The bracket of $\alpha,\beta\in \Omega^1_{\mathrm{Ham}}(M)$ is defined as the 1-form $\{\alpha,\beta\}_{\omega}:=i_{X_{\beta}}i_{X_{\alpha}}\omega\in \Omega^1_{\mathrm{Ham}}(M)$. There is a semistrict Lie 2-algebra $L_{\infty}(M,\omega)$ whose underlying complex is given by the de Rham differential $d:C^{\infty}(M)\to \Omega^1_{\mathrm{Ham}}(M)$, the only non-zero skew-symmetric brackets are $\{\cdot,\cdot\}_{\omega}:\Omega^1_{\mathrm{Ham}}(M)\times \Omega^1_{\mathrm{Ham}}(M)\to \Omega^1_{\mathrm{Ham}}(M)$ and

\begin{align*}
[\cdot]^3:\Omega^1_{\mathrm{Ham}}(M)\times \Omega^1_{\mathrm{Ham}}(M)\times \Omega^1_{\mathrm{Ham}}&(M)\to C^{\infty}(M)\\
(\alpha,\beta,\gamma)\mapsto & \qquad \omega(X_{\alpha},X_{\beta},X_{\gamma}).
\end{align*}

The Lie algebra $\mathfrak{X}_{\mathrm{Ham}}(M)$ is regarded as a Lie 2-algebra with underlying complex $0\to \mathfrak{X}_{\mathrm{Ham}}(M)$. As shown in [53], one can see $L_{\infty}(M,\omega)$ as a central extension of $\mathfrak{X}_{\mathrm{Ham}}(M)$ by the abelian Lie 2-algebra $\mathfrak{n}$ whose underlying complex is $d:C^{\infty}(M)\to \Omega^1_{cl}(M)$. The extension is given by

\begin{equation}\label{eq:extension2plectic}
\begin{tikzcd}
0 \arrow[r] & C^{\infty}(M) \arrow[d, "d"] \arrow[r, "\id"] & C^{\infty}(M) \arrow[d, "d"] \arrow[r, "0"] & 0 \arrow[d, "d=0"] \arrow[r] & 0 \\
0 \arrow[r] & \Omega^1_{cl}(M) \arrow[r, "\iota"] & \Omega^{1}_{\mathrm{Ham}}(M) \arrow[r, "p"] & \mathfrak{X}_{\mathrm{Ham}}(M) \ar[r] & 0.
\end{tikzcd}
\end{equation}

A splitting $h_0:\mathfrak{X}_{\mathrm{Ham}}(M)\to \Omega^1_{\mathrm{Ham}}(M)$ induces a splitting $h$ of (<ref>) with $h_{-1}:=0$. Hamiltonian vector fields act on $C^{\infty}(M)\to \Omega^1_{cl}(M)$ by the Lie derivative, hence as seen in Example <ref>, one has a Chern-Weil-Lecomte map

$$cw:\underline{\Hom}^{\bullet}(\mathfrak{n}[1],\mathfrak{n}[1])^{\hat{\mathfrak{g}}}  \to H^{\bullet+1}(\mathfrak{X}_{\mathrm{Ham}}(M),\mathfrak{n}[1]),$$

with $cw(\id)=[K_h]$. There are only two non-vanishing components of the curvature $K_h$, namely the ordinary curvature of the splitting $h_0$

$$K_h:\wedge^2\mathfrak{X}_{\mathrm{Ham}}(M)\to \Omega^1_{\mathrm{Ham}}(M),$$

and $K^{-3}_h:\wedge^3\mathfrak{X}_{\mathrm{Ham}}(M)\to C^{\infty}(M)$ which is given by the 2-plectic structure

[Courant algebroids]

