aa r X i v : . [ m a t h . A T ] M a y 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 − g -valued differential forms introduced by V. Hinich [9]. Introduction
The principal result of the present note compares two spaces (sim-plicial sets) naturally associated with a nilpotent differential gradedLie algebra (DGLA) subject to certain restrictions. Our interest inthis problem has its origins in formal deformation theory of associa-tive algebras and, more generally, algebroid stacks ([1]). The results ofthe present note are used in [2] to deduce a quasi-classical descriptionof the deformation theory of a gerbe from the formality theorem ofM. Kontsevich.To a nilpotent DGLA h which satisfies the additional condition(1.1) h i = 0 for i < − ( h ) and refer to as the Deligne 2-groupoid.Our principal result (Theorem 4.1) compares the simplicial nerve N MC ( h ) of the 2-groupoid MC ( h ), h a nilpotent DGLA satisfying(1.1), to another simplicial set, denoted Σ( h ), introduced by V. Hinich[9]: Theorem. (Main theorem) Suppose that h is a nilpotent DGLA suchthat h i = 0 for i < − . Then, the simplicial sets N MC ( h ) and Σ( h ) are homotopy equivalent. A. Gorokhovsky was partially supported by NSF grant DMS-0900968. B. Tsy-gan was partially supported by NSF grant DMS-0906391. R. Nest was partiallysupported by the Danish National Research Foundation through the Centre forSymmetry and Deformation (DNRF92).
In the case when the nilpotent DGLA h satisfies h i = 0 for i < ( h ) is an ordinary groupoid a homotopy equivalencebetween Σ( h ) and the nerve of MC ( h ) was constructed by V. Hinichin [9].Differential grade Lie algebras satisfying (1.1) arise in formal defor-mation theory of algebraic structures such as Lie algebras, commuta-tive algebras, associative algebras to name a few. In what follows weshall concentrate on the latter example. Let k denote an algebraicallyclosed field of characteristic zero. For an associative algebra A over k the shifted Hochschild cochain complex C • ( A )[1] has a canonicalstructure of a DGLA under the Gerstenhaber bracket; we denote thisDGLA by g ( A ) for short. Suppose that m is a nilpotent commuta-tive k -algebra (without unit). Then, g ( A ) ⊗ k m is a nilpotent DGLAwhich satisfies (1.1). Thus, the Deligne 2-groupoid MC ( g ( A ) ⊗ k m )is defined. For an Artin k -algebra R with maximal ideal m R the 2-groupoid MC ( 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 ) controlsthe formal deformation theory of A .The reason for considering the space Σ( h ) is that it is defined notjust for a DGLA (V. Hinich, [9]), but, more generally, for any nilpotent L ∞ algebra (E. Getzler, [6]). Homotopy invariance properties of thefunctor Σ (Proposition 3.7), the theory of J.W. Duskin ([5]) and thetheorem above yield the following result. If h is a DGLA satisfying(1.1), g is a L ∞ algebra L ∞ -quasi-isomorphic to h and m is a nilpotentcommutative k -algebra, then N MC ( h ⊗ k m ) is homotopy equivalentto Σ( g ⊗ k m ). Thus, the 2-groupoid MC ( h ⊗ k m ) can be reconstructed,up to equivalence, from the space Σ( g ⊗ k m ). The situation envisagedabove 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 variousconstructions of nerves of 2-groupoids and their properties. In section3 we recall the definitions of the functor Σ (3.2) and of the Deligne2-groupoid (3.3) and prove basic properties thereof. The proof of themain theorem (Theorem 4.1) given in Section 4 proceeds by exhibitingcanonical homotopy equivalences from Σ( h ) and N MC ( h ) to a thirdnaturally defined simplicial set.2. The homotopy type of a strict 2-groupoid
Nerves of simplicial groupoids.
Simplicial groupoids.
In what follows a simplicial category is acategory enriched over the category of simplicial sets. A small simplicial
ELIGNE GROUPOID REVISITED 3 category consists of a set of objects and a simplicial set of morphismsfor 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 G p is constant.A simplicial category is a simplicial groupoid if it is a groupoid ineach (simplicial) degree.2.1.2. The na¨ıve nerve.
