arXiv:1906.11977v1 [math.AT] 27 Jun 2019 Comparison of spaces associated to DGLA via higher holonomy Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, and Boris Tsygan Abstract. For a differential graded Lie algebra g whose components vanish in degrees below −1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of g-valued differential forms introduced by V. Hinich. The construction uses the theory of non-abelian multiplicative integration. 1. Introduction The principal result of the present note is a construction of a weak equivalence of two spaces (simplicial sets) naturally associated with a nilpotent differential grade Lie algebra subject to certain restrictions. To a nilpotent DGLA g which satisfies the additional condition (1.1) gi = 0 for i < −1 P. Deligne [3] and, independently, E. Getzler [4] associated a (strict) 2-groupoid which we denote MC2 (g) and refer to as the Deligne 2groupoid. In an earlier paper by the authors ([2]) it was shown that the simplicial nerve N MC2 (g) of the 2-groupoid MC2 (g), g a nilpotent DGLA satisfying (1.1) is weakly equivalent to another simplicial set, denoted Σ(g), introduced by V. Hinich [6] by constructing natural maps from N MC2 (g) and Σ(g) to a third simplicial set and showing that they are equivalences. In this note we outline a construction of the map (1.2) I(g) : Σ(g) → N MC2 (g) omitting technical details. The main tool in our construction is the theory of non-abelian multiplicative integration on surfaces. R. Nest was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). 1 2 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN In [9], Yekutieli constructed a theory of multiplicative integration on surfaces based on Riemann products(cf. also Aref’eva’s work [1]; a sketch of a construction, also based on Riemann products, is due to Kontsevich, cf. [8]) and does not apply in the context at hand. In [7] Kapranov develops a theory of higher holonomy based on Chen’s iterated integrals. In the present work we take the approach based on the fundamental theorem of calculus. The paper is organized as follows. In Section 2 we recall some basic notions of differential and integral calculus. In Section 3 we review the theory of connections and holonomy. In Section 4 we define the two-dimensional holonomy of connection-curvature pairs and discuss some of its properties. In Section 5 we recall the definitions of the simplicial sets N MC2 (g) and Σ(g), construct the map (1.2) and state its properties in Theorem 5.2 and Theorem 5.3. 1.1. Notation. Throughout the paper • k is a field of characteristic zero n • for a variety X over k: An X := X × Ak ; for a commutative n n k-algebra R: AR := ASpec R • for a variety prX : Y → X over X and a OX -module E we denote by EY the OY -module pr∗X E • for an X-variety : Y → X over X we denote by d/X : OY → Ω1Y/X the relative differential • for an nilpotent OX -Lie algebra g we denote by exp(g) the algebraic group over X defined as exp(g)(Y) = gY (Y) equipped with the group structure given by the Baker-Campbell-Hausdorff series • for a group G