j) = {I ⊂ ℤ | (i, j ∈ I) & (k ∈ I ⟹ i ⩽ k ⩽ j)}.

The composition in Δ_{𝔑}^{p} is induced by functors

P(i, j) × P(j, k) → P(i, k): (I, J) ↦ I ∪ J.

In particular, Δ_{𝔑}^{0} = [0] and Δ_{𝔑}^{1} = [1]

We refer the reader to [Hinich, 2007] for applications to deformation theory and to [Lurie, 2009] for the connection with higher category theory. The simplicial nerve of a simplicial groupoid is a Kan complex which reduces to the usual nerve for ordinary groupoids.

Since Δ_{𝔑}^{0} = [0] (respectively, Δ_{𝔑}^{1} = [1]) it follows that 𝔑_{0}G (respectively, 𝔑_{1}G) is the set of objects (respectively, the set of morphisms) of G_{0}.

2.1.4. Comparison of nerves. We refer the reader to [Hinich, 2007] for the definition of the canonical map of simplicial sets N G → 𝔑G. In what follows we will make use of the following result of loc. cit.

2.2. Theorem. ([Hinich, 2007], Corollary 2.6.3) For any simplicial groupoid G the canonical map N G → 𝔑G is a weak homotopy equivalence.

2.3. Strict 2-groupoids.

2.3.1. From strict 2-groupoids to simplicial groupoids. Suppose that G is a strict 2-groupoid, i.e. a groupoid enriched over the category of groupoids. Thus, for every g, g′ ∈ G, we have the groupoid Hom_{G}(g, g′) and the composition is strictly associative.

The nerve functor [p] ↦ N_{p}(·) := Hom_{Cat}([p], ·) commutes with products. Let G_{p} denote the category with the same objects as G and with morphisms defined by Hom_{G_{p}}(g, g′) = N_{p} Hom_{G}(g, g′); the composition of morphisms is induced by the composition in G. Note that the groupoid G_{0} is obtained from G by forgetting the 2-morphisms.

The assignment [p] ↦ G_{p} defines a simplicial object in groupoids with the constant simplicial set of objects, i.e. a simplicial groupoid which we denote by G̃.

2.4. Lemma. The simplicial nerve 𝔑G̃ admits the following explicit description:

1. There is a canonical bijection between 𝔑_{0}G̃ and the set of objects of G.

2. For n ≥ 1 there is a canonical bijection between 𝔑_{n}G̃ and the set of data of the form ((μ_{i})_{0≤i≤n}, (g_{ij})_{0≤i<j≤n}, (c_{ijk})_{0≤i<j<k≤n}), where (μ_{i}) is an (n + 1)-tuple of objects of G, (g_{ij}) is a collection of 1-morphisms g_{ij}: μ_{j} → μ_{i} and (c_{ijk}) is a collection of 2-morphisms c_{ijk} : g_{ij}g_{jk} → g_{ik} which satisfies

c_{ijl}c_{jkl} = c_{ikl}c_{ijk}  (2)

(in the set of 2-morphisms g_{ij}g_{jk}g_{kl} → g_{il}).

For a morphism f: [m] → [n] in Δ the induced structure map f^{∗}: 𝔑_{n}G̃ → 𝔑_{m}G̃ is given (under the above bijection) by f^{∗}((μ_{i}), (g_{ij}), (c_{ijk})) = ((ν_{i}), (h_{ij}), (d_{ijk})), where ν_{i} = μ_{f(i)}, h_{ij} = g_{f(i),f(j)}, d_{ijk} = c_{f(i),f(j),f(k)} (cf. [Duskin, 2001/02]).

Proof. An n-simplex of 𝔑G̃ is the following collection of data:

1. objects μ_{0}, …, μ_{n} of G;

2. morphisms of simplicial sets NP(i, j)) → N Hom_{G}(μ_{i}, μ_{j})) intertwining the maps induced on the nerves by composition functors P(i, j) × P(j, k) → P(i, k) and Hom_{G}(μ_{i}, μ_{j}) × Hom_{G}(μ_{j}, μ_{k}) → Hom_{G}(μ_{i}, μ_{k}).

Since the nerve functor is fully faithful, the above data are equivalent to the following:

1. objects μ_{0}, …, μ_{n} of G;

2. for any I ∈ N_{0}P(i, j), a 1-morphism g_{I} : μ_{j} → μ_{i} in G;

3. for any morphism J → I in P(i, j), a 2-morphism c_{IJ}: g_{J} → g_{I}, such that

c_{IJ}c_{JK} = c_{IK}  (3)

These data have to be compatible with the composition pairings P(i, j)×P(j, k) → P(i, k) and Hom_{G}(μ_{i}, μ_{j}) × Hom_{G}(μ_{j}, μ_{k}) → Hom_{G}(μ_{i}, μ_{k}).

Let g_{ij}: μ_{j} → μ_{i} denote the morphism g_{{i,j}}. By compatibility with compositions, if I = {i, i_{1}, …, i_{k}, j} then g_{I} = g_{ii_{1}} … g_{i_{k}j}. Let c_{ijk} denote the two-morphism c_{{i,j,k},{i,k}}: g_{ik} → g_{ij}g_{jk}. Now, by virtue of (3) and of compatibility with compositions, c_{ijk} satisfy the twococycle identity (3) and determine c_{IJ} for any I, J.

