Comparison of spaces associated to DGLA via higher holonomy

TL;DR

This paper establishes an explicit equivalence between two mathematical structures related to DGLA, using non-abelian multiplicative integration, advancing understanding of higher holonomy in differential graded Lie algebras.

Contribution

It constructs an explicit equivalence between the nerve of the Deligne 2-groupoid and a simplicial set of differential forms for certain nilpotent DGLAs, utilizing non-abelian multiplicative integration.

Findings

Established an explicit equivalence between the nerve of the Deligne 2-groupoid and differential form simplicial sets.

Applied non-abelian multiplicative integration to DGLA in the context of higher holonomy.

Enhanced the understanding of the relationship between DGLA structures and higher categorical models.

Abstract

Fof a nilpotent differential graded Lie algebra 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 differential forms with values in the Lie algebra introduced by V.Hinich. The construction uses the theory of non-abelian multiplicative integration.

Equations

$${\mathfrak{g}}^i=0$$

$$\mathbb{I}(\mathfrak{g}):\Sigma (\mathfrak{g})\to \mathfrak{N}{\mathrm{MC}}^2(\mathfrak{g})$$

$$D\log (a):=(Da)\cdot a^{-1}=\left(\frac{\exp (\mathrm{ad}(x))-1}{\mathrm{ad}(x)}\right)(Dx).$$

$$D\log (ab)$$

$$D\log (a^{-1})$$

$${\mathrm{ad}}_x\log (a)$$

$$\begin{gathered} D_1(D_2\log (a))-D_2(D_1\log (a))= \\ [D_1,D_2]\log (a)+[D_1\log (a),D_2\log (a)] \end{gathered}$$

$$\{\begin{array}{l}\frac{\partial u}{\partial s}+A(u)=0\\ {\sigma }^{\ast }(u)=e.\end{array}$$

$$\underset{x}{\overset{y}{\int }}\alpha \in \Gamma ({\mathbb{A}}_X^1{\times }_X{\mathbb{A}}_X^1;{\mathcal{O}}_{{\mathbb{A}}_X^1{\times }_X{\mathbb{A}}_X^1})$$

$$\{\begin{array}{l}\frac{\partial u}{\partial y}={\mathrm{pr}}_2^{\ast }\alpha \\ u{|}_{\Delta }=0\end{array}$$

$$d\log (u):=\left(\frac{\exp (\mathrm{ad}(x))-1}{\mathrm{ad}(x)}\right)(\mathtt{d}x)$$

$$\nabla \log (u):=d\log u+\mathrm{A}.$$

$$\begin{gathered} {\nabla }_{{\xi }_1}^{\mathfrak{g}}{\nabla }_{{\xi }_2}\log u-{\nabla }_{{\xi }_2}^{\mathfrak{g}}{\nabla }_{{\xi }_1}\log u= \\ [{\nabla }_{{\xi }_1}\log u,{\nabla }_{{\xi }_2}\log u]+{\iota }_{{\xi }_1}{\iota }_{{\xi }_2}F \end{gathered}$$

$$(x,y)\mapsto \text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_x^y\nabla \in \exp (\mathfrak{g})({\mathbb{A}}_X^1{\times }_X{\mathbb{A}}_X^1)$$

$$\{\begin{array}{l}{\mathrm{pr}}_2^{\ast }{\nabla }_{\frac{\partial }{\partial y}}\log (u)=0\\ u{|}_{\Delta }=1,\end{array}$$

$$\exp (\mathfrak{g})({\mathbb{A}}_X^1{\times }_X{\mathbb{A}}_X^1)\to \mathrm{Isom}({\mathrm{pr}}_1^{\ast }{\mathcal{E}}_{{\mathbb{A}}_X^1},{\mathrm{pr}}_2^{\ast }{\mathcal{E}}_{{\mathbb{A}}_X^1}).$$

$$\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_x^y{\nabla }^{\mathcal{E}}\right)\frac{\partial }{\partial y}{\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_x^y{\nabla }^{\mathcal{E}}\right)}^{-1}={\nabla }_{\frac{\partial }{\partial y}}^{\mathcal{E}}$$

$$\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_x^y\nabla \right)\cdot \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_y^z\nabla \right)=\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_x^z\nabla$$

$$((x_1,y_2),(y_1,y_2))\mapsto \text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\overline{\nabla }\in \exp (\mathfrak{g})({\mathbb{A}}_X^2{\times }_{{\mathbb{A}}_X^1}{\mathbb{A}}_X^2)$$

$${\nabla }_{\frac{\partial }{\partial y_2}}\log \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\nabla \right)=\underset{t=x_1}{\overset{t=y_1}{\int }}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{y_1}^t{\nabla }^{\mathfrak{g}}{)}^{-1}({\iota }_{\frac{\partial }{\partial y_2}}F)$$

$$\{\begin{array}{l}{\nabla }_{\frac{\partial }{\partial y_1}}^{\mathfrak{g}}\Phi ={\iota }_{\frac{\partial }{\partial y_1}}{\iota }_{\frac{\partial }{\partial y_2}}F\\ \Phi {|}_{x_1=y_1}=0.\end{array}$$