Suppose that G is a simplicial category. Ap-plying the nerve functor degree-wise we obtain the bi-simplicial set N G : ([ p ] , [ q ]) N q G p whose diagonal we denote by N G and refer to asthe na¨ıve nerve of G .2.1.3. The simplicial nerve.
For a simplicial category G the simplicialnerve , also known as the homotopy coherent nerve, N G is representedby the cosimplicial object in [ p ] ∆ p N ∈ Cat ∆ , i.e N p G = Hom Cat ∆ (∆ p N , G ) . Here, ∆ p N is the canonical free simplicial resolution of [ p ] which admitsthe following explicit description ([3]).The set of objects of ∆ p N is { , , . . . , p } . For 0 ≤ i ≤ j ≤ p thesimplicial set of morphisms is given by Hom ∆ p N ( i, j ) = N P ( i, j ). Thecategory P ( i, j ) is a sub-poset of 2 { ,...,p } (with the induced partialordering whereby viewed as a category) given by P ( i, j ) = { I ⊂ Z | ( i, j ∈ I ) & ( k ∈ I = ⇒ i k j ) } . The composition in ∆ p N is induced by functors P ( i, j ) × P ( j, k ) → P ( i, k ) : ( I, J ) I ∪ J. In particular, ∆ N = [0] and ∆ N = [1]We refer the reader to [10] for applications to deformation theory andto [13] for the connection with higher category theory. The simplicialnerve of a simplicial groupoid is a Kan complex which reduces to theusual nerve for ordinary groupoids.Since ∆ N = [0] (respectively, ∆ N = [1]) it follows that N G (respec-tively, N G ) is the set of objects (respectively, the set of morphisms) of G .2.1.4. Comparison of nerves.
We refer the reader to [10] for the defini-tion of the canonical map of simplicial sets N G → N G . In what followswe will make use of the following result of loc. cit. Theorem 2.1 ([10]) . For any simplicial groupoid G the canonical map N G → N G is an equivalence. Strict 2-groupoids.
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
From strict -groupoids to simplicial groupoids. Suppose that G is a strict 2-groupoid, i.e. a groupoid enriched over the category ofgroupoids. 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 prod-ucts. Let G p denote the category with the same objects as G and withmorphisms defined by Hom G p ( g, g ′ ) = N p Hom G ( g, g ′ ); the composi-tion of morphisms is induced by the composition in G . Note that thegroupoid G is obtained from G by forgetting the 2-morphisms.The assignment [ p ] G p defines a simplicial object in groupoidswith the constant simplicial set of objects, i.e. a simplicial groupoidwhich we denote by e G . Lemma 2.2.
The simplicial nerve N e G admits the following explicitdescription: (1) There is a canonical bijection between N e G and the set of objectsof G . (2) For n ≥ there is a canonical bijection between N n e G and theset of data of the form (( µ i ) ≤ i ≤ n , ( g ij ) ≤ i Maurer-Cartan elements. Suppose that g is a nilpotent L ∞ -algebra. For µ ∈ g let(3.1) F ( µ ) = δµ + ∞ X k =2 k ! [ µ ∧ k ] . The element F ( µ ) of g is called the curvature of µ . For any µ ∈ g the curvature F ( µ ) satisfies the Bianchi identity ([6], Lemma 4.5)(3.2) δ F ( µ ) + ∞ X k =1 k ! [ µ ∧ k , F ( µ )] = 0 . An element µ ∈ g is called a Maurer-Cartan element (of g ) if itsatisfies the condition F ( µ ) = 0. The set of Maurer-Cartan elementsof g will be denoted MC( g ):MC( g ) := { µ ∈ g | F ( µ ) = 0 } . The set MC( g ) is pointed by the distinguished element 0 ∈ g .Suppose that a is an abelian L ∞ -algebra. Then,MC( a ) = Z ( a ) := ker( δ : a → a ) , 