In what follows, for a strict 2-groupoid G, we will denote by N G (respectively 𝔑G) the naïve (respectively simplicial) nerve of the associated simplicial groupoid G̃.

3. Homotopy types associated with L_{∞}-algebras

3.1. L_{∞}-algebras. We follow the notation of [Getzler, 2009] and refer the reader to loc. cit. for details.

Recall that an L_{∞}-algebra is a graded vector space 𝔤 equipped with operations

⋀^{k}𝔤 → 𝔤[2 − k]: x_{1} ∧ … ∧ x_{k} ↦ [x_{1}, …, x_{k}]

defined for k = 1, 2, . . .. which satisfy a sequence of Jacobi identities.

It follows from the Jacobi identities that the unary operation [.]: 𝔤 → 𝔤[1] is a differential, which we will denote by δ.

An L_{∞}-algebra is abelian if all operations with valency two and higher (i.e. all operations except for δ) vanish. In other words, an abelian L_{∞}-algebra is a complex. An L_{∞}-algebra structure with vanishing operations of valency three and higher reduces to a structure of a DGLA.

The lower central series of an L_{∞}-algebra 𝔤 is the canonical decreasing filtration F^{•}𝔤 with F^{i}𝔤 = 𝔤 for i ≤ 1 and defined recursively for i ≥ 1 by

F^{i+1}𝔤 = ∑_{k=2}^{∞} ∑_{\begin{matrix}i=i_{1}+···+i_{k} \\ i_{k}⩽i \end{matrix}}[F^{i_{1}}𝔤, …, F^{i_{k}}𝔤].

An L_{∞}-algebra is nilpotent if there exists an i such that F^{i}𝔤 = 0.

3.1.1. Maurer-Cartan elements. Suppose that 𝔤 is a nilpotent L_{∞}-algebra. For μ ∈ 𝔤^{1} let

F(μ) = δμ + ∑_{k=2}^{∞} \frac{1}{k!}[μ^{∧k}].  (4)

The element F(μ) of 𝔤^{2} is called the curvature of μ. For any μ ∈ 𝔤^{1} the curvature F(μ) satisfies the Bianchi identity ([Getzler, 2009], Lemma 4.5)

δF(μ) + ∑_{k=1}^{∞} \frac{1}{k!}[μ^{∧k}, F(μ)] = 0.  (5)

An element μ ∈ 𝔤^{1} is called a Maurer-Cartan element (of 𝔤) if it satisfies the condition F(μ) = 0. The set of Maurer-Cartan elements of 𝔤 will be denoted MC(𝔤):

MC(𝔤) := {μ ∈ 𝔤^{1} | F(μ) = 0}.

The set MC(𝔤) is pointed by the distinguished element 0 ∈ 𝔤^{1}.

Suppose that 𝔞 is an abelian L_{∞}-algebra. Then,

MC(𝔞) = Z^{1}(𝔞) := ker(δ: 𝔞^{1} → 𝔞^{2}),

hence is equipped with a canonical structure of an abelian group.

3.1.2. Central extensions. Suppose that 𝔤 is a L_{∞}-algebra and 𝔞 is a subcomplex of (𝔤, δ) such that [𝔞 ∧ 𝔤^{∧k}] = 0 for all k ≥ 1. In this case we will say that 𝔞 is central in 𝔤.

If 𝔞 is central in 𝔤, then there is a unique structure of an L_{∞}-algebra on 𝔤/𝔞 such that the projection 𝔤 → 𝔤/𝔞 is a map of L_{∞}-algebras. If 𝔤 is nilpotent, then so is 𝔤/𝔞.

In what follows we assume that 𝔤 is a nilpotent L_{∞}-algebra and 𝔞 is central in 𝔤.

3.2. Lemma.

1. The addition operation on 𝔤^{1} restricts to a free action of the abelian group MC(𝔞) on the set MC(𝔤).

2. The map MC(𝔤) → MC(𝔤/𝔞) is constant on the orbits of the action.

3. The induced map MC(𝔤)/ MC(𝔞) → MC(𝔤/𝔞) is injective.

Proof. Suppose that α ∈ 𝔞^{1} and μ ∈ 𝔤^{1}. Since 𝔞 is central in 𝔤, [(α + μ)^{∧k}] = [μ^{∧k}] for k ≥ 2 and F(α + μ) = δα + F(μ) (in the notation of (4)). Therefore, MC(𝔞) + MC(𝔤) = MC(𝔤). In other words, the addition operation in 𝔤^{1} restricts to an action of the abelian group MC(𝔞) on the set MC(𝔤) which is obviously free. Since the map MC(𝔤) → MC(𝔤/𝔞) is the restriction of the map 𝔤 → 𝔤/𝔞 constant on the orbits of the action, i.e. factors through MC(𝔤)/ MC(𝔞), and the induced map MC(𝔤)/ MC(𝔞) → MC(𝔤/𝔞) is injective.

3.2.1. The obstruction map. The image of the map MC(𝔤) → MC(𝔤/𝔞) may be described in terms of the obstruction map (6) which we construct presently.