and g ∈ G we denote Conj(g) the automorphism h 7→ ghg−1 2. Calculus 2.1. Logarithmic derivatives. Suppose that g is a nilpotent Lie algebra. For D ∈ Der(g), a = exp(x) ∈ exp(g), x ∈ g. Define D log(a) ∈ g by   exp(ad(x)) − 1 −1 D log(a) := (Da) · a = (Dx). ad(x) For D ∈ Der(g), a, b ∈ exp(g), x ∈ g, (2.1) (2.2) (2.3) D log(ab) = D log(a) + Ada (D log(b)) D log(a−1 ) = − Ada−1 (D log(a)) adx log(a) = x − Ada (x) COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY3 For D1 , D2 ∈ Der(g), a ∈ exp(g) (2.4) D1 (D2 log(a)) − D2 (D1 log(a)) = [D1, D2 ] log(a) + [D1 log(a), D2 log(a)] 2.2. Initial value problems. Let prX : A1X → X denote the projection so that prX∗ OA1X = OX [s]. Suppose that E is a vector bundle on X. For • A ∈ EndOA1 (EA1X ) nilpotent X • a section σ : X → A1X of the projection prX • a section e ∈ Γ (X; E) there exists a unique section u ∈ Γ (A1X ; EA1X ) which satisfies   ∂u + A(u) = 0 ∂s σ∗ (u) = e. 2.3. Integration. Let x = s ⊗ 1 and y = 1 ⊗ s denote the coordinate functions on A1X ×X A1X . Let ∆ denote the subvariety defined by the equation x = y. For α ∈ Ω1A1 /X we denote by X Zy α ∈ Γ (A1X ×X A1X ; OA1X ×X A1X ) x the solution of the initial value problem   ∂u = pr∗ α 2 ∂y  u|∆ = 0 in the unknown u = u(x, y) ∈ Γ (A1X ×X A1X ; OA1X ×X A1X ). 3. Differential geometry 3.1. Connections and curvature. Suppose that g is a nilpotent OX -Lie algebra. For A ∈ Γ (AkX ; Ω1Ak /X ⊗ g) we consider connection X 1 ∇ := d/X +A. Its curvature is defined by F = F(∇) := d/X A+ [A, A] ∈ 2 Γ (AkX; Ω2Ak /X ⊗ g). X Suppose now that E is an OX -module and ρ : g → EndOX (E) is a representation of g. The connection ∇ gives rise to the connection ∇E on EAkX . 4 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN The sheaf Ω1Ak /X of relative differentials is equipped with the canonX ical relative connection. Hence, the connection ∇ induces a connection on the OAkX -module Ω1Ak /X ⊗ E which will be denoted ∇E as well. X An example of the above situation is given by the adjoint representation ad : g → Der(g) giving rise to the connection ∇g . 3.2. Logarithmic covariant derivative. For u = exp(x) ∈ exp(g)(AkX ), let   exp(ad(x)) − 1 (dx) d log(u) := ad(x) and ∇ log(u) := d log u + A. Lemma 3.1. For ξ1 , ξ2 ∈ TAkX /X such that [ξ1 , ξ2 ] = 0 the identity (3.1) ∇gξ1 ∇ξ2 log u − ∇gξ2 ∇ξ1 log u = [∇ξ1 log u, ∇ξ2 log u] + ιξ1 ιξ2 F holds. 3.3. Holonomy. Suppose that g is a nilpotent OX -Lie algebra. For A ∈ Γ (A1X ; Ω1A1 /X ⊗ g), ∇ := d/X + A we denote by X y P (x, y) 7→ ∇ ∈ exp(g)(A1X ×X A1X ) x the solution of the initial value problem  pr∗2 ∇ ∂ log(u) = 0 ∂y u|∆ = 1, where ∆ is the subvariety defined by the equation x = y. Suppose that E is an OX -module and ρ : g → EndOX (E) is a representation of g which is nilpotent in the sense that there exists N such that Im(ρ)N = 0. The representation ρ induces the map of groups pri exp(ρ) : exp(g) → Aut(E). Since the two compositions A1X ×X A1X −→ prX A1X −−→ X, where pri are the canonical projections, are equal it follows that HomO 1 1 (pr∗1 EA1X , pr∗2 EA1X ) = EndO 1 1 (EA1X ×X A1X ) and