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 asubcomplex of ( g , δ ) such that [ a , g , . . . , g ] = 0 for all k ≥ 2. In thiscase we will say that a is central in g .If a is central in g , then there is a unique structure of an L ∞ -algebraon 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 iscentral in g . Lemma 3.1. (1) The addition operation on g restricts to a free action of theabelian group MC( a ) on the set MC( g ) . (2) The map MC( g ) → MC( g / a ) is constant on the orbits of theaction. (3) The induced map MC( g ) / MC( a ) → MC( g / a ) is injective.Proof. Suppose that α ∈ a and µ ∈ g . Since a is central in g ,[( α + µ ) ∧ k ] = [ µ ∧ k ] for k ≥ F ( α + µ ) = δα + F ( µ ) (in thenotation of (3.1)). Therefore, MC( a ) + MC( g ) = MC( g ). In otherwords, the addition operation in g restricts to an action of the abeliangroup MC( a ) on the set MC( g ) which is obviously free. Since the mapMC( g ) → MC( g / a ) is the restriction of the map g → g / a constant on ELIGNE GROUPOID REVISITED 7 the orbits of the action, i.e. factors through MC( g ) / MC( a ), and theinduced map MC( g ) / MC( a ) → MC( g / a ) is injective. (cid:3) The obstruction map. The image of the map MC( g ) → MC( g / a )may be described in terms of the obstruction map (3.3) which we con-struct presently.If µ ∈ g and µ + a ∈ MC( g / a ), then F ( µ + a ) = F ( µ ) + δ a ⊂ a and the Bianchi identity (3.2) reduces to δ F ( µ + a ) = 0, i.e. theassignment µ + a 7→ F ( µ + a ) gives rise to a well-defined map(3.3) o : MC( g / a ) → H ( a )(notation borrowed from [8], 2.6). Lemma 3.2. The sequence of pointed sets (3.4) 0 → MC( g ) / MC( a ) → MC( g / a ) o −→ H ( a ) is exact.Proof. If F ( µ + a ) ⊂ δ a , then there exists α ∈ a such that F ( µ + α ) =0, i.e. µ + a is in the image of MC( g ) → MC( g / a ). (cid:3) The functor Σ . In what follows we denote by Ω n , n = 0 , , , . . . the commutative differential graded algebra over Q with generators t , . . . , t n of degree zero and dt , . . . , dt n of degree one subject to therelations t + · · · + t n = 1 and dt + · · · + 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 differ-ential graded algebra.3.2.1. The simplicial set Σ( g ) . For a nilpotent L ∞ -algebra g and anon-negative integer n letΣ 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 Hinich in [9] for DGLAand used by Getzler in [6] (where it is denoted MC • ( g )) for generalnilpotent L ∞ -algebras.3.2.2. Abelian algebras. If a is an abelian algebra, then Σ( a ) is givenby Σ n ( a ) = Z (Ω n ⊗ a ) = Z (Ω n ⊗ a [1]) and has a canonical structureof a simplicial abelian group. In particular, it is a Kan simplicial set.Recall that the Dold-Kan correspondence associates to a complex ofabelian groups A a simplicial abelian group K ( A ) defined by K ( A ) n = P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN Z ( C • ([ n ]; A )), the group of cocycles of (total) degree zero in the com-plex of simplicial cochains on the n -simplex with coefficients in A .The integration map R : Ω n ⊗ a → C • ([ n ]; a ) induces a homotopyequivalence(3.5) Z : Σ( a ) → K ( a [1]) . Thus, π i Σ( a ) ∼ = H − i ( a ).3.2.3. Central extensions. Suppose that g is a nilpotent L ∞ -algebraand a is a central subalgebra in g . Then, for n = 0 , , . . . , Ω n ⊗ a iscentral in Ω n ⊗ g . Lemma 3.3. (1) The addition operation on (Ω n ⊗ g ) induces a principal actionof 