If μ ∈ 𝔤^{1} and μ + 𝔞^{1} ∈ MC(𝔤/𝔞), then F(μ + 𝔞^{1}) = F(μ) + δ𝔞^{1} ⊂ 𝔞^{2} and the Bianchi identity (5) reduces to δF(μ + 𝔞^{1}) = 0, i.e. the assignment μ + 𝔞^{1} ↦ F(μ + 𝔞^{1}) gives rise to a well-defined map

o_{2} : MC(𝔤/𝔞) → H^{2}(𝔞)  (6)

(notation borrowed from [Goldman, Millson, 1988], 2.6).

3.3. Lemma. The sequence of pointed sets

0 → MC(𝔤)/ MC(𝔞) → MC(𝔤/𝔞) ⟶^{o_{2}} H^{2}(𝔞)  (7)

is exact.

Proof. If F(μ + 𝔞^{1}) ⊂ δ𝔞^{1}, then there exists α ∈ 𝔞^{1} such that F(μ + α) = 0, i.e. μ + 𝔞^{1} is in the image of MC(𝔤) → MC(𝔤/𝔞).

3.4. The functor Σ. In what follows we denote by Ω_{n}, n = 0, 1, 2, … the commutative differential graded algebra over ℚ with generators t_{0}, …, t_{n} of degree zero and dt_{0}, …, dt_{n} of degree one subject to the relations t_{0}+⋯+t_{n} = 1 and dt_{0}+⋯+dt_{n} = 0. The differential d: Ω_{n} → Ω_{n}[1] is defined by t_{i} ↦ dt_{i} and dt_{i} ↦ 0. The assignment [n] ↦ Ω_{n} extends in a natural way to a simplicial commutative differential graded algebra.

3.4.1. The simplicial set Σ(𝔤). For a nilpotent L_{∞}-algebra 𝔤 and a non-negative integer n let

Σ_{n}(𝔤) = MC(𝔤 ⊗ Ω_{n}).

Equipped with structure maps induced by those of Ω_{•} the assignment n ↦ Σ_{n}(𝔤) defines a simplicial set denoted Σ(𝔤).

The simplicial set Σ(𝔤) was introduced by V. Hinich in [Hinich, 1997] for DGLA and used by E. Getzler in [Getzler, 2009] (where it is denoted MC_{•}(𝔤)) for general nilpotent L_{∞}-algebras.

3.4.2. Abelian DGLA. If 𝔞 is an abelian L_{∞}-algebra, then Σ(𝔞) is given by Σ_{n}(𝔞) = Z^{1}(Ω_{n} ⊗ 𝔞) = Z^{0}(Ω_{n} ⊗ 𝔞[1]) and has a canonical structure of a simplicial abelian group. In particular, it is a Kan simplicial set.

Recall that the Dold-Kan correspondence associates to a complex of abelian groups A a simplicial abelian group K(A) defined by K(A)_{n} = Z^{0}(C^{•}([n]; A)), the group of cocycles of (total) degree zero in the complex of simplicial cochains on the n-simplex with coefficients in A.

The integration map ∫ : Ω_{n} ⊗ 𝔞 → C^{•}([n]; 𝔞) induces a homotopy equivalence

∫ : Σ(𝔞) → K(𝔞[1]);  (8)

see [Getzler, 2009], Section 3. Thus, π_{i}Σ(𝔞) ≅ H^{1−i}(𝔞).

3.4.3. Central extensions. Suppose that 𝔤 is a nilpotent L_{∞}-algebra and 𝔞 is a central subalgebra in 𝔤. Then, for n = 0, 1, …, Ω_{n} ⊗ 𝔞 is central in Ω_{n} ⊗ 𝔤.

3.5. Lemma.

1. The addition operation on (Ω_{n} ⊗ 𝔤)^{1} induces a principal action of the simplicial abelian group Σ(𝔞) on the simplicial set Σ(𝔤).

2. The map Σ(𝔤) → Σ(𝔤/𝔞) factors through Σ(𝔤)/Σ(𝔞).

3. The induced map Σ(𝔤)/Σ(𝔞) → Σ(𝔤/𝔞) is injective.

Proof. Follows from Lemma 3.2 and the naturality properties of the constructions in 3.1.2.

For n = 0, 1, … the map ([n] → [0])^{∗}: ℚ → Ω_{n} is a quasi-isomorphism, with the quasiinverse provided by the map induced by any morphism [0] → [n]. Therefore, the map 𝔞 → Ω_{n}⊗𝔞 is a quasi-isomorphism as well. The induced isomorphisms H^{2}(𝔞) ≅ H^{2}(Ω_{n}⊗𝔞) give rise to the isomorphism of the constant simplicial set H^{2}(𝔞) and n ↦ H^{2}(Ω_{n} ⊗ 𝔞).

The maps

o_{2,n}: Σ_{n}(𝔤/𝔞) = MC(Ω_{n} ⊗ 𝔤/𝔞) → H^{2}(Ω_{n} ⊗ 𝔞) ≅ H^{2}(𝔞)

assemble into the map of simplicial sets

o_{2} : Σ(𝔤/𝔞) → H^{2}(𝔞).  (9)

which factors as Σ(𝔤/𝔞) → π_{0}Σ(𝔤/𝔞) → H^{2}(𝔞).

Let Σ(𝔤/𝔞)_{0} = o_{2}^{−1}(0). Thus, by (7), Σ(𝔤/𝔞)_{0} is a union of connected components of Σ(𝔤/𝔞) equal to the range of the map Σ(𝔤)/Σ(𝔞) → Σ(𝔤/𝔞).