$$\begin{gathered} {\nabla }_{\frac{\partial }{\partial y_1}}^{\mathfrak{g}}{\nabla }_{\frac{\partial }{\partial y_2}}\log \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\nabla \right)-{\nabla }_{\frac{\partial }{\partial y_2}}^{\mathfrak{g}}{\nabla }_{\frac{\partial }{\partial y_1}}\log \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\nabla \right)= \\ [{\nabla }_{\frac{\partial }{\partial y_1}}\log \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\nabla \right),{\nabla }_{\frac{\partial }{\partial y_2}}\log \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\nabla \right)]+{\iota }_{\frac{\partial }{\partial y_1}}{\iota }_{\frac{\partial }{\partial y_2}}F \end{gathered}$$

$${\nabla }_{\frac{\partial }{\partial y_1}}^{\mathfrak{g}}{\nabla }_{\frac{\partial }{\partial y_2}}\log \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}\nabla \right)={\iota }_{\frac{\partial }{\partial y_1}}{\iota }_{\frac{\partial }{\partial y_2}}F$$

$$\begin{gathered} {\nabla }_{\frac{\partial }{\partial y_1}}^{\mathfrak{g}}\underset{x_1}{\overset{y_1}{\int }}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{y_1}^y{\nabla }^{\mathfrak{g}}{)}^{-1}({\iota }_{\frac{\partial }{\partial y_2}}F) \\ =\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}{\nabla }^{\mathfrak{g}}\right)\frac{\partial }{\partial y_1}{\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}{\nabla }^{\mathfrak{g}}\right)}^{-1}\underset{x_1}{\overset{y_1}{\int }}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{y_1}^y{\nabla }^{\mathfrak{g}}{)}^{-1}({\iota }_{\frac{\partial }{\partial y_2}}F) \\ =\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}{\nabla }^{\mathfrak{g}}\right)\frac{\partial }{\partial y_1}\underset{x_1}{\overset{y_1}{\int }}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^y{\nabla }^{\mathfrak{g}}{)}^{-1}({\iota }_{\frac{\partial }{\partial y_2}}F)=\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}{\nabla }^{\mathfrak{g}}\right){\iota }_{\frac{\partial }{\partial y_1}}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}{\nabla }^{\mathfrak{g}}{)}^{-1}({\iota }_{\frac{\partial }{\partial y_2}}F) \\ ={\iota }_{\frac{\partial }{\partial y_1}}{\iota }_{\frac{\partial }{\partial y_2}}F \end{gathered}$$

$$\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_j}^{y_j}\nabla :=\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_j}^{y_j}{\overline{\nabla }}^j,$$

$$\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{\gamma }\nabla :=\left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{p_{j_{n-1}}^{(n-1)}}^{p_{j_{n-1}}^{(n)}}\nabla \right)\cdot \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{p_{j_{n-2}}^{(n-1)}}^{p_{j_{n-2}}^{(n-1)}}\nabla \right)\cdot \cdots \cdot \left(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{p_{j_0}^{(0)}}^{p_{j_0}^{(1)}}\nabla \right)$$

$$\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{-\gamma }\nabla =(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{\gamma }\nabla {)}^{-1}$$

$$\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{\tau \gamma }\nabla =\mathrm{Conj}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{p_{j_0}^{(0)}}^{p_{j_0}^{(1)}}\nabla )(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{\gamma }\nabla )$$

$$\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_2}^{y_2}\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_1}^{y_1}(\nabla ,\beta )\in \exp (\mathfrak{h})({\mathbb{A}}_X^2{\times }_X{\mathbb{A}}_X^2)$$

$$\{\begin{array}{l}{\nabla }_{\frac{\partial }{\partial y_2}}\log (u)=\underset{x_1}{\overset{y_1}{\int }}(\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{y_1}^y{\nabla }^{\mathfrak{h}}{)}^{-1}({\iota }_{\frac{\partial }{\partial y_2}}\beta )\\ u{|}_{{\Delta }_1}=1\end{array}$$

$$\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_{j_2}}^{y_{j_2}}\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_{j_1}}^{y_{j_1}}(\nabla ,\beta ):=\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_{j_2}}^{y_{j_2}}\text{vbox}\text{leavevmode}\text{resizebox}5.69054pt13.00018pt\text{mathbfcal}P{\text{slimits@}}_{x_{j_1}}^{y_{j_1}}({\overline{\nabla }}^{(j_1,j_2)},{\overline{\beta }}^{(j_1,j_2)})$$

Full text

R. Nest was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

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 $\mathfrak{g}$ which satisfies the additional condition

[TABLE]

P. Deligne [3] and, independently, E. Getzler [4] associated a (strict) $2$-groupoid which we denote $\operatorname{MC}^{2}(\mathfrak{g})$ and refer to as the Deligne $2$-groupoid. In an earlier paper by the authors ([2]) it was shown that the simplicial nerve $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g})$ of the $2$-groupoid $\operatorname{MC}^{2}(\mathfrak{g})$, $\mathfrak{g}$ a nilpotent DGLA satisfying (1.1) is weakly equivalent to another simplicial set, denoted $\Sigma(\mathfrak{g})$, introduced by V. Hinich [6] by constructing natural maps from $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g})$ and $\Sigma(\mathfrak{g})$ to a third simplicial set and showing that they are equivalences.

In this note we outline a construction of the map

[TABLE]

omitting technical details. The main tool in our construction is the theory of non-abelian multiplicative integration on surfaces.