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 3.1 and the naturality properties of theconstructions in 3.1.2. (cid:3) For n = 0 , , . . . the map ([ n ] → [0]) ∗ : Q → Ω n is a quasi-isomorphism,with the quasi-inverse provided by the map induced by any morphism[0] → [ n ]. Therefore, the map a → Ω n ⊗ a is a quasi-isomorphism aswell. The induced isomorphisms H ( a ) ∼ = H (Ω n ⊗ a ) give rise to theisomorphism of the constant simplicial set H ( a ) and n H (Ω n ⊗ a ).The maps o ,n : Σ n ( g / a ) = MC(Ω n ⊗ g / a ) → H (Ω n ⊗ a ) ∼ = H ( a )assemble into the map of simplicial sets(3.6) o : Σ( g / a ) → H ( a ) . which factors as Σ( g / a ) → π Σ( g / a ) → H ( a ).Let Σ( g / a ) = o − (0). Thus, by (3.4), Σ( g / a ) is a union of con-nected 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 ) is a principal fibration withgroup Σ( a ), in particular, a Kan fibration ([14], Lemma 18.2). Lemma 3.4. Suppose that g is a nilpotent L ∞ -algebra. Then, Σ( g ) isa Kan simplicial set.Proof. If that g is abelian then Σ( g ) is a simplicial group and thereforea Kan simplicial set. ELIGNE GROUPOID REVISITED 9 Let F • g denote the lower central series. Assume that Gr iF g = 0 ifand only if 0 ≤ i ≤ n ; that is, g is nilpotent of length n . By inductionassume that Σ( h ) is a Kan simplicial set for any nilpotent L ∞ -algebra h of length at most n − g is nilpotent of length n , it follows that F n g = Gr n g is centralin g and g /F n g is nilpotent of length n − 1. Therefore, Σ( g /F n g is aKan simplicial set and so is Σ( g /F n g ) . Since Σ( g ) → Σ( g /F n g ) is aKan fibration it follows that Σ( g ) is a Kan simplicial set as well. (cid:3) Lemma 3.5. 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 abelian. Then, π i Σ( g ) ∼ = H − i ( g ). For g notnecessarily abelian the statement follows by induction on the nilpotencylength, the abelian case establishing the base of the induction.Let F • g denote the lower central series. Assume that Gr iF g = 0 ifand only if 0 ≤ i ≤ n ; that is, g is nilpotent of length n . By inductionassume that the conclusion holds for all nilpotent L ∞ -algebras of lengthat most n − g is nilpotent of length n , it follows that F n g = Gr n g is centralin g and g /F n g is nilpotent of length n − 1. Let X ⊆ Σ( g ) be aconnected 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 componentof the identity in Σ( F n g ). The desired vanishing of higher homotopygroups of X follows from the induction hypotheses using the long exactsequence of homotopy groups. (cid:3) Homotopy invariance. Lemma 3.6. Suppose that f : a → b is a quasi-isomorphism of abelianalgebras. Then, the induced map Σ( f ) : Σ( a ) → Σ( b ) is an equivalence.Proof. Note that Σ( f ) is a morphism of simplicial abelian groups. Itis sufficient to show that the maps π n Σ( f ) : π n Σ( a ) → π n Σ( b ) areisomorphisms for n > 0. To this end note that π n Σ( f ) factors as thecomposition of isomorphisms π n Σ( a ) ∼ = H − n ( a ) H − n (Σ( f )) −−−−−−−→ H − n ( b ) ∼ = π n Σ( b ) . (cid:3) Proposition 3.7 ([6], Proposition 4.9) . Suppose that f : g → h isa quasi-isomorphism 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 an equivalence.Proof. We use induction on the nilpotency length of m R , which is tosay the largest integer l such that m lR = 0.If m R = 0, then f ⊗ Id : g ⊗ m R → h ⊗ m R is a quasi-isomorphism ofabelian algebras and the claim follows