It follows that the map Σ(𝔤) → Σ(𝔤/𝔞)_{0} is a principal fibration with group Σ(𝔞), in particular, a Kan fibration ([May, 1967], Lemma 18.2).

3.6. Lemma. Suppose that 𝔤 is a nilpotent L_{∞}-algebra. Then, Σ(𝔤) is a Kan simplicial set.

Proof. If 𝔤 is an abelian L_{∞}-algebra then Σ(𝔤) is a simplicial group and therefore a Kan simplicial set.

Let F^{•}𝔤 denote the lower central series. Assume that Gr_{F}^{i} 𝔤 ≠ 0 if and only if 0 ≤ i ≤ n; that is, 𝔤 is nilpotent of length n. By induction assume that Σ(𝔥) is a Kan simplicial set for any nilpotent L_{∞}-algebra 𝔥 of length at most n − 1.

Since 𝔤 is nilpotent of length n, it follows that F^{n}𝔤 = Gr^{n}𝔤 is central in 𝔤 and 𝔤/F^{n}𝔤 is nilpotent of length n−1. Therefore, Σ(𝔤/F^{n}𝔤) is a Kan simplicial set and so is Σ(𝔤/F^{n}𝔤)_{0}. Since Σ(𝔤) → Σ(𝔤/F^{n}𝔤)_{0} is a Kan fibration it follows that Σ(𝔤) is a Kan simplicial set as well.

3.7. Lemma. Suppose that 𝔤 is a nilpotent L_{∞}-algebra such that 𝔤^{q} = 0 for q ≤ −k, k a positive integer. Then, for any connected component X of Σ(𝔤), π_{i}(X) = 0 for i > k.

Proof. Suppose that 𝔤 is an abelian L_{∞}-algebra. Then, π_{i}Σ(𝔤) ≅ H^{1−i}(𝔤). For an L_{∞}-algebra 𝔤 which is not necessarily abelian the statement follows by induction on the nilpotency length, the abelian case establishing the base of the induction.

Let F^{•}𝔤 denote the lower central series. Assume that Gr_{F}^{i} 𝔤 ≠ 0 if and only if 0 ≤ i ≤ n; that is, 𝔤 is nilpotent of length n. By induction assume that the conclusion holds for all nilpotent L_{∞}-algebras of length at most n − 1.

Since 𝔤 is nilpotent of length n, it follows that F^{n}𝔤 = Gr^{n}𝔤 is central in 𝔤 and 𝔤/F^{n}𝔤 is nilpotent of length n − 1. Let X ⊆ Σ(𝔤) be a connected component of Σ(𝔤) and let Y ⊆ Σ(𝔤/F^{n}𝔤) be the image of X under the map induced by the quotient map 𝔤 → 𝔤/F^{n}𝔤. Then, X → Y is a principal fibration with group the connected component of the identity in Σ(F^{n}𝔤). The desired vanishing of higher homotopy groups of X follows from the induction hypotheses using the long exact sequence of homotopy groups.

3.7.1. Homotopy invariance.

3.8. Lemma. Suppose that f: 𝔞 → 𝔟 is a quasi-isomorphism of abelian L_{∞}-algebras. Then, the induced map Σ(f): Σ(𝔞) → Σ(𝔟) is a weak homotopy equivalence.

Proof. Note that Σ(f) is a morphism of simplicial abelian groups. It is sufficient to show that the maps π_{n}Σ(f): π_{n}Σ(𝔞) → π_{n}Σ(𝔟) are isomorphisms for n ⩾ 0. To this end note that π_{n}Σ(f) factors as the composition of isomorphisms

π_{n}Σ(𝔞) ≅ H^{1−n}(𝔞) ⟶^{H^{1−n}(Σ(f))} H^{1−n}(𝔟) ≅ π_{n}Σ(𝔟).

3.9. Proposition. ([Getzler, 2009], Proposition 4.9) Suppose that f: 𝔤 → 𝔥 is a quasiisomorphism of L_{∞}-algebras and R is an Artin algebra with maximal ideal 𝔪_{R}. Then, the map Σ(f ⊗ Id): Σ(𝔤 ⊗ 𝔪_{R}) → Σ(𝔥 ⊗ 𝔪_{R}) is a weak homotopy equivalence.

Proof. We use induction on the nilpotency length of 𝔪_{R}, which is to say the largest integer l such that 𝔪_{R}^{l} ≠ 0.

If 𝔪_{R}^{2} = 0, then f ⊗Id: 𝔤⊗𝔪_{R} → 𝔥⊗𝔪_{R} is a quasi-isomorphism of abelian L_{∞}-algebras and the claim follows from Lemma 3.8.

Suppose that 𝔪_{R}^{l+1} = 0. By the induction hypothesis

• the map Σ(𝔤 ⊗ 𝔪_{R}/𝔪_{R}^{l}) → Σ(𝔥 ⊗ 𝔪_{R}/𝔪_{R}^{l}) is a weak homotopy equivalence and

• the map π_{0}Σ(𝔤 ⊗ 𝔪_{R}/𝔪_{R}^{l}) → π_{0}Σ(𝔥 ⊗ 𝔪_{R}/𝔪_{R}^{l}) is a bijection.