there A ×X A X X A ×X A X X is a canonical map (3.2) exp(g)(A1X ×X A1X ) → Isom(pr∗1 EA1X , pr∗2 EA1X ). y y We denote by P ∇E the image of P ∇ under (3.2). x x COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY5 Lemma 3.2. The following useful identities hold. y  −1 y  ∇E  ∂  ∇E  = ∇E∂ ∂y ∂y x x y   z  z P P P P P x y x  ∇ ·  ∇ = ∇ 3.4. Holonomy and curvature. Suppose that g is a nilpotent OX -Lie algebra, A ∈ Γ (A2X ; Ω1A2 /X ⊗ g), ∇ := d/X + A, F = F(∇) := X 1 2 2 d/X A + [A, A] ∈ Γ (AX ; ΩA2 /X ⊗ g). X 2 The second projection A2X → A1X identifies A2X with A1A1 . Let A ∈ X Γ (A2X; Ω1A2 /A1 ⊗ g) denote the image of A, ∇ := d/A1X + A. Hence, the X X holonomy y1 P ((x1 , y2 ), (y1, y2 )) 7→ ∇ ∈ exp(g)(A2X ×A1X A2X ) x1 y1 is defined and will be denoted, by abuse of notation, by P ∇. x1 Lemma 3.3. (3.3) y ∇ ∂ log  ∂y2  1 P x1 ∇ = t=y Z 1 t ( P ∇g )−1 (ι ∂ F) ∂y2 t=x1 y1 Proof. We will show that both sides of (3.3) are solutions of the initial value problem  g ∇ ∂ Φ = ι ∂y ∂ ι ∂ F ∂y ∂y1 1 2 Φ|x1 =y1 = 0. It is clear that both sides of (3.3) satisfy the initial condition. By (3.1), y  y  1 1 g ∂ ∇ ∂ ∇ ∂y log  ∂y1 2 P x1 g log  ∇ − ∇ ∂ ∇ ∂y ∂ ∂y2 y [∇ ∂y ∂ log  1 1 1 P x1  P x1 ∇ = y ∇ , ∇ ∂y log  ∂ 2 1 P x1  ∇] + ι ∂y ∂ ι ∂ F ∂y 1 2 6 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN y1 By definition, ∇ ∂y ∂ log 1 P∇ x1 ! = 0. Hence, y ∇g∂ ∇ ∂y log  ∂ ∂y1 2  1 P x1 ∇ = ι ∂y ∂ ι ∂ F ∂y 1 2 which is to say, the left-hand side satisfies the differential equation. Differentiating the right-hand side we obtain y Z1 y g ( ∇∂ ∂y1 P ∇g )−1 (ι ∂y ∂ F) 2 x1 y1 y y 1 P = ∇ x1 =  g 1 P x1 ∂ ∂y1  ∇g  y Z1 y P y ∂  ∂y1 1 P x1 −1 y y Z1 g ∇ ∂ F) ( ∇g )−1 (ι ∂y P 2 x1 y1 y g −1 ( ∇ ) (ι ∂y ∂ F) =  2 x1 x1 1 P x1 ∇  g y1 P ι ∂y ∂ ( ∇g )−1 (ι ∂y ∂ F) 1 2 x1 = ι ∂y ∂ ι ∂ F ∂y 1 2 Thus, the right-hand side satisfies the differential equation as well.  For a nilpotent OX -Lie algebra g, A ∈ Γ (AkX; Ω1Ak /X ⊗ g), ∇ := X d/X + A, a pair of points (x1 , . . . , xk ), (y1 , . . . , yk ) ∈ AkX (X) and an index 1 6 j 6 k such that xi = yi for i 6= j let yj yj P P ∇ := xj j ∇, xj k−1 where Y = AX and AkX → Y is the projection along the jth coordinate, j j j ∇ = d/Y + A , A ∈ Γ (A1Y ; Ω1Ak /Y ⊗ g) is the image of A. Y 3.5. Special broken lines. Definition 3.4. A special broken line (of length n) γ in AkX is a (i) (i) (i) sequence of points (p(i) )n = (p1 , . . . , pk ) such that for any i=0 , p (i) (i+1) 0 6 i < n − 1 there exists a 1 6 ji 6 k such that pl = pl for all l 6= ji . A special broken line is closed if p(0) = p(n) . COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY7 For a nilpotent OAkX -Lie algebra g, A ∈ Γ (AkX; Ω1Ak /X ⊗ g), ∇ := X k d/X + A and a special broken line γ = (p(i) )n i=0 in AX , we define the holonomy of ∇ along γ as the product  (n)   (n−1)   (1)  pj P γ  ∇ :=   pj n−1 P (n−1) n−1 pj    ∇ · pj n−2 P (n−1) n−2 pj    ∇ · · · · ·   0 P (0) 0 pj  ∇  For a special broken line γ = (p(i) )n i=0 we denote by −γ the special (i) n (i) (n−i) broken line (q )i=0 with q = p . Lemma 3.5. P P ∇ = ( ∇)−1 −γ γ For a closed special broken line γ = (p(i) )n i=0 we denote by τγ the (i) n (i) closed special broken line (q )i=0 with q = p(i+1 (mod n+1)) . Lemma 3.6. (1) 0 pj P ∇ = Conj( τγ P P ∇)( ∇) (0) 0 pj γ where j0 is as in Definition 3.4. 