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 $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g})$ and $\Sigma(\mathfrak{g})$, 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

• •

for a variety $X$ over $k$: $\mathbb{A}^{n}_{X}:=X\times\mathbb{A}^{n}_{k}$; for a commutative $k$-algebra $R$: $\mathbb{A}^{n}_{R}:=\mathbb{A}^{n}_{\operatorname{Spec}R}$

• •

for a variety $\mathrm{pr}_{X}\colon Y\to X$ over $X$ and a $\mathcal{O}_{X}$-module $\mathcal{E}$ we denote by $\mathcal{E}_{Y}$ the $\mathcal{O}_{Y}$-module $\mathrm{pr}_{X}^{*}\mathcal{E}$

• •

for an $X$-variety $\colon Y\to X$ over $X$ we denote by $\mathtt{d}_{/X}\colon\mathcal{O}_{Y}\to\Omega^{1}_{Y/X}$ the relative differential

• •

for an nilpotent $\mathcal{O}_{X}$-Lie algebra $\mathfrak{g}$ we denote by $\exp(\mathfrak{g})$ the algebraic group over $X$ defined as $\exp(\mathfrak{g})(Y)=\mathfrak{g}_{Y}(Y)$ equipped with the group structure given by the Baker-Campbell-Hausdorff series

• •

for a group $G$ and $g\in G$ we denote $\operatorname{Conj}(g)$ the automorphism $h\mapsto ghg^{-1}$

2. Calculus

2.1. Logarithmic derivatives

Suppose that $\mathfrak{g}$ is a nilpotent Lie algebra. For $D\in\operatorname{Der}(\mathfrak{g})$, $a=\exp(x)\in\exp(\mathfrak{g})$, $x\in\mathfrak{g}$. Define $D\log(a)\in\mathfrak{g}$ by

[TABLE]

For $D\in\operatorname{Der}(\mathfrak{g})$, $a,b\in\exp(\mathfrak{g})$, $x\in\mathfrak{g}$,

[TABLE]

For $D_{1},D_{2}\in\operatorname{Der}(\mathfrak{g})$, $a\in\exp(\mathfrak{g})$

[TABLE]

2.2. Initial value problems

Let $\mathrm{pr}_{X}\colon\mathbb{A}^{1}_{X}\to X$ denote the projection so that $\mathrm{pr}_{X*}\mathcal{O}_{\mathbb{A}^{1}_{X}}=\mathcal{O}_{X}[s]$.

Suppose that $\mathcal{E}$ is a vector bundle on $X$. For

• •

$A\in\operatorname{End}_{\mathcal{O}_{\mathbb{A}^{1}_{X}}}(\mathcal{E}_{\mathbb{A}^{1}_{X}})$ nilpotent

• •

a section $\sigma\colon X\to\mathbb{A}^{1}_{X}$ of the projection $\mathrm{pr}_{X}$

• •

a section $e\in\Gamma(X;\mathcal{E})$

there exists a unique section $u\in\Gamma(\mathbb{A}^{1}_{X};\mathcal{E}_{\mathbb{A}^{1}_{X}})$ which satisfies

[TABLE]

2.3. Integration

Let $x=s\otimes 1$ and $y=1\otimes s$ denote the coordinate functions on $\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X}$. Let $\Delta$ denote the subvariety defined by the equation $x=y$.

For $\alpha\in\Omega^{1}_{\mathbb{A}^{1}_{X}/X}$ we denote by

[TABLE]

the solution of the initial value problem

[TABLE]

in the unknown $u=u(x,y)\in\Gamma(\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X};\mathcal{O}_{\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X}})$.

3. Differential geometry

3.1. Connections and curvature

Suppose that $\mathfrak{g}$ is a nilpotent $\mathcal{O}_{X}$-Lie algebra. For $\mathrm{A}\in\Gamma(\mathbb{A}^{k}_{X};\Omega^{1}_{\mathbb{A}^{k}_{X}/X}\otimes\mathfrak{g})$ we consider connection $\nabla:=\mathtt{d}_{/X}+\mathrm{A}$. Its curvature is defined by $F=F(\nabla):=\mathtt{d}_{/X}\mathrm{A}+\dfrac{1}{2}[\mathrm{A},\mathrm{A}]\in\Gamma(\mathbb{A}^{k}_{X};\Omega^{2}_{\mathbb{A}^{k}_{X}/X}\otimes\mathfrak{g})$.

Suppose now that $\mathcal{E}$ is an $\mathcal{O}_{X}$-module and $\rho\colon\mathfrak{g}\to\operatorname{\underline{End}}_{\mathcal{O}_{X}}(\mathcal{E})$ is a representation of $\mathfrak{g}$. The connection $\nabla$ gives rise to the connection $\nabla^{\mathcal{E}}$ on $\mathcal{E}_{\mathbb{A}^{k}_{X}}$.

The sheaf $\Omega^{1}_{\mathbb{A}^{k}_{X}/X}$ of relative differentials is equipped with the canonical relative connection. Hence, the connection $\nabla$ induces a connection on the $\mathcal{O}_{\mathbb{A}^{k}_{X}}$-module $\Omega^{1}_{\mathbb{A}^{k}_{X}/X}\otimes\mathcal{E}$ which will be denoted $\nabla^{\mathcal{E}}$ as well.

An example of the above situation is given by the adjoint representation $\operatorname{\mathtt{ad}}\colon\mathfrak{g}\to\operatorname{Der}(\mathfrak{g})$ giving rise to the connection $\nabla^{\mathfrak{g}}$.

3.2. Logarithmic covariant derivative

For $u=\exp(x)\in\exp(\mathfrak{g})(\mathbb{A}^{k}_{X})$, let

[TABLE]

and

[TABLE]

Lemma 3.1**.**

For $\xi_{1},\xi_{2}\in\mathcal{T}_{\mathbb{A}^{k}_{X}/X}$ such that $[\xi_{1},\xi_{2}]=0$ the identity

[TABLE]

holds.