The map f ⊗ Id_{𝔪_{R}^{l}} is a quasi-isomorphism of abelian L_{∞}-algebras, therefore the map H^{2}(𝔤 ⊗ 𝔪_{R}^{l}) → H^{2}(𝔥 ⊗ 𝔪_{R}^{l}) is an isomorphism. The commutativity of

π_{0}Σ(𝔤 ⊗ 𝔪_{R}/𝔪_{R}^{l}) ⟶ π_{0}Σ(𝔥 ⊗ 𝔪_{R}/𝔪_{R}^{l})
↓  ↓
H^{2}(𝔤 ⊗ 𝔪_{R}^{l}) ⟶ H^{2}(𝔥 ⊗ 𝔪_{R}^{l})

implies that the map

π_{0}Σ(𝔤 ⊗ 𝔪_{R}/𝔪_{R}^{l})_{0} → π_{0}Σ(𝔥 ⊗ 𝔪_{R}/𝔪_{R}^{l})_{0}

is a bijection. Therefore, the map

Σ(𝔤 ⊗ 𝔪_{R}/𝔪_{R}^{l})_{0} → Σ(𝔥 ⊗ 𝔪_{R}/𝔪_{R}^{l})_{0}

is a weak homotopy equivalence. The map Σ(f) restricts to a map of principal fibrations

Σ(𝔤 ⊗ 𝔪_{R}) ⟶ Σ(𝔥 ⊗ 𝔪_{R})
↓  ↓
Σ(𝔤 ⊗ 𝔪_{R}/𝔪_{R}^{l})_{0} ⟶ Σ(𝔥 ⊗ 𝔪_{R}/𝔪_{R}^{l})_{0}

relative to the map of simplicial groups Σ(𝔤 ⊗ 𝔪_{R}^{l}) → Σ(𝔥 ⊗ 𝔪_{R}^{l}). The latter is a weak homotopy equivalence by Lemma 3.8. Therefore, so is the map Σ(𝔤⊗𝔪_{R}) → Σ(𝔥⊗𝔪_{R}).

3.10. Deligne groupoids.

3.10.1. Gauge transformations. Suppose that 𝔥 is a nilpotent DGLA. Then, 𝔥^{0} is a nilpotent Lie algebra. The unipotent group exp 𝔥^{0} acts on the space 𝔥^{1} by affine transformations. The action of exp X, X ∈ 𝔥^{0}, on γ ∈ 𝔥^{1} is given by the formula

(exp X) · γ = γ − ∑_{i=0}^{∞} \frac{(ad X)^{i}}{(i + 1)!} (δX + [γ, X]).  (10)

The effect of the above action on the curvature F(γ) = δγ + \frac{1}{2}[γ, γ] is given by

F((exp X) · γ) = exp(ad X)(F(γ)).  (11)

3.10.2. The functor MC^{1}. Suppose that 𝔥 is a nilpotent DGLA. It follows from (11) that gauge transformations (10) preserve the subset of Maurer-Cartan elements MC(𝔥) ⊂ 𝔥^{1}.

We denote by MC^{1}(𝔥) the Deligne groupoid (denoted C(𝔥) in [Hinich, 1997]) defined as the groupoid associated with the action of the group exp 𝔥^{0} by gauge transformations on the set MC(𝔥).

Thus, MC^{1}(𝔥) is the category with the set of objects MC(𝔥). For γ_{1}, γ_{2} ∈ MC(𝔥), Hom_{MC^{1}(𝔥)}(γ_{1}, γ_{2}) is the set of gauge transformations between γ_{1}, γ_{2}. The composition

Hom_{MC^{1}(𝔥)}(γ_{2}, γ_{3}) × Hom_{MC^{1}(𝔥)}(γ_{1}, γ_{2}) → Hom_{MC^{1}(𝔥)}(γ_{1}, γ_{3})

is given by the product in the group exp(𝔥^{0}).

3.10.3. The functor MC^{2}. For 𝔥 as above satisfying the additional vanishing condition 𝔥^{i} = 0 for i < −1 we denote by MC^{2}(𝔥) the Deligne 2-groupoid as defined by P. Deligne [Deligne, 1994] and independently by E. Getzler, [Getzler, 2009]. Below we review the construction of Deligne 2-groupoid of a nilpotent DGLA following [Getzler, 2009, Getzler, 2002] and references therein.

The objects and the 1-morphisms of MC^{2}(𝔥) are those of MC^{1}(𝔥). That is, for γ_{1}, γ_{2} ∈ MC(𝔥) the set Hom_{MC^{1}(𝔥)}(γ_{1}, γ_{2}) is the set of objects of the groupoid Hom_{MC^{2}(𝔥)}(γ_{1}, γ_{2}). The morphisms in Hom_{MC^{2}(𝔥)}(γ_{1}, γ_{2}) (i.e. the 2-morphisms of MC^{2}(𝔥)) are defined as follows.