from Lemma 3.6.Suppose that m l +1 R = 0. By the induction hypothesis • the map Σ( g ⊗ m R / m lR ) → Σ( h ⊗ m R / m lR ) is an equivalence and • the map π Σ( g ⊗ m R / m lR ) → π Σ( h ⊗ m R / m lR ) is a bijection.The map f ⊗ Id m lR is a quasi-isomorphism of abelian L ∞ -algebras,therefore the map H ( g ⊗ m lR ) → H ( h ⊗ m lR ) is an isomorphism. Thecommutativity of π Σ( g ⊗ m R / m lR ) −−−→ π Σ( h ⊗ m R / m lR ) y y H ( g ⊗ m lR ) −−−→ H ( h ⊗ m lR )implies that the map π Σ( g ⊗ m R / m lR ) → π Σ( h ⊗ m R / m lR ) is a bijection. Therefore, the mapΣ( g ⊗ m R / m lR ) → Σ( h ⊗ m R / m lR ) is an equivalence. The map Σ( f ) restricts to a map of principal fibra-tions Σ( g ⊗ m R ) −−−→ Σ( h ⊗ m R ) y y Σ( g ⊗ m R / m lR ) −−−→ Σ( h ⊗ m R / m lR ) relative to the map of simplicial groups Σ( g ⊗ m lR ) → Σ( h ⊗ m lR ).The latter is an equivalence by Lemma 3.6. Therefore, so is the mapΣ( g ⊗ m R ) → Σ( h ⊗ m R ). (cid:3) Deligne groupoids. Gauge transformations. Suppose that h is a nilpotent DGLA.Then, h is a nilpotent Lie algebra. The unipotent group exp h actson the space h by affine transformations. The action of exp X , X ∈ h ,on γ ∈ h is given by the formula(3.7) (exp X ) · γ = γ − ∞ X i =0 ( ad X ) i ( i + 1)! ( δX + [ γ, X ]) . ELIGNE GROUPOID REVISITED 11 The effect of the above action on the curvature F ( γ ) = δγ + 12 [ γ, γ ] isgiven by(3.8) F ((exp X ) · γ ) = exp( ad X )( F ( γ )) . The functor MC . Suppose that h is a nilpotent DGLA. It fol-lows from (3.8) that gauge transformations (3.7) preserve the subset ofMaurer-Cartan elements MC( h ) ⊂ h .We denote by MC ( h ) the Deligne groupoid (denoted C ( h ) in [9])defined as the groupoid associated with the action of the group exp h by gauge transformations on the set MC( h ).Thus, MC ( h ) is the category with the set of objects MC( h ). For γ , γ ∈ MC( h ), Hom MC ( h ) ( γ , γ ) is the set of gauge transformationsbetween γ , γ . The compositionHom MC ( h ) ( γ , γ ) × Hom MC ( h ) ( γ , γ ) → Hom MC ( h ) ( γ , γ )is given by the product in the group exp( h ).3.3.3. The functor MC . For h as above satisfying the additional van-ishing condition h i = 0 for i < − ( h ) the Deligne2-groupoid as defined by P. Deligne [4] and independently by E. Get-zler, [6]. Below we review the construction of Deligne 2-groupoid of anilpotent DGLA following [6, 7] and references therein.The objects and the 1-morphisms of MC ( h ) are those of MC ( h ).That is, for γ , γ ∈ MC( h ) the set Hom MC ( h ) ( γ , γ ) is the set of ob-jects of the groupoid Hom MC ( h ) ( γ , γ ). The morphisms in Hom MC ( h ) ( γ , γ )(i.e. the 2-morphisms of MC ( h )) are defined as follows.For γ ∈ MC( h ) let [ · , · ] γ denote the Lie bracket on h − defined by(3.9) [ a, b ] γ = [ a, δb + [ γ, b ]] . Equipped with this bracket, h − becomes a nilpotent Lie algebra. Wedenote by exp γ h − the corresponding unipotent group, and byexp γ : h − → exp γ h − the corresponding exponential map. If γ , γ are two Maurer-Cartanelements, then the group exp γ h − acts on Hom MC ( h ) ( γ , γ ). Forexp γ t ∈ exp γ h − and Hom MC ( h ) ( γ , γ ) the action is given by(exp γ t ) · (exp X ) = exp( δt + [ γ , t ]) exp X ∈ exp h . By definition, Hom MC ( h ) ( γ , γ ) is the groupoid associated with theabove action. The horizontal composition in MC ( h ), i.e. the map of groupoids ⊗ : Hom MC ( h ) (exp X , exp Y ) × Hom MC ( h ) (exp X , exp Y ) → Hom MC ( h ) (exp X exp X , exp X exp Y ) , where γ i ∈ MC( h ), exp X ij , exp Y ij , 1 ≤ i, j ≤ γ t ⊗ exp γ t = exp γ t exp γ (exp( ad X )( t )) , where exp γ j t ij ∈ Hom MC ( h ) (exp X ij , exp Y ij ). Remark . There is a canonical map of 2-groupoids MC ( h ) → MC ( h )which induces a bijection π (MC ( h )) → π (MC ( h )) on sets of isomor-phism classes of objects.3.4. Properties of N MC . Abelian algebras. Lemma 3.9. Suppose that a is an abelian DGLA satisfying a i = 0 for i < − . Then, the simplicial sets N MC ( a ) and K ( a [1]) are isomor-phic naturally in a .Proof. The claim is an immediate consequence of the definitions andthe explicit description of the nerve of MC ( a ) given in Lemma 2.2. (cid:3) Combining Lemma 3.9 with the integration map (3.5) we obtain themap of simplicial abelian groups(3.10) Z : Σ( a ) → N MC ( a )which is a homotopy equivalence.3.4.2. Central extensions. Suppose that g is a nilpotent DGLA satis-fying g i = 0 for i < − a is a central subalgebra in g . Note thatMC commutes with products, N commutes with products and the ad-dition map + : a × g → g is a morphism of DGLA. Thus, we obtain anaction of the simplicial abelian group N MC ( a ) on the simplicial set N MC ( g ) N MC (+) : N MC ( a ) × N MC ( g ) → N MC ( g ) . Note that the group structure on N MC ( a ) is obtained from the case a = g . Clearly, the action is free and the map N MC ( g ) → N MC ( g / a )factors through N MC ( g ) / N MC ( a ). ELIGNE GROUPOID REVISITED 13 The obstruction map. Lemma 3.10. The obstruction map (3.3) factors as MC( g / a ) → π MC ( g / a ) → H ( a ) Proof. Suppose µ + a ∈ MC( g / a ). It follows from the formula (3.7)that exp( X + a ) · ( µ + a ) = (exp X ) · µ + a . The formula (3.8) implies that F (exp( X + a ) · ( µ + a )) = F ((exp X ) · µ )+ δ a = exp( ad X )( F ( µ )+ δ a ) . Since F ( µ ) + δ a ⊂ a , it follows that exp( ad X )( F ( µ ) + δ a ) = F ( µ ) + δ a or, equivalently, o (exp( X + a ) · ( µ + a )) = o ( µ + a ). (cid:3) Recall (Lemma 2.2) that an n -simplex of N MC ( g / a ), i.e. an ele-ment of N n MC ( g / a ) includes, among other things, a collection of n +1gauge-equivalent Maurer-Cartan elements of g / a . By Lemma 3.10 allof these Maurer-Cartan elements give rise to the same element of H ( a )under the map (3.3). Therefore, the assignment of this common valueto an element of N n MC ( g / a ) give rise to a well-defined map(3.11) o ,n : N n MC ( g / a ) → H ( a )for each n = 0 , , , . . . such that the sequence of pointed sets0 → N n MC ( g ) / N n MC ( a ) → N n MC ( g / a ) o ,n −−→ H ( a )is exact. The maps (3.11) assemble into a map of simplicial sets o : N MC ( g / a ) o −→ H ( a ) , where H ( a ) is constant. Let N MC ( g / a ) = o − (0). The simplicialsubset N MC ( g / a ) is a union of connected components of N MC ( g / a )equal to the range of the map N MC ( g ) / N MC ( a ) → N MC ( g / a ).It follows that N MC ( g ) → N MC ( g / a ) is a principal fibrationwith the group N MC ( a ). 