3.3. Holonomy

Suppose that $\mathfrak{g}$ is a nilpotent $\mathcal{O}_{X}$-Lie algebra. For $\mathrm{A}\in\Gamma(\mathbb{A}^{1}_{X};\Omega^{1}_{\mathbb{A}^{1}_{X}/X}\otimes\mathfrak{g})$, $\nabla:=\mathtt{d}_{/X}+\mathrm{A}$ we denote by

[TABLE]

the solution of the initial value problem

[TABLE]

where $\Delta$ is the subvariety defined by the equation $x=y$.

Suppose that $\mathcal{E}$ is an $\mathcal{O}_{X}$-module and $\rho\colon\mathfrak{g}\to\operatorname{\underline{End}}_{\mathcal{O}_{X}}(\mathcal{E})$ is a representation of $\mathfrak{g}$ which is nilpotent in the sense that there exists $N$ such that $\operatorname{Im}(\rho)^{N}=0$. The representation $\rho$ induces the map of groups $\exp(\rho)\colon\exp(\mathfrak{g})\to\operatorname{\underline{Aut}}(\mathcal{E})$. Since the two compositions $\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X}\xrightarrow{\mathrm{pr}_{i}}\mathbb{A}^{1}_{X}\xrightarrow{\mathrm{pr}_{X}}X$, where $\mathrm{pr}_{i}$ are the canonical projections, are equal it follows that $\operatorname{\underline{Hom}}_{\mathcal{O}_{\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X}}}(\mathrm{pr}_{1}^{*}\mathcal{E}_{\mathbb{A}^{1}_{X}},\mathrm{pr}_{2}^{*}\mathcal{E}_{\mathbb{A}^{1}_{X}})=\operatorname{\underline{End}}_{\mathcal{O}_{\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X}}}(\mathcal{E}_{\mathbb{A}^{1}_{X}\times_{X}\mathbb{A}^{1}_{X}})$ and there is a canonical map

[TABLE]

We denote by \mathop{\vphantom{\int}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{13.00018pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{10.00012pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{7.00009pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{5.00006pt}{\mathbfcal{P}}}}}}\slimits@\limits_{x}^{y}\nabla^{\mathcal{E}} the image of \mathop{\vphantom{\int}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{13.00018pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{10.00012pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{7.00009pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{5.00006pt}{\mathbfcal{P}}}}}}\slimits@\limits_{x}^{y}\nabla under (3.2).

Lemma 3.2**.**

The following useful identities hold.

[TABLE]

[TABLE]

3.4. Holonomy and curvature

Suppose that $\mathfrak{g}$ is a nilpotent $\mathcal{O}_{X}$-Lie algebra, $\mathrm{A}\in\Gamma(\mathbb{A}^{2}_{X};\Omega^{1}_{\mathbb{A}^{2}_{X}/X}\otimes\mathfrak{g})$, $\nabla:=\mathtt{d}_{/X}+\mathrm{A}$, $F=F(\nabla):=\mathtt{d}_{/X}\mathrm{A}+\dfrac{1}{2}[\mathrm{A},\mathrm{A}]\in\Gamma(\mathbb{A}^{2}_{X};\Omega^{2}_{\mathbb{A}^{2}_{X}/X}\otimes\mathfrak{g})$.

The second projection $\mathbb{A}^{2}_{X}\to\mathbb{A}^{1}_{X}$ identifies $\mathbb{A}^{2}_{X}$ with $\mathbb{A}^{1}_{\mathbb{A}^{1}_{X}}$. Let $\overline{\mathrm{A}}\in\Gamma(\mathbb{A}^{2}_{X};\Omega^{1}_{\mathbb{A}^{2}_{X}/\mathbb{A}^{1}_{X}}\otimes\mathfrak{g})$ denote the image of $\mathrm{A}$, $\overline{\nabla}:=d_{/\mathbb{A}^{1}_{X}}+\overline{\mathrm{A}}$. Hence, the holonomy

[TABLE]

is defined and will be denoted, by abuse of notation, by \displaystyle{\mathop{\vphantom{\int}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{13.00018pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{10.00012pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{7.00009pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{5.00006pt}{\mathbfcal{P}}}}}}\slimits@_{x_{1}}^{y_{1}}\nabla}.

Lemma 3.3**.**

[TABLE]

Proof.

We will show that both sides of (3.3) are solutions of the initial value problem

[TABLE]

It is clear that both sides of (3.3) satisfy the initial condition. By (3.1),

[TABLE]

By definition, \nabla_{\frac{\partial~{}}{\partial y_{1}}}\log\left(\mathop{\vphantom{\int}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{13.00018pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{10.00012pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{7.00009pt}{\mathbfcal{P}}}}}{\vbox{\hbox{\leavevmode\resizebox{5.69054pt}{5.00006pt}{\mathbfcal{P}}}}}}\slimits@\limits_{x_{1}}^{y_{1}}\nabla\right)=0. Hence,

[TABLE]

which is to say, the left-hand side satisfies the differential equation. Differentiating the right-hand side we obtain

[TABLE]