For γ ∈ MC(𝔥) let [·, ·]_{γ} denote the Lie bracket on 𝔥^{−1} defined by

[a, b]_{γ} = [a, δb + [γ, b]].  (12)

Equipped with this bracket, 𝔥^{−1} becomes a nilpotent Lie algebra. We denote by exp_{γ} 𝔥^{−1} the corresponding unipotent group, and by

exp_{γ}: 𝔥^{−1} → exp_{γ} 𝔥^{−1}

the corresponding exponential map. If γ_{1}, γ_{2} are two Maurer-Cartan elements, then the group exp_{γ_{2}} 𝔥^{−1} acts on Hom_{MC^{1}(𝔥)}(γ_{1}, γ_{2}). For exp_{γ_{2}} t ∈ exp_{γ_{2}} 𝔥^{−1} and Hom_{MC^{1}(𝔥)}(γ_{1}, γ_{2}) the action is given by

(exp_{γ_{2}} t) · (exp X) = exp(δt + [γ_{2}, t]) exp X ∈ exp 𝔥^{0}.

By definition, Hom_{MC^{2}(𝔥)}(γ_{1}, γ_{2}) is the groupoid associated with the above action. The horizontal composition in MC^{2}(𝔥), i.e. the map of groupoids

⊗ : Hom_{MC^{2}(𝔥)}(exp X_{23}, exp Y_{23}) × Hom_{MC^{2}(𝔥)}(exp X_{12}, exp Y_{12}) →

Hom_{MC^{2}(𝔥)}(exp X_{23} exp X_{12}, exp X_{23} exp Y_{12}),

where γ_{i} ∈ MC(𝔥), exp X_{ij}, exp Y_{ij}, 1 ≤ i, j ≤ 3 is defined by

exp_{γ_{3}} t_{23} ⊗ exp_{γ_{2}} t_{12} = exp_{γ_{3}} t_{23} exp_{γ_{3}}(exp(ad X_{23})(t_{12})),

where exp_{γ_{j}} t_{ij} ∈ Hom_{MC^{2}(𝔥)}(exp X_{ij}, exp Y_{ij}).

3.11. Remark. There is a canonical map of 2-groupoids MC^{1}(𝔥) → MC^{2}(𝔥) which induces a bijection π_{0}(MC^{1}(𝔥)) → π_{0}(MC^{2}(𝔥)) on sets of isomorphism classes of objects.

3.12. Properties of 𝔑 MC^{2}.

3.12.1. Abelian DGLA.

3.13. Lemma. Suppose that 𝔞 is an abelian DGLA satisfying 𝔞^{i} = 0 for i < −1. Then, the simplicial sets 𝔑 MC^{2}(𝔞) and K(𝔞[1]) are isomorphic naturally in 𝔞.

Proof. The claim is an immediate consequence of the definitions and the explicit description of the nerve of MC^{2}(𝔞) given in Lemma 2.4.

Combining Lemma 3.13 with the integration map (8) we obtain the map of simplicial abelian groups

∫ : Σ(𝔞) → 𝔑 MC^{2}(𝔞)  (13)

which is a weak homotopy equivalence.

3.13.1. Central extensions. Suppose that 𝔤 is a nilpotent DGLA satisfying 𝔤^{i} = 0 for i < −1 and 𝔞 is a central subalgebra in 𝔤. Note that MC^{2} commutes with products, 𝔑 commutes with products and the addition map +: 𝔞 × 𝔤 → 𝔤 is a morphism of DGLAs. Thus, we obtain an action of the simplicial abelian group 𝔑 MC^{2}(𝔞) on the simplicial set 𝔑 MC^{2}(𝔤)

𝔑 MC^{2}(+): 𝔑 MC^{2}(𝔞) × 𝔑 MC^{2}(𝔤) → 𝔑 MC^{2}(𝔤).

Note that the group structure on 𝔑 MC^{2}(𝔞) is obtained from the case 𝔞 = 𝔤. Clearly, the action is free and the map 𝔑 MC^{2}(𝔤) → 𝔑 MC^{2}(𝔤/𝔞) factors through 𝔑 MC^{2}(𝔤)/𝔑 MC^{2}(𝔞).

3.13.2. The obstruction map.

3.14. Lemma. The obstruction map (6) factors as

MC(𝔤/𝔞) → π_{0} MC^{2}(𝔤/𝔞) → H^{2}(𝔞)

Proof. Suppose μ + 𝔞^{1} ∈ MC(𝔤/𝔞). It follows from the formula (10) that

exp(X + 𝔞^{0}) · (μ + 𝔞^{1}) = (exp X) · μ + 𝔞^{1}.

The formula (11) implies that

F(exp(X + 𝔞^{0}) · (μ + 𝔞^{1})) = F((exp X) · μ) + δ𝔞^{1} = exp(ad X)(F(μ) + δ𝔞^{1}).

Since F(μ)+δ𝔞^{1} ⊂ 𝔞^{2}, it follows that exp(ad X)(F(μ)+δ𝔞^{1}) = F(μ)+δ𝔞^{1} or, equivalently, o_{2}(exp(X + 𝔞^{0}) · (μ + 𝔞^{1})) = o_{2}(μ + 𝔞^{1}).