Let $E\to M$ be a Courant algebroid with pairing $\langle\cdot,\cdot\rangle:E\times E\to \R$, skew-symmetric bracket $\Cour{\cdot,\cdot}:\Gamma(E)\times \Gamma(E)\to \Gamma(E)$ and anchor map $\rho:E\to TM$, see [41]. We say that $E$ is exact if the sequence of vector bundles

\begin{equation}\label{eq:exactCourant}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

is exact. It is well known that an exact Courant algebroid has an isotropic splitting $\nabla:TM\to E$, which induces a closed 3-form $H\in \Omega^3_{cl}(M)$ defined as

$$H(X,Y,Z):=\langle\Cour{\nabla X,\nabla Y},\nabla Z\rangle,$$

for every $X,Y,Z\in \mathfrak{X}(M)$. The cohomology class $[H]\in H^3_{dR}(M)$ is called the Ševera class of the exact Courant algebroid $E$. We will see that the Ševera class can be described by means of an appropriate Chern-Weil-Lecomte map. Indeed, as shown in [55], every Courant algebroid has an associated semistrict 2-term $L_{\infty}$-algebra $\hat{\mathfrak{g}}$ with underlying complex

$$\mathcal{D}:C^{\infty}(M)\to \Gamma(E); \langle \mathcal{D}f,e\rangle=\frac{1}{2}\Lie_{\rho(e)}f,$$

and higher skew-symmetric brackets

$$[e_1,e_2]^2:=\Cour{e_1,e_2}; \quad [e,f]^2:=\langle e,\mathcal{D}e\rangle; \quad [e_1,e_2,e_3]^3:=-T(e_1,e_2,e_3),$$

where $T(e_1,e_2,e_3):=\frac{1}{3}\langle \Cour{e_1,e_2},e_3\rangle + c.p.$

Let $\mathfrak{g}:=\mathfrak{X}(M)$ be the Lie algebra of vector fields on $M$ viewed as a strict Lie 2-algebra with complex $0\to \mathfrak{X}(M)$.  Note that $\hat{\mathfrak{g}}$ is an extension of $\mathfrak{X}(M)$ by the abelian Lie 2-algebra $\mathfrak{n}$ whose underlying complex is $\frac{1}{2}d:C^{\infty}(M)\to \Omega^1(M)$. In terms of the corresponding 2-term complexes, the extension is given by

\begin{equation}\label{eq:extensionCourant}
\begin{tikzcd}
0 \arrow[r] & C^{\infty}(M) \arrow[d, "\frac{1}{2}d"] \arrow[r, "\id"] & C^{\infty}(M) \arrow[d, "\mathcal{D}"] \arrow[r, "0"] & 0 \arrow[d, "d=0"] \arrow[r] & 0 \\
0 \arrow[r] & \Omega^1(M) \arrow[r, "\rho^*"] & \Gamma(E) \arrow[r, "\rho"] & \mathfrak{X}(M) \ar[r] & 0.
\end{tikzcd}
\end{equation}

An isotropic splitting $\nabla:TM\to E$ of (<ref>) induces a splitting $h:\mathfrak{g}\to \hat{\mathfrak{g}}$ of (<ref>) whose homogeneous components are $h_{-1}:=0$ and $h_0:=\nabla:\mathfrak{X}(M)\to \Gamma(E)$. In particular, there is an induced representation of vector fields $\mathfrak{X}(M)$ on $C^{\infty}(M)\to \Omega^{1}(M)$ given by the Lie derivative. As observed in Example <ref>, one has a Chern-Weil-Lecomte map

$$cw:\underline{\Hom}^{\bullet}(\mathfrak{n}[1],\mathfrak{n}[1])^{\hat{\mathfrak{g}}}  \to H^{\bullet+1}(\mathfrak{g},\mathfrak{n}[1]),$$

which maps $\id\in \underline{\Hom}^0(\mathfrak{n}[1],\mathfrak{n}[1])^{\hat{\mathfrak{g}}}$ to the cohomology class $cw(\id)=[K_h]\in H^1(\mathfrak{g},\mathfrak{n}[1])$ of the curvature of $h$. Due to Remark <ref>, the curvature $K_h\in C^1(\mathfrak{g},\mathfrak{n}[1])$ has only two non-zero components, namely the ordinary curvature of the isotropic splitting

$$K_h:\wedge^2\mathfrak{X}(M)\to \Omega^1(M); K_h(X,Y)=\nabla[X,Y]-\Cour{\nabla X, \nabla Y},$$

and the curvature component coming from the bracket of order 3

$$K_h:\wedge^3\mathfrak{X}(M)\to C^{\infty}(M); K_h(X,Y,Z)=[\nabla X, \nabla Y, \nabla Z]^3.$$