Thus, the right-hand side satisfies the differential equation as well. ∎

For a nilpotent $\mathcal{O}_{X}$-Lie algebra $\mathfrak{g}$, $\mathrm{A}\in\Gamma(\mathbb{A}^{k}_{X};\Omega^{1}_{\mathbb{A}^{k}_{X}/X}\otimes\mathfrak{g})$, $\nabla:=\mathtt{d}_{/X}+\mathrm{A}$, a pair of points $(x_{1},\ldots,x_{k}),(y_{1},\ldots,y_{k})\in\mathbb{A}^{k}_{X}(X)$ and an index $1\leqslant j\leqslant k$ such that $x_{i}=y_{i}$ for $i\neq j$ let

[TABLE]

where $Y=\mathbb{A}^{k-1}_{X}$ and $\mathbb{A}^{k}_{X}\to Y$ is the projection along the $j^{\text{th}}$ coordinate, $\overline{\nabla}^{j}=\mathtt{d}_{/Y}+\overline{\mathrm{A}}^{j}$, $\overline{\mathrm{A}}^{j}\in\Gamma(\mathbb{A}^{1}_{Y};\Omega^{1}_{\mathbb{A}^{k}_{Y}/Y}\otimes\mathfrak{g})$ is the image of $\mathrm{A}$.

3.5. Special broken lines

Definition 3.4**.**

A special broken line (of length $n$) $\gamma$ in $\mathbb{A}^{k}_{X}$ is a sequence of points $(p^{(i)})_{i=0}^{n}$, $p^{(i)}=(p^{(i)}_{1},\ldots,p^{(i)}_{k})$ such that for any $0\leqslant i<n-1$ there exists a $1\leqslant j_{i}\leqslant k$ such that $p^{(i)}_{l}=p^{(i+1)}_{l}$ for all $l\neq j_{i}$.

A special broken line is closed if $p^{(0)}=p^{(n)}$.

For a nilpotent $\mathcal{O}_{\mathbb{A}^{k}_{X}}$-Lie algebra $\mathfrak{g}$, $\mathrm{A}\in\Gamma(\mathbb{A}^{k}_{X};\Omega^{1}_{\mathbb{A}^{k}_{X}/X}\otimes\mathfrak{g})$, $\nabla:=\mathtt{d}_{/X}+\mathrm{A}$ and a special broken line $\gamma=(p^{(i)})_{i=0}^{n}$ in $\mathbb{A}^{k}_{X}$, we define the holonomy of $\nabla$ along $\gamma$ as the product

[TABLE]

For a special broken line $\gamma=(p^{(i)})_{i=0}^{n}$ we denote by $-\gamma$ the special broken line $(q^{(i)})_{i=0}^{n}$ with $q^{(i)}=p^{(n-i)}$.

Lemma 3.5**.**

[TABLE]

For a closed special broken line $\gamma=(p^{(i)})_{i=0}^{n}$ we denote by $\tau\gamma$ the closed special broken line $(q^{(i)})_{i=0}^{n}$ with $q^{(i)}=p^{(i+1\pmod{n+1})}$.

Lemma 3.6**.**

[TABLE]

where $j_{0}$ is as in Definition 3.4.

4. Two-dimensional holonomy

4.1. Connection-curvature pairs

Data of a connection-curvature pair on $\mathbb{A}^{k}_{X}$ consists, by definition, of

• •

a nilpotent crossed module111A 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. in $\mathcal{O}_{X}$-Lie algebras $\mathfrak{h}\xrightarrow{\delta}\mathfrak{g}\xrightarrow{\rho}\operatorname{Der}(\mathfrak{h})$

• •

$\mathrm{A}\in\Gamma(\mathbb{A}^{k}_{X};\Omega^{1}_{\mathbb{A}^{k}_{X}/X}\otimes\mathfrak{g})$

• •

$\beta\in\Gamma(\mathbb{A}^{k}_{X};\Omega^{2}_{\mathbb{A}^{k}_{X}/X}\otimes\mathfrak{h})$

Let $\nabla:=\mathtt{d}_{/X}+\mathrm{A}$, $F=F(\nabla):=\mathtt{d}_{/X}\mathrm{A}+\dfrac{1}{2}[\mathrm{A},\mathrm{A}]$.

These data are required to satisfy

• •

$\rho(F)=\operatorname{\mathtt{ad}}(\beta)$,

• •

the Bianchi identity $\nabla\beta=0$.

4.2. Two-dimensional holonomy

For a connection-curvature pair $(\nabla,\beta)$ on $\mathbb{A}^{2}_{X}$ we denote by

[TABLE]

the solution of the initial value problem

[TABLE]

where $\Delta_{1}$ is given by the equation $x_{1}=y_{1}$.

For a connection-curvature pair $(\nabla,\beta)$ on $\mathbb{A}^{k}_{X}$ and

• •

points $(x_{1},\ldots,x_{k}),(y_{1},\ldots,y_{k})\in\mathbb{A}^{k}_{X}(X)$;

• •

indices $1\leqslant j_{1},j_{2}\leqslant k$ such that $x_{i}=y_{i}$ for $i\neq j_{1}$ and $i\neq j_{2}$

let

[TABLE]

where $Y=\mathbb{A}^{k-2}_{X}$ and $\mathbb{A}^{k}_{X}\to Y$ is the projection along the $j_{1}^{\text{th}}$ and $j_{2}^{\text{th}}$ coordinates, $\overline{\nabla}^{(j_{1},j_{2})}=\mathtt{d}_{/Y}+\overline{\mathrm{A}}^{(j_{1},j_{2})}$, $\overline{\mathrm{A}}^{(j_{1},j_{2})}\in\Gamma(\mathbb{A}^{2}_{Y};\Omega^{1}_{\mathbb{A}^{k}_{Y}/Y}\otimes\mathfrak{g})$ is the image of $\mathrm{A}$, $\overline{\beta}^{(j_{1},j_{2})}\in\Gamma(\mathbb{A}^{2}_{Y};\Omega^{1}_{\mathbb{A}^{k}_{Y}/Y}\otimes\mathfrak{h})$ is the image of $\beta$.