Recall (Lemma 2.4) that an n-simplex of 𝔑 MC^{2}(𝔤/𝔞), i.e. an element of 𝔑_{n} MC^{2}(𝔤/𝔞) includes, among other things, a collection of n + 1 gauge-equivalent Maurer-Cartan elements of 𝔤/𝔞. By Lemma 3.14 all of these Maurer-Cartan elements give rise to the same element of H^{2}(𝔞) under the map (6). Therefore, the assignment of this common value to an element of 𝔑_{n} MC^{2}(𝔤/𝔞) give rise to a well-defined map

o_{2,n}: 𝔑_{n} MC^{2}(𝔤/𝔞) → H^{2}(𝔞)  (14)

for each n = 0, 1, 2, … such that the sequence of pointed sets

0 → 𝔑_{n} MC^{2}(𝔤)/𝔑_{n} MC^{2}(𝔞) → 𝔑_{n} MC^{2}(𝔤/𝔞) ⟶^{o_{2,n}} H^{2}(𝔞)

is exact. The maps (14) assemble into a map of simplicial sets

o_{2} : 𝔑 MC^{2}(𝔤/𝔞) ⟶^{o_{2}} H^{2}(𝔞),

where H^{2}(𝔞) is constant. Let 𝔑 MC^{2}(𝔤/𝔞)_{0} = o_{2}^{−1}(0). The simplicial subset 𝔑 MC^{2}(𝔤/𝔞)_{0} is a union of connected components of 𝔑 MC^{2}(𝔤/𝔞) equal to the range of the map 𝔑 MC^{2}(𝔤)/𝔑 MC^{2}(𝔞) → 𝔑 MC^{2}(𝔤/𝔞).

It follows that 𝔑 MC^{2}(𝔤) → 𝔑 MC^{2}(𝔤/𝔞)_{0} is a principal fibration with the group 𝔑 MC^{2}(𝔞).

4. 𝔑 MC^{2} vs. Σ

In this section we show that for a DGLA 𝔥 satisfying 𝔥^{i} = 0 for i < −1 the simplicial sets 𝔑 MC^{2}(𝔥) and Σ(𝔥) are isomorphic in the homotopy category of simplicial sets.

4.1. The main theorem. Let Σ_{n}^{2}(𝔥) = { \atop MC^{2}(Ω_{n} ⊗ 𝔥)}, where the latter is the simplicial groupoid associated with the strict 2-groupoid MC^{2}(Ω_{n} ⊗𝔥) (see 2.3.1). Let Σ^{2}(𝔥): [n] ↦ Σ_{n}^{2}(𝔥) denote the corresponding simplicial object in simplicial groupoids. Note that Σ(𝔥) is the simplicial set of objects of Σ^{2}(𝔥), hence there is a canonical map

Σ(𝔥) → 𝔑Σ^{2}(𝔥).  (15)

The map ℚ → Ω_{•} of simplicial DGA induces the map of simplicial objects in simplicial groupoids

MC^{2}(𝔥) → Σ^{2}(𝔥).  (16)

Consider the diagram

Σ(𝔥) ⟶^{(15)} 𝔑Σ^{2}(𝔥) ⟵^{𝔑((16))} 𝔑 MC^{2}(𝔥).  (17)

4.2. Theorem. Suppose that 𝔥 is a nilpotent DGLA satisfying 𝔥^{i} = 0 for i < −1. Then, the morphisms (15) and 𝔑((16)) are weak homotopy equivalences so that the diagram (17) represents an isomorphism Σ(𝔥) ≅ 𝔑 MC^{2}(𝔥) in the homotopy category of simplicial sets.

The rest of Section 4 is devoted to a proof of Theorem 4.2 which borrows techniques from the proof of Proposition 3.2.1 of [Hinich, 2004].

4.3. The map (15) is a weak homotopy equivalence. Let Σ^{1}(𝔥) denote the simplicial object in groupoids defined by Σ_{n}^{1}(𝔥) = MC^{1}(Ω_{n} ⊗𝔥). Note that Σ(𝔥) is the simplicial set of objects of Σ^{1}(𝔥) and hence there is a canonical map

Σ(𝔥) → N Σ^{1}(𝔥);  (18)

by Remark 3.11 there is a canonical map of simplicial objects in simplicial groupoids

Σ^{1}(𝔥) → Σ^{2}(𝔥).  (19)

The map (15) is equal to the composition

Σ(𝔥) ⟶^{(18)} N Σ^{1}(𝔥) ⟶^{N ((19))} N Σ^{2}(𝔥) → 𝔑Σ^{2}(𝔥),

where the last map is the weak homotopy equivalence of Theorem 2.2.

4.4. Lemma. ([Hinich, 2004], Proposition 3.2.1) The map (18) is a weak homotopy equivalence.

Proof. Let G_{n}(𝔥) := exp((Ω_{n} ⊗ 𝔥)^{0}). Then, G(𝔥): [n] ↦ G_{n}(𝔥) is a simplicial group acting on Σ(𝔥), and Σ(𝔥) is the associated groupoid. Therefore,

N_{q}Σ(𝔥) = Σ(𝔥) × G(𝔥)^{×q}

and the map

Σ(𝔥) → N_{q}Σ(𝔥)

is a weak homotopy equivalence because G(𝔥) is contractible.

4.5. Proposition. The map N ((19)) is a weak homotopy equivalence.

Proof. Let Γ^{1}(𝔥) (respectively, Γ^{2}(𝔥)) denote the full subcategory of Σ^{1}(𝔥) (respectively, of Σ^{2}(𝔥)) whose set of objects is MC(𝔥) (a constant simplicial set). There is a commutative diagram

Γ^{1}(𝔥) ⟶ Γ^{2}(𝔥)
↓  ↓
Σ^{1}(𝔥) ⟶^{(19)} Σ^{2}(𝔥)

The vertical arrows induce weak homotopy equivalences on respective nerves since, for each n the functors Γ^{1}(𝔥)_{n} → Σ^{1}(𝔥)_{n} = MC^{1}(Ω_{n} ⊗ 𝔥) and Γ^{2}(𝔥)_{n} → Σ^{2}(𝔥)_{n} = MC^{2}(Ω_{n} ⊗ 𝔥) are equivalences by [Hinich, 2001], Proposition 8.2.5.