It follows from the definition of the bracket $[\cdot,\cdot]^3$, that

\begin{align*}
[\nabla X, \nabla Y, \nabla Z]^3=&-T(\nabla X, \nabla Y, \nabla Z)\\
=& -\frac{1}{3}\langle\Cour{\nabla X, \nabla Y}, \nabla Z \rangle + c.p.\\
=& -\frac{1}{3}H(X,Y,Z) + c.p.\\
=& -H(X,Y,Z).
\end{align*}

In particular, the Ševera class $[H]\in H^3_{dR}(M)$ appears as one of the components of the cohomology class $[K_h]\in H^1(\mathfrak{g},\mathfrak{n}[1])$.

Examples <ref> and <ref> are related by Ševera's classification of exact Courant algebroids. Every 2-plectic structure yields an exact Courant algebroid (unique up to isomorphism) together with an isotropic splitting and whose Ševera class is precisely the 2-plectic structure [53]. Hence, the Chern-Weil-Lecomte maps of Examples <ref> and <ref> commute with the construction of  the exact Courant algebroid defined by a 2-plectic manifold.

§ CHERN-WEIL MAP FOR PRINCIPAL 2-BUNDLES

In this section we show how the Chern-Weil-Lecomte of an $L_{\infty}$-algebra extension can be applied to introduce a Chern-Weil map for principal 2-bundles over Lie groupoids.

§.§ Principal 2-bundles over a Lie groupoid

We start by introducing some terminology that shall be used throughout the whole section. For a Lie 2-group we use the notation $\mathbb{G}=(G_1\rightrightarrows G_0)$ and similarly a Lie groupoid will be denoted by $\mathbb{X}=(X_1\rightrightarrows X_0)$.

Let $\mathbb{G}$ be a Lie 2-group and $\mathbb{X}$ be a Lie groupoid. A right action of $\mathbb{G}$ on $\mathbb{X}$ is a Lie groupoid morphism $a:\mathbb{X}\times \GG\to \mathbb{X}$ which defines Lie group actions $a_1:X_1\times G_1\to X_1$ and $a_0:X_0\times G_0\to X_0$ on arrows and objects, respectively. Left actions are defined in a similar manner.

Right and left actions are denoted $\mathbb{X}\curvearrowleft \GG$ and $\GG\curvearrowright \mathbb{X}$, respectively. Similarly,  $xg$ and $gx$ denote the corresponding actions of $g\in \GG$ on $x\in \mathbb{X}$.

Given a right action of $\mathbb{G}$ on $\mathbb{X}$,  the structural maps of $\mathbb{X}$ are equivariant with respect to the structural maps of $\mathbb{G}$. That is, if $g\in G_1, g_0\in G_0$ and $x\in X_1, x_0\in X_0$, then

$$s_X(xg)=s_X(x)s_G(g),\quad t_X(xg)=t_X(x)t_G(g),\quad 1_{x_0g_0}=1_{x_0}1_{g_0}$$

and for every composable arrows $(x,y)\in X_2$ and $(g,g') \in G_2$ the following interchange law is fulfilled

\begin{equation}\label{eq:2interchangelaw}
\end{equation}
where $x*y$ and $g*h$ denote the groupoid multiplication in $\mathbb{X}$ and $\GG$, respectively.