4. N MC vs. ΣIn this section we show that for a DGLA h satisfying h i = 0 for i < − N MC ( h ) and Σ( h ) are isomorphic in thehomotopy category of simplicial sets. The main theorem. Let Σ n ( h ) = ^ MC (Ω n ⊗ h ), where the lat-ter is the simplicial groupoid associated with the strict 2-groupoidMC (Ω n ⊗ h ) (see 2.2.1). Let Σ ( h ) : [ n ] Σ n ( h ) denote the cor-responding simplicial object in simplicial groupoids. Note that Σ( h ) isthe simplicial set of objects of Σ ( h ), hence there is a canonical map(4.1) Σ( h ) → N Σ ( h ) . The map Q → Ω • of simplicial DGA induces the map of simplicialobjects in simplicial groupoids(4.2) MC ( h ) → Σ ( h ) . Consider the diagram(4.3) Σ( h ) (4.1) −−−→ N Σ ( h ) N (4.2) ←−−− N MC ( h ) . Theorem 4.1. Suppose that h is a nilpotent DGLA satisfying h i = 0 for i < − . Then, the morphisms (4.1) and N (4.2) are equivalences sothat the diagram (4.3) represents an isomorphism Σ( h ) ∼ = N MC ( h ) inthe homotopy category of simplicial sets. The rest of Section 4 is devoted to a proof of Theorem 4.1 whichborrows techniques from the proof of Proposition 3.2.1 of [12].4.2. The map (4.1) is an equivalence. Let Σ ( h ) denote the sim-plicial object in groupoids defined by Σ n ( h ) = MC (Ω n ⊗ h ). Notethat Σ( h ) is the simplicial set of objects of Σ ( h ) and hence there is acanonical map(4.4) Σ( h ) → N Σ ( h );by Remark 3.8 there is a canonical map of simplicial objects in simpli-cial groupoids(4.5) Σ ( h ) → Σ ( h ) . The map (4.1) is equal to the compositionΣ( h ) (4.4) −−→ N Σ ( h ) N (4.5) −−−−→ N Σ ( h ) → N Σ ( h ) , where the last map is the equivalence of Theorem 2.1. Lemma 4.2 ([12], Proposition 3.2.1) . The map (4.4) is an equivalence.Proof. Let G n ( h ) := exp((Ω n ⊗ h ) ). Then, G ( h ) : [ n ] G n ( h ) is asimplicial group acting on Σ( h ), and Σ ( h ) is the associated groupoid.Therefore, N q Σ ( h ) = Σ( h ) × G ( h ) × q ELIGNE GROUPOID REVISITED 15 and the map Σ( h ) → N q Σ ( h )is an equivalence because G ( h ) is contractible. (cid:3) Proposition 4.3. The map N (4.5) is an equivalence.Proof. Let Γ ( h ) (respectively, Γ ( h )) denote the full subcategory of Σ ( h ) (respectively, of Σ ( h )) whose set of objects is MC( h ) (a constantsimplicial set). There is a commutative diagram Γ ( h ) −−−→ Γ ( h ) y y Σ ( h ) (4.5) −−−→ Σ ( h )The vertical arrows induce equivalences on respective nerves since, foreach n the functors Γ ( h ) n → Σ ( h ) n = MC (Ω n ⊗ h ) and Γ ( h ) n → Σ ( h ) n = MC (Ω n ⊗ h ) are equivalences by [11], Proposition 8.2.5.The map Γ ( h ) → Γ ( h ) induces a bijection between sets of iso-morphism classes of objects. For µ ∈ MC( h ), Hom Γ ( h ) ( µ, µ ) is natu-rally identified with the nerve of the groupoid associated to the actionof the simplicial group H ( h , µ ) : [ n ] exp((Ω n ⊗ h ) µ ) on the simpli-cial set Hom Γ ( h ) ( µ, µ ). Since the group H ( h , µ ) is contractible (it isisomorphic as a simplicial set to [ n ] Ω n ⊗ h − ) the induced mapHom Γ ( h ) ( µ, µ ) → Hom Γ ( h ) ( µ, µ ) is an equivalence. (cid:3) The map N ((4.2)) : N MC ( h ) → N Σ ( h ) is an equivalence. It suffices to show that the map N MC ( h ) → N MC (Ω n ⊗ h )is an equivalence for all n . This follows from Proposition 4.4. Proposition 4.4. Suppose that h is a nilpotent DGLA concentrated indegrees greater than or equal to − . The functor (4.6) MC ( h ) → MC (Ω n ⊗ h ) is an equivalence.Proof. The induced map π ((4.6)) is a bijection by Remark 3.8 and(the proof of) [9], Lemma 2.2.1. The result now follows from Lemma4.5 below. (cid:3) Lemma 4.5. Suppose µ ∈ MC( h ) . The functor (4.7) Hom MC ( h ) ( µ, µ ) → Hom MC (Ω n ⊗ h ) ( µ, µ ) is an equivalence. Proof. According to the description given in 3.3.3, for any nilpotentDGLA ( g , δ ) with g i = 0 for i < − µ ∈ MC( g ) the groupoidHom MC ( g ) ( µ, µ ) is isomorphic to the groupoid associated with the ac-tion of the group exp µ g − on the set exp(ker( δ − µ )) ⊂ exp( g ) where δ µ = δ + [ µ, . ].Note that, for any X ∈ ker( δ − µ ), the automorphism group Aut(exp( X ))is isomorphic to (the additive group) ker( δ − µ ).The map(4.8) ([ n ] → [0]) ∗ ⊗ Id : ( h , δ ) → (Ω n ⊗ h , d + δ )is a quasi-isomorphism of DGLA with the quasi-inverse given by theevaluation map ev := ([0] → [ n ]) ∗ ⊗ Id : Ω n ⊗ h → h (for any choiceof a morphism [0] → [ n ]) which is a morphism of DGLA as well. Thesame maps are mutually quasi-inverse quasi-isomorphisms of DGLA( h , δ µ ) ⇄ (Ω n ⊗ h , d + δ µ ) . Since (4.8) is a quasi-isomorphism and both DGLA are concentratedin degrees greater than or equal to − 1, the induced map ker( δ − µ ) → ker(( d + δ µ ) − ) an isomorphism, hence so are the maps of automorphismgroups.Since the map (4.7) admits a left inverse (namely, ev ) it remains toshow that the induced map on sets of isomorphism classes is surjective.Note that, since ev is a surjective quasi-isomorphism, the map d + δ µ : ker(ev ) − → ker(ev ) T ker(( d + δ µ ) ) is an isomorphism.Consider X ∈ (Ω n ⊗ g ) . Then, X = ev ( X ) + Y with Y ∈ ker(ev ),and ( d + δ µ ) X = 0 if and only if δ µ ev ( X ) = 0 and ( d + δ µ ) Y = 0.Suppose X ∈ ker(( d + δ µ ) ). Then, exp( X ) = exp(ev ( X )) · exp( Z )where Z ∈ ker(ev ) T ker(( d + δ µ ) ), and, therefore, Z = ( d + δ µ ) U fora uniquely determined U . (cid:3) References [1] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan. Deformation quantiza-tion of gerbes. Adv. Math. , 214(1):230–266, 2007.[2] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan. Formality theorem forgerbes. Adv. Math. , 273:215-241,2015. Preprint arXiv:1308.3951[3] J.-M. Cordier. Sur la notion de diagramme homotopiquement coh´erente. Cahiers Topologie G´eom. Diff´erentielle , 23(1), pages 93–112, 1982.[4] P. Deligne. Letter to L. Breen, 1994.[5] J. W. Duskin. Simplicial matrices and the nerves of weak n -categories. I. Nervesof bicategories. Theory Appl. Categ. , 9, pages 198–308 (electronic), 2001/02.CT2000 Conference (Como).[6] E. Getzler. Lie theory for nilpotent L-infinity algebras. Ann. of Math. (2), (2009), no. 1, 271–301. ELIGNE GROUPOID REVISITED 17 [7] E. Getzler. A Darboux theorem for Hamiltonian operators in the formal cal-culus of variations. Duke Math. J. , 111(3), pages 535–560, 2002.[8] W.M. Goldman, J.J. Millson. The deformation theory of representations offundamental groups of compact K¨ahler manifolds. Bull. Amer. Math. Soc.(N.S.) Volume 18, Number 2 (1988), 153–158.[9] V. Hinich. Descent of Deligne groupoids. Internat. Math. Res. Notices 1997,no. 5, pages 223-239.[10] V. Hinich. Homotopy coherent nerve in deformation theory. arXiv:0704.2503.[11] V. Hinich. DG-coalgebras as formal stacks. J. Pure Appl. Algebra 162 (2001),209-250.[12] V. Hinich. Deformations of homotopy algebras. Communication in Algebra Departamento de Matem´aticas, Universidad de Los Andes E-mail address : [email protected] Department of Mathematics, UCB 395, University of Colorado,Boulder, CO 80309-0395, USA E-mail address : [email protected] Department of Mathematics, Copenhagen University, Universitetsparken5, 2100 Copenhagen, Denmark E-mail address : [email protected] Department of Mathematics, Northwestern University, Evanston,IL 60208-2730, USA E-mail address ::