The map Γ^{1}(𝔥) → Γ^{2}(𝔥) induces a bijection between sets of isomorphism classes of objects. For μ ∈ MC(𝔥), Hom_{Γ^{2}(𝔥)}(μ, μ) is naturally identified with the nerve of the groupoid associated to the action of the simplicial group H(𝔥, μ): [n] ↦ exp((Ω_{n}⊗𝔥)_{μ}) on the simplicial set Hom_{Γ^{1}(𝔥)}(μ, μ). Since the group H(𝔥, μ) is contractible (it is isomorphic as a simplicial set to [n] ↦ Ω_{n}^{0} ⊗ 𝔥^{−1}) the induced map Hom_{Γ^{1}(𝔥)}(μ, μ) → Hom_{Γ^{2}(𝔥)}(μ, μ) is an equivalence.

4.6. The map 𝔑((16)): 𝔑 MC^{2}(𝔥) → 𝔑Σ^{2}(𝔥) is a weak homotopy equivalence. It suffices to show that the map

𝔑 MC^{2}(𝔥) → 𝔑 MC^{2}(Ω_{n} ⊗ 𝔥)

is a weak homotopy equivalence for all n. This follows from Proposition 4.7.

4.7. Proposition. Suppose that 𝔥 is a nilpotent DGLA concentrated in degrees greater than or equal to −1. The functor

MC^{2}(𝔥) → MC^{2}(Ω_{n} ⊗ 𝔥)  (20)

is an equivalence.

Proof. The induced map π_{0}((20)) is a bijection by Remark 3.11 and (the proof of) [Hinich, 1997], Lemma 2.2.1. The result now follows from Lemma 4.8 below.

4.8. Lemma. Suppose μ ∈ MC(𝔥). The functor

Hom_{MC^{2}(𝔥)}(μ, μ) → Hom_{MC^{2}(Ω_{n}⊗𝔥)}(μ, μ)  (21)

is an equivalence.

Proof. According to the description given in 3.10.3, for any nilpotent DGLA (𝔤, δ) with 𝔤^{i} = 0 for i < −1 and μ ∈ MC(𝔤) the groupoid Hom_{MC^{2}(𝔤)}(μ, μ) is isomorphic to the groupoid associated with the action of the group exp_{μ} 𝔤^{−1} on the set exp(ker(δ_{μ}^{−1})) ⊂ exp(𝔤^{0}) where δ_{μ} = δ + [μ, .].

Note that, for any X ∈ ker(δ_{μ}^{−1}), the automorphism group Aut(exp(X)) is isomorphic to (the additive group) ker(δ_{μ}^{−1}).

The map

([n] → [0])^{∗} ⊗ Id: (𝔥, δ) → (Ω_{n} ⊗ 𝔥, d + δ)  (22)

is a quasi-isomorphism of DGLA with the quasi-inverse given by the evaluation map ev_{0} := ([0] → [n])^{∗}⊗Id: Ω_{n}⊗𝔥 → 𝔥 (for any choice of a morphism [0] → [n]) which is a morphism of DGLA as well. The same maps are mutually quasi-inverse quasi-isomorphisms of DGLA

(𝔥, δ_{μ}) ⇄ (Ω_{n} ⊗ 𝔥, d + δ_{μ}).

Since (22) is a quasi-isomorphism and both DGLA are concentrated in degrees greater than or equal to −1, the induced map ker(δ_{μ}^{−1}) → ker((d + δ_{μ})^{−1}) an isomorphism, hence so are the maps of automorphism groups.

Since the map (21) admits a left inverse (namely, ev_{0}) it remains to show that the induced map on sets of isomorphism classes is surjective. Note that, since ev_{0} is a surjective quasi-isomorphism, the map d + δ_{μ}: ker(ev_{0})^{−1} → ker(ev_{0})^{0} ⋂ ker((d + δ_{μ})^{0}) is an isomorphism.

Consider X ∈ (Ω_{n} ⊗𝔤)^{0}. Then, X = ev_{0}(X)+Y with Y ∈ ker(ev_{0}), and (d+δ_{μ})X = 0 if and only if δ_{μ} ev_{0}(X) = 0 and (d + δ_{μ})Y = 0.

Suppose X ∈ ker((d + δ_{μ})^{0}). Then, exp(X) = exp(ev_{0}(X)) · exp(Z) where Z ∈ ker(ev_{0})^{0} ⋂ ker((d + δ_{μ})^{0}), and, therefore, Z = (d + δ_{μ})U for a uniquely determined U.

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Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia

Department of Mathematics, UCB 395, University of Colorado, Boulder, CO 80309-0395, USA

Department of Mathematics, Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark

Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA

Email: paul.bressler@gmail.com
      Alexander.Gorokhovsky@colorado.edu
      rnest@math.ku.dk
      b-tsygan@northwestern.edu
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