4. Two-dimensional holonomy 4.1. Connection-curvature pairs. Data of a connection-curvature pair on AkX consists, by definition, of δ ρ • a nilpotent crossed module1 in OX -Lie algebras h → − g → − Der(h) • A ∈ Γ (AkX ; Ω1Ak /X ⊗ g) X • β ∈ Γ (AkX ; Ω2Ak /X ⊗ h) X 1 Let ∇ := d/X + A, F = F(∇) := d/X A + [A, A]. 2 These data are required to satisfy • ρ(F) = ad(β), • the Bianchi identity ∇β = 0. 1A crossed module is the same thing as a dgla concentrated in degrees 0 and -1. A nilpotent crossed module is one which corresponds to a nilpotent dgla. 8 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 4.2. Two-dimensional holonomy. For a connection-curvature pair (∇, β) on A2X we denote by y2 y1 PP (∇, β) ∈ exp(h)(A2X ×X A2X ) x2 x1 the solution of the initial value problem  y Z1 y   ∇ ∂ log(u) = ( ∇h )−1 (ι ∂ β) ∂y2 ∂y2  x y 1 1   u|∆1 = 1 P where ∆1 is given by the equation x1 = y1 . For a connection-curvature pair (∇, β) on AkX and • points (x1 , . . . , xk ), (y1, . . . , yk ) ∈ AkX (X); • indices 1 6 j1 , j2 6 k such that xi = yi for i 6= j1 and i 6= j2 let y j2 y j1 PP y j2 y j1 PP (∇, β) := x j2 x j1 (∇ (j1 ,j2 ) (j1 ,j2 ) ,β ) x j2 x j1 k−2 th where Y = AX and AkX → Y is the projection along the jth 1 and j2 (j1 ,j2 ) (j1 ,j2 ) (j1 ,j2 ) = d/Y + A ,A ∈ Γ (A2Y ; Ω1Ak /Y ⊗ g) is the coordinates, ∇ (j1 ,j2 ) image of A, β Y ∈ Γ (A2Y ; Ω1Ak /Y Y ⊗ h) is the image of β. 4.3. Rectangles. A rectangle R in A2X is a sequence of points (i) (i) (p(i) )3i=0 , p(i) = (p1 , p2 ) ∈ A2X (X), such that (i) (i+1) p1 = p1 (i+1) ⇔ p2 (i+2) = p2 for all i (mod 4). Rectangles are in one-to-one correspondence with pairs of points in A2X (X). The pair ((x1 , x2 ), (y1, y2 )) determines the rectangle ((x1 , x2 ), (y1 , x2 )(y1 , y2 )(x1 , y2 )). For a rectangle R = ((x1 , x2 ), (y1 , x2 )(y1 , y2 )(x1 , y2 )) the closed special broken line of length four ∂R := ((x1 , x2 ), (y1 , x2 )(y1 , y2 )(x1 , y2 ), (x1 , x2 )) , will be called the boundary of R. For a rectangle R = ((x1 , x2 ), (y1, x2 )(y1 , y2 )(x1 , y2 )) in A2X let y2 y1 P (∇, β) := R PP x2 x1 (∇, β). COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY9 4.4. Chains of rectangles. A chain of rectangles S = (γ, x, y) in is given by the following data: k+1 AX (i) (i) (i) • A special broken line γ = (p(i) )n = (p1 , . . . , pk ) in i=0 , p AkA1 . X • Points x, y ∈ A1X (X). The chain of rectangles S consists of rectangles   (i−1) (i−1) (i) (i) Ri := (x, pji−1 ), (y, pji−1 ), (y, pji−1 ), (x, pji−1 ) , 1 6 i 6 n, in the notations of Definition 3.4. The boundary of S, denoted ∂S is the closed special broken line ∂S := ((p(0) , x), (p(0) , y), (p(1) , y), . . . , (p(n) , y), (p(n), x), . . . , (p(0) , x)) k+1 . in AX For 1 6 i 6 n let γ6i = (p(l) )il=0 , S6i = (γ6i , x, y). Note that γ6n = γ and S6n = S. Define P (∇, β) := S61 P (∇, β) R1 and for 2 6 i 6 n let P S6i   (∇, β) := Conj  P γ6i−1  P   ∇  (∇, β) · Ri P (∇, β) S6i−1 4.5. Green’s theorem for chains of rectangles. Theorem 4.1 (Green’s Theorem). Suppose that S is a chain of k+1 and (∇, β) is a connection-curvature pair as in 4.1. rectangles in AX Then, P ∂S P ∇ = exp(δ)( (∇, β)). S 4.6. Parallelepipeds. A parallelepiped is a pair of points in A3X . In what follows we use that first coordinate projection to identify A3X with A2A1 . A parallelepiped Q = ((x1 , x2 , x3 ), (y1, y2 , y3 )) ∈ A3X ×X X A3X (X) gives rise to the rectangle R = ((x2 , x3 ), (y2 , x3 )(y2 , y3 )(x2 , y3 )) in A2A1 and the chain of rectangles S = (∂R, x1 , y1 ). X Theorem 4.2 (Gauss-Ostrogradsky Theorem). Suppose that Q is a parallelepiped in A3X and (∇, β) is a connection-curvature pair as in 10 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 4.1.  y1 P (∇, β) = Conj( P ∇ x1 S (x2 ,x3 ) P  )  (∇, β) R     (∇, β) · −1 P y1 R x1  4.7. Parametrization of simplexes. Recall that the n-simplex ∆n is defined as the hyperplane in An+1 given by the equation t0 + · · · + tn = 1. We parameterize ∆1 , ∆2 and ∆3 , respectively, by ∆1 : t0 = 1 − x, t1 = x; ∆2 : t0 = 1 − x1 , t1 = x1 (1 − x2 ), t2 = x1 x2 ; ∆3 : t0 = 1 − x1 ,t1 = x1 (1 − x2 ), t2 = x1 x2 (1 − x3 ), t3 = x1 x2 x3 . Let Φi denote the parameterization of ∆i , i = 1, 2, 3. 1 P P P PP ∇ := Φ1∗ ∇ 0 ∆1 1 1 (Φ2∗ ∇, Φ2∗ β) (∇, β) := 0 0 ∆2 With these notations, for a connection-curvature pair (∇, β) on A3X Gauss-Ostrogradsky Theorem 4.2 says   (4.1) Conj( P ∆1 f∗ ∇)  P ∆2   P ∆2 ∂∗0 (∇, β) = −1  ∂∗2 (∇, β) · P ∆2   ∂∗1 (∇, β) ·  P ∆2  ∂∗3 (∇, β) , where f = ∂3 ◦ ∂2 = ∂2 ◦ ∂2 is the map ∆1 → ∆3 induced by 0 7→ 0, 1 7→ 1. 5. From Σ to N MC2 Throughout this section we assume that g is a nilpotent DGLA which satisfies gi = 0 for i 6 −2. Recall that in [2] we showed that the simplicial nerve N MC2 (g) of the 2-groupoid MC2 (g) is weakly equivalent to another simplicial set Σ(g). In this section we apply the theory of multiplicative integration outlined above to construct the map I(g) : Σ(g) → N MC2 (g) COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY 11 omitting technical details. We will rely heavily on results and notations from [2]. In what follows we denote by Ωn , n = 0, 1, 2, . . . the commutative differential graded algebra over Q with generators t0 , . . . , tn of degree zero and dt0 , . . . , dtn of degree one subject to the relations t0 + · · · + tn = 1 and dt0 + · · · + dtn = 0. The differential d : Ωn → Ωn [1] is defined by ti 7→ dti and dti 7→ 0. The assignment [n] 7→ Ωn extends in a natural way to a simplicial commutative differential graded algebra. Recall that the set of Maurer-Cartan elements of g, denoted MC(g) is defined as 1 MC(g) := γ ∈ g1 | δγ + [γ, γ] = 0 . 2 5.1. The simplicial set Σ(g). For a nilpotent L∞ -algebra g and a non-negative integer n let Σn (g) = MC(g ⊗ Ωn ). Equipped with structure maps induced by those of Ω• the assignment n 7→ Σn (g) defines a simplicial set denoted Σ(g). The simplicial set Σ(g) was introduced by V. Hinich in [6] for DGLA and used by E. Getzler in [4] (where it is denoted MC• (g)) for general nilpotent L∞ -algebras. 