4.3. Rectangles

A rectangle $R$ in $\mathbb{A}^{2}_{X}$ is a sequence of points $(p^{(i)})_{i=0}^{3}$, $p^{(i)}=(p^{(i)}_{1},p^{(i)}_{2})\in\mathbb{A}^{2}_{X}(X)$, such that

[TABLE]

for all $i\pmod{4}$. Rectangles are in one-to-one correspondence with pairs of points in $\mathbb{A}^{2}_{X}(X)$. The pair $((x_{1},x_{2}),(y_{1},y_{2}))$ determines the rectangle $\left((x_{1},x_{2}),(y_{1},x_{2})(y_{1},y_{2})(x_{1},y_{2})\right)$.

For a rectangle $R=\left((x_{1},x_{2}),(y_{1},x_{2})(y_{1},y_{2})(x_{1},y_{2})\right)$ the closed special broken line of length four

[TABLE]

will be called the boundary of $R$.

For a rectangle $R=\left((x_{1},x_{2}),(y_{1},x_{2})(y_{1},y_{2})(x_{1},y_{2})\right)$ in $\mathbb{A}^{2}_{X}$ let

[TABLE]

4.4. Chains of rectangles

A chain of rectangles $S=(\gamma,x,y)$ in $\mathbb{A}^{k+1}_{X}$ is given by the following data:

• •

A special broken line $\gamma=(p^{(i)})_{i=0}^{n}$, $p^{(i)}=(p^{(i)}_{1},\ldots,p^{(i)}_{k})$ in $\mathbb{A}^{k}_{\mathbb{A}^{1}_{X}}$.

• •

Points $x,y\in\mathbb{A}^{1}_{X}(X)$.

The chain of rectangles $S$ consists of rectangles

[TABLE]

$1\leqslant i\leqslant n$, in the notations of Definition 3.4. The boundary of $S$, denoted $\partial S$ is the closed special broken line

[TABLE]

in $\mathbb{A}^{k+1}_{X}$.

For $1\leqslant i\leqslant n$ let $\gamma_{\leqslant i}=(p^{(l)})_{l=0}^{i}$, $S_{\leqslant i}=(\gamma_{\leqslant i},x,y)$. Note that $\gamma_{\leqslant n}=\gamma$ and $S_{\leqslant n}=S$. Define

[TABLE]

and for $2\leqslant i\leqslant n$ let

[TABLE]

4.5. Green’s theorem for chains of rectangles

Theorem 4.1** (Green’s Theorem).**

Suppose that $S$ is a chain of rectangles in $\mathbb{A}^{k+1}_{X}$ and $(\nabla,\beta)$ is a connection-curvature pair as in 4.1. Then,

[TABLE]

4.6. Parallelepipeds

A parallelepiped is a pair of points in $\mathbb{A}^{3}_{X}$.

In what follows we use that first coordinate projection to identify $\mathbb{A}^{3}_{X}$ with $\mathbb{A}^{2}_{\mathbb{A}^{1}_{X}}$. A parallelepiped $Q=((x_{1},x_{2},x_{3}),(y_{1},y_{2},y_{3}))\in\mathbb{A}^{3}_{X}\times_{X}\mathbb{A}^{3}_{X}(X)$ gives rise to the rectangle $R=((x_{2},x_{3}),(y_{2},x_{3})(y_{2},y_{3})(x_{2},y_{3}))$ in $\mathbb{A}^{2}_{\mathbb{A}^{1}_{X}}$ and the chain of rectangles $S=(\partial R,x_{1},y_{1})$.

Theorem 4.2** (Gauss-Ostrogradsky Theorem).**

Suppose that $Q$ is a parallelepiped in $\mathbb{A}^{3}_{X}$ and $(\nabla,\beta)$ is a connection-curvature pair as in 4.1.

[TABLE]

4.7. Parametrization of simplexes

Recall that the $n$-simplex $\Delta^{n}$ is defined as the hyperplane in $\mathbb{A}^{n+1}$ given by the equation $t_{0}+\dots+t_{n}=1$.

We parameterize $\Delta^{1}$, $\Delta^{2}$ and $\Delta^{3}$, respectively, by

1. $\Delta^{1}$:

$t_{0}=1-x$, $t_{1}=x$; 2. $\Delta^{2}$:

$t_{0}=1-x_{1}$, $t_{1}=x_{1}(1-x_{2})$, $t_{2}=x_{1}x_{2}$; 3. $\Delta^{3}$:

$t_{0}=1-x_{1}$,$t_{1}=x_{1}(1-x_{2})$, $t_{2}=x_{1}x_{2}(1-x_{3})$, $t_{3}=x_{1}x_{2}x_{3}$ .

Let $\Phi^{i}$ denote the parameterization of $\Delta^{i}$, $i=1,2,3$.

[TABLE]

[TABLE]

With these notations, for a connection-curvature pair $(\nabla,\beta)$ on $\mathbb{A}^{3}_{X}$ Gauss-Ostrogradsky Theorem 4.2 says

[TABLE]

where $f=\partial_{3}\circ\partial_{2}=\partial_{2}\circ\partial_{2}$ is the map $\Delta^{1}\to\Delta^{3}$ induced by $0\mapsto 0$, $1\mapsto 1$.

5. From $\Sigma$ to $\mathfrak{N}\operatorname{MC}^{2}$

Throughout this section we assume that $\mathfrak{g}$ is a nilpotent DGLA which satisfies $\mathfrak{g}^{i}=0$ for $i\leqslant-2$. Recall that in [2] we showed that the simplicial nerve $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g})$ of the $2$-groupoid $\operatorname{MC}^{2}(\mathfrak{g})$ is weakly equivalent to another simplicial set $\Sigma(\mathfrak{g})$. In this section we apply the theory of multiplicative integration outlined above to construct the map

[TABLE]

omitting technical details. We will rely heavily on results and notations from [2].

In what follows we denote by $\Omega_{n}$, $n=0,1,2,\ldots$ the commutative differential graded algebra over $\mathbb{Q}$ with generators $t_{0},\ldots,t_{n}$ of degree zero and $dt_{0},\ldots,dt_{n}$ of degree one subject to the relations $t_{0}+\cdots+t_{n}=1$ and $dt_{0}+\cdots+dt_{n}=0$. The differential $d\colon\Omega_{n}\to\Omega_{n}[1]$ is defined by $t_{i}\mapsto dt_{i}$ and $dt_{i}\mapsto 0$. The assignment $[n]\mapsto\Omega_{n}$ extends in a natural way to a simplicial commutative differential graded algebra.

Recall that the set of Maurer-Cartan elements of $\mathfrak{g}$, denoted $\operatorname{MC}(\mathfrak{g})$ is defined as

[TABLE]

5.1. The simplicial set $\Sigma(\mathfrak{g})$

For a nilpotent $L_{\infty}$-algebra $\mathfrak{g}$ and a non-negative integer $n$ let

[TABLE]

Equipped with structure maps induced by those of $\Omega_{\bullet}$ the assignment $n\mapsto\Sigma_{n}(\mathfrak{g})$ defines a simplicial set denoted $\Sigma(\mathfrak{g})$.

The simplicial set $\Sigma(\mathfrak{g})$ was introduced by V. Hinich in [6] for DGLA and used by E. Getzler in [4] (where it is denoted $\operatorname{MC}_{\bullet}(\mathfrak{g})$) for general nilpotent $L_{\infty}$-algebras.

5.2. The Deligne 2-groupoid

We denote by $\operatorname{MC}^{2}(\mathfrak{g})$ 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 $\mathfrak{g}$ is a nilpotent DGLA it follows that $\mathfrak{g}^{0}$ is a nilpotent Lie algebra. The unipotent group $\exp\mathfrak{g}^{0}$ acts on the space $\mathfrak{g}^{1}$ by affine transformations. The action of $\exp X$, $X\in\mathfrak{g}^{0}$, on $\gamma\in\mathfrak{g}^{1}$ is given by the formula

[TABLE]

The action of gauge transformations (5.1) preserves the subset of Maurer-Cartan elements $\operatorname{MC}(\mathfrak{g})\subset\mathfrak{g}^{1}$.

We denote by $\operatorname{MC}^{1}(\mathfrak{g})$ the Deligne groupoid defined as the groupoid associated with the action of the group $\exp\mathfrak{g}^{0}$ by gauge transformations on the set $\operatorname{MC}(\mathfrak{g})$.

We denote by $\operatorname{MC}^{2}(\mathfrak{g})$ the Deligne $2$-groupoid whose objects and the $1$-morphisms of $\operatorname{MC}^{2}(\mathfrak{g})$ are those of $\operatorname{MC}^{1}(\mathfrak{g})$. The 2-morphisms of $\operatorname{MC}^{2}(\mathfrak{h})$are defined as follows.

For $\gamma\in\operatorname{MC}(\mathfrak{g})$ let $[\cdot,\cdot]_{\gamma}$ denote the Lie bracket on $\mathfrak{g}^{-1}$ defined by

[TABLE]

Equipped with this bracket, $\mathfrak{g}^{-1}$ becomes a nilpotent Lie algebra. We denote by $\exp_{\gamma}\mathfrak{g}^{-1}$ the corresponding unipotent group, and by

[TABLE]

the corresponding exponential map. If $\gamma_{1}$, $\gamma_{2}$ are two Maurer-Cartan elements, then the group $\exp_{\gamma_{2}}\mathfrak{g}^{-1}$ acts on $\operatorname{Hom}_{\operatorname{MC}^{1}(\mathfrak{g})}(\gamma_{1},\gamma_{2})$. For $\exp_{\gamma_{2}}t\in\exp_{\gamma_{2}}\mathfrak{g}^{-1}$ and $\operatorname{Hom}_{\operatorname{MC}^{1}(\mathfrak{g})}(\gamma_{1},\gamma_{2})$ the action is given by

[TABLE]

By definition, $\operatorname{Hom}_{\operatorname{MC}^{2}(\mathfrak{g})}(\gamma_{1},\gamma_{2})$ 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 $\operatorname{MC}^{2}(\mathfrak{g})$. See [2] for details.