5.2. The Deligne 2-groupoid. We denote by MC2 (g) the Deligne 2-groupoid as defined by P. Deligne [3] and independently by E. Getzler, [4]. Below we review the construction of Deligne 2-groupoid of a nilpotent DGLA following [4, 5] and references therein. Since g is a nilpotent DGLA it follows that g0 is a nilpotent Lie algebra. The unipotent group exp g0 acts on the space g1 by affine transformations. The action of exp X, X ∈ g0 , on γ ∈ g1 is given by the formula ∞ X (ad X)i (5.1) (exp X) · γ = γ − (δX + [γ, X]). (i + 1)! i=0 The action of gauge transformations (5.1) preserves the subset of MaurerCartan elements MC(g) ⊂ g1 . We denote by MC1 (g) the Deligne groupoid defined as the groupoid associated with the action of the group exp g0 by gauge transformations on the set MC(g). We denote by MC2 (g) the Deligne 2-groupoid whose objects and the 1-morphisms of MC2 (g) are those of MC1 (g). The 2-morphisms of MC2 (h)are defined as follows. 12 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN For γ ∈ MC(g) let [·, ·]γ denote the Lie bracket on g−1 defined by (5.2) [a, b]γ = [a, δb + [γ, b]]. Equipped with this bracket, g−1 becomes a nilpotent Lie algebra. We denote by expγ g−1 the corresponding unipotent group, and by expγ : g−1 → expγ g−1 the corresponding exponential map. If γ1 , γ2 are two Maurer-Cartan elements, then the group expγ2 g−1 acts on HomMC1 (g) (γ1 , γ2 ). For expγ2 t ∈ expγ2 g−1 and HomMC1 (g) (γ1 , γ2 ) the action is given by (expγ2 t) · (exp X) = exp(δt + [γ2, t]) exp X ∈ exp g0 . By definition, HomMC2 (g) (γ1, γ2 ) is the groupoid associated with the above action. 5.3. The simplicial nerve of the Deligne 2-groupoid. We will need the following explicit description of the simplicial nerve of MC2 (g). See [2] for details. Lemma 5.1. The simplicial nerve N MC2 (g) admits the following explicit description: (1) N0 MC2 (g) = MC(g) (2) For n > 1 there is a canonical bijection between Nn MC2 (g) and the set of data of the form ((µi)06i6n , (gij )06i<j6n , (cijk )06i<j<k6n ), where • (µi) is an (n + 1)-tuple of objects of Maurer-Cartan elements of g, • (gij ) is a collection of 1-morphisms (gauge transformations) gij : µj → µi • (cijk ) is a collection of 2-morphisms cijk : gij gjk → gik which satisfies (5.3) cijl cjkl = cikl cijk (in the set of 2-morphisms gij gjk gkl → gil ). For a morphism f : [m] → [n] in ∆ the induced structure map f : Nn MC2 (g) → Nm MC2 (g) is given (under the above bijection) by f∗ ((µi ), (gij ), (cijk )) = ((νi), (hij ), (dijk )), where νi = µf(i) , hij = gf(i),f(j), dijk = cf(i),f(j),f(k). ∗ COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY 13 5.4. From Σ to N MC2 . Recall that, for n = 0, 1, 2, . . ., Σn (g) = 1 MC(Ωn ⊗ g) ⊂ (Ωn ⊗ g) = Ω0n ⊗ g1 ⊕ Ω1n ⊗ g0 ⊕ Ω2n ⊗ g−1 . Thus, every element µ ∈ Σn (g) is a triple: µ = (µ0,1 , µ1,0, µ2,−1 ). For n = 0, 1, 2, . . . the map (5.4) In (g) : Σn (g) → Nn MC2 (g) is defined as follows: n = 0 The map I0 (g) : Σ0 (g) = MC(g) → MC(g) = N0 MC2 (g) is defined to be the identity. n = 1 For µ = (µ0,1 , µ1,0) ∈ Σ1 (g), I1 (g)(µ) = ((∂∗i µ0,1 )i=0, , P (d + µ1,0 )). ∆1 n > 2 For µ = (µ0,1 , µ1,0, µ2,−1 ) ∈ Σn (g), P P ∆1 ∆2 1,0 2,−1 1,0 In (g)(µ) = ((µ0,1 i )06i6n , ( (d+µij )06i<j6n , ( (d+µijk , µijk ))06i<j<k6n ), where ∗ 0,1 0 n • µ0,1 i = f µ , where f : ∆ → ∆ is the map induced by [0] → [n] : 0 7→ i; ∗ 1,0 1 n • µ1,0 ij = f µ , where f : ∆ → ∆ is the map induced by [1] → [n] : 0 7→ i, 1 7→ j; 2,−1 • µijk = f∗ µ2,−1 , where f : ∆2 → ∆n is the map induced by [1] → [n] : 0 7→ i, 1 7→ j, 2 7→ k. Theorem 5.2. Suppose that g is a nilpotent DGLA which satisfies gi = 0 for i 6 −2. The collection of maps In (g), n = 0, 1, 2, . . ., defines a morphism of simplicial sets (5.5) I(g) : Σ(g) → N MC2 (g) natural in g which satisfies (1) I0 (g) : Σ0 (g) → N0 MC2 (g) is the identity map after the identification Σ0 (g) = MC(g) = N0 MC2 (g). (2) The restriction of I to the subcategory of abelian algebras (a.k.a. complexes) coincides with the integration map Z (5.6) : Σ(a) → K(a[1]) = N MC2 (a), a abelian. Note that the integration map is a morphism of simplicial groups. 14 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN (3) If a ֒→ g is a central subalgebra the diagram Σ(g)   y I −−−→ I N MC2 (g)   y Σ(g/a)0 −−−→ N MC2 (g/a)0 is a morphism of principal fibrations relative to the morphism of groups (5.6), where Σ(g/a)0 (respectively, N MC2 (g/a)0 ) denotes the image of the map Σ(g) → Σ(g/a) (respectively, N MC2 (g) → N MC2 (g/a)). Theorem 5.2 and induction on the nilpotency length imply the following statement. Theorem 5.3. The map I(g) is an equivalence. References [1] I. Ja. Aref’eva. Nonabelian Stokes formula. Teoret. Mat. Fiz., 43(1):111–116, 1980. [2] Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, and Boris Tsygan. Deligne groupoid revisited. Theory Appl. Categ., 30:Paper No. 29, 1001–1017, 2015. [3] Pierre Deligne. Letter to L. Breen, 1994. [4] Ezra Getzler. A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J., 111(3):535–560, 2002. [5] Ezra Getzler. Lie theory for nilpotent L∞ -algebras. Ann. of Math. (2), 170(1):271–301, 2009. [6] Vladimir Hinich. Descent of Deligne groupoids. Internat. Math. Res. Notices, (5):223–239, 1997. [7] Mikhail Kapranov. Membranes and higher groupoids. arXiv:1502.06166, Feb 2015. [8] Maxim Kontsevich. Deformation quantization of algebraic varieties. Lett. Math. Phys., 56(3):271–294, 2001. EuroConférence Moshé Flato 2000, Part III (Dijon). [9] Amnon Yekutieli. Nonabelian multiplicative integration on surfaces. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. COMPARISON OF SPACES ASSOCIATED TO DGLA VIA HIGHER HOLONOMY 15 Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia E-mail address: paul.bressler@gmail.com Department of Mathematics, UCB 395, University of Colorado, Boulder, CO 80309-0395, USA E-mail address: Alexander.Gorokhovsky@colorado.edu Department of Mathematics, Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark E-mail address: rnest@math.ku.dk Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA E-mail address: b-tsygan@northwestern.edu