Lemma 5.1**.**

The simplicial nerve $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g})$ admits the following explicit description:

1. (1)

$\mathfrak{N}_{0}\operatorname{MC}^{2}(\mathfrak{g})=\operatorname{MC}(\mathfrak{g})$** 2. (2)

For $n\geq 1$ there is a canonical bijection between $\mathfrak{N}_{n}\operatorname{MC}^{2}(\mathfrak{g})$ and the set of data of the form

[TABLE]

where

• •

$(\mu_{i})$* is an $(n+1)$-tuple of objects of Maurer-Cartan elements of $\mathfrak{g}$,*

• •

$(g_{ij})$* is a collection of $1$-morphisms (gauge transformations) $g_{ij}\colon\mu_{j}\to\mu_{i}$*

• •

$(c_{ijk})$* is a collection of $2$-morphisms $c_{ijk}:g_{ij}g_{jk}\to g_{ik}$ which satisfies*

[TABLE]

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

For a morphism $f\colon[m]\to[n]$ in $\Delta$ the induced structure map $f^{*}\colon\mathfrak{N}_{n}\operatorname{MC}^{2}(\mathfrak{g})\to\mathfrak{N}_{m}\operatorname{MC}^{2}(\mathfrak{g})$ is given (under the above bijection) by $f^{*}((\mu_{i}),(g_{ij}),(c_{ijk}))=((\nu_{i}),(h_{ij}),(d_{ijk}))$, where $\nu_{i}=\mu_{f(i)}$, $h_{ij}=g_{f(i),f(j)}$, $d_{ijk}=c_{f(i),f(j),f(k)}$.

5.4. From $\Sigma$ to $\mathfrak{N}\operatorname{MC}^{2}$

Recall that, for $n=0,1,2,\ldots$, $\Sigma_{n}(\mathfrak{g})=\operatorname{MC}(\Omega_{n}\otimes\mathfrak{g})\subset\left(\Omega_{n}\otimes\mathfrak{g}\right)^{1}=\Omega_{n}^{0}\otimes\mathfrak{g}^{1}\oplus\Omega_{n}^{1}\otimes\mathfrak{g}^{0}\oplus\Omega_{n}^{2}\otimes\mathfrak{g}^{-1}$. Thus, every element $\mu\in\Sigma_{n}(\mathfrak{g})$ is a triple: $\mu=(\mu^{0,1},\mu^{1,0},\mu^{2,-1})$.

For $n=0,1,2,\ldots$ the map

[TABLE]

is defined as follows:

1. $n=0$

The map $\mathbb{I}_{0}(\mathfrak{g})\colon\Sigma_{0}(\mathfrak{g})=\operatorname{MC}(\mathfrak{g})\to\operatorname{MC}(\mathfrak{g})=\mathfrak{N}_{0}\operatorname{MC}^{2}(\mathfrak{g})$ is defined to be the identity. 2. $n=1$

For $\mu=(\mu^{0,1},\mu^{1,0})\in\Sigma_{1}(\mathfrak{g})$,

[TABLE] 3. $n\geq 2$

For $\mu=(\mu^{0,1},\mu^{1,0},\mu^{2,-1})\in\Sigma_{n}(\mathfrak{g})$,

[TABLE]

where

• •

$\mu^{0,1}_{i}=f^{*}\mu^{0,1}$, where $f\colon\Delta^{0}\to\Delta^{n}$ is the map induced by $[0]\to[n]\colon 0\mapsto i$;

• •

$\mu^{1,0}_{ij}=f^{*}\mu^{1,0}$, where $f\colon\Delta^{1}\to\Delta^{n}$ is the map induced by $[1]\to[n]\colon 0\mapsto i,1\mapsto j$;

• •

$\mu^{2,-1}_{ijk}=f^{*}\mu^{2,-1}$, where $f\colon\Delta^{2}\to\Delta^{n}$ is the map induced by $[1]\to[n]\colon 0\mapsto i,1\mapsto j,2\mapsto k$.

Theorem 5.2**.**

Suppose that $\mathfrak{g}$ is a nilpotent DGLA which satisfies $\mathfrak{g}^{i}=0$ for $i\leqslant-2$. The collection of maps $\mathbb{I}_{n}(\mathfrak{g})$, $n=0,1,2,\ldots$, defines a morphism of simplicial sets

[TABLE]

natural in $\mathfrak{g}$ which satisfies

1. (1)

$\mathbb{I}_{0}(\mathfrak{g})\colon\Sigma_{0}(\mathfrak{g})\to\mathfrak{N}_{0}\operatorname{MC}^{2}(\mathfrak{g})$* is the identity map after the identification $\Sigma_{0}(\mathfrak{g})=\operatorname{MC}(\mathfrak{g})=\mathfrak{N}_{0}\operatorname{MC}^{2}(\mathfrak{g})$.* 2. (2)

The restriction of $\mathbb{I}$ to the subcategory of abelian algebras (a.k.a. complexes) coincides with the integration map

[TABLE]

$\mathfrak{a}$* abelian. Note that the integration map is a morphism of simplicial groups.* 3. (3)

If $\mathfrak{a}\hookrightarrow\mathfrak{g}$ is a central subalgebra the diagram

[TABLE]

is a morphism of principal fibrations relative to the morphism of groups (5.6), where $\Sigma(\mathfrak{g}/\mathfrak{a})_{0}$ (respectively, $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g}/\mathfrak{a})_{0}$) denotes the image of the map $\Sigma(\mathfrak{g})\to\Sigma(\mathfrak{g}/\mathfrak{a})$ (respectively, $\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g})\to\mathfrak{N}\operatorname{MC}^{2}(\mathfrak{g}/\mathfrak{a})$).

Theorem 5.2 and induction on the nilpotency length imply the following statement.

Theorem 5.3**.**

The map $\mathbb{I}(\mathfrak{g})$ is an equivalence.