It is worth observing that a right action $\mathbb{X}\curvearrowleft \GG$ is not necessarily given by Lie groupoid automorphism. However, the Lie group $G_0$ acts on $\mathbb{X}$ by Lie groupoid automorphisms via $1_{g_0}:\mathbb{X}\to \mathbb{X}$ for every $g_0\in G_0$. Also, every arrow $g\in G_1$ defines a map $\tau_g:X_0\to X_1; x_0\mapsto 1_{x_0}g$ which is a natural isomorphism $\tau_g:1_{s(g)}\Rightarrow 1_{t(g)}$. Hence $\mathbb{X}\curvearrowleft \GG$ can be seen as a 2-group morphism $\GG\to 2\mathrm{Aut}(\mathbb{X})$ where $2\mathrm{Aut}(\mathbb{X})$ denotes the 2-group of automorphisms of $\mathbb{X}$.

Let us see some examples of Lie 2-group actions.

A usual action of a Lie group $G$ on a manifold $X$ can be seen as an action of the Lie 2-group $G\rightrightarrows G$ on the unit groupoid $X\rightrightarrows X$.

Every Lie 2-group $\mathbb{G}$ acts on itself by right translations. More generally,  any Lie 2-subgroup $\mathbb{H}$ of $\mathbb{G}$ acts on $\mathbb{G}$ by right translations.

Any Lie 2-group $\mathbb{G}$ acts on itself on the left by conjugation at both the level of arrows and the level of objects. This induces an action $\mathbb{G}\curvearrowright (\mathfrak{g}_1\rightrightarrows \mathfrak{g}_0)$ on its Lie 2-algebra, called the adjoint representation of $\GG$.

A right action of $\mathbb{X}\curvearrowleft \GG$ induces a right action $T\mathbb{X}\curvearrowleft\mathbb{G}$ on the tangent groupoid $T\mathbb{X}=(TX_1\rightrightarrows TX_0)$ given by the tangent lift of the action of $G_i$ on $X_i$, for $i=0,1$.

Let $\mathbb{X}=(X_1\rightrightarrows X_0)$ and $\mathbb{E}=(E_1\rightrightarrows E_0)$ be two Lie groupoids. A Lie groupoid morphism $\pi: \mathbb{E}\to \mathbb{X}$ is said to be  a fibration if it satisfies the following conditions

* the base map $\pi_0:E_0\to X_0$ is a surjective submersion, and

* the map $\tilde{\pi}:E_1\to X_1\pullback{s}{\pi_0}E_0, e\mapsto (\pi_1(e),s_{E}(e))$ is a surjective submersion.

Let $\pi:\p\to \mathbb{X}$ be a groupoid fibration together with a right action $\mathbb{P}\curvearrowleft \GG$. We say that the action is  principal along $\pi$ if

\begin{equation}\label{eq:principality}
\mathbb{P}\times \GG\to \mathbb{P}\times_{\mathbb{X}}\mathbb{P}; (p,g)\mapsto (p,pg)
\end{equation}

is a Lie groupoid isomorphism. In this case, we say that $\mathbb{P}\to \mathbb{X}$ is a principal $\GG$-2-bundle.

Our main motivation to consider principal 2-bundles as in Definition <ref> comes from the study of principal actions on differentiable stacks as in [13], where principal actions of stacky Lie groupoids are studied. Since every Lie groupoid $\mathbb{X}=(X_1\to X_0)$ defines a quotient stack $[X_0/X_1]$ [7], principal 2-bundles $\mathbb{P}\curvearrowleft \mathbb{G}\to \mathbb{X}$ in the sense of Definition <ref> define principal actions of the stacky Lie group $[G_0/G_1]$ along the stack morphism $[P_0/P_1]\to [X_0/X_1]$ induced by $\pi:\p\to \mathbb{X}$. In [13], a stacky Lie group is not necessarily strict and actions are weak. Hence, the notion of principal 2-bundle considered here can be seen as a strict version of a stacky principal bundle as in [13]. Several notions of principal 2-bundles have been studied by different authors, in [63] weak 2-groups act in a principal manner along a map $\mathcal{P}\to M$ between smooth 2-spaces. Here the base $M$ is a manifold seen as a smooth 2-space. Similarly, in [61], Waldorf studies principal 2-bundles as in Definition <ref> with $\mathbb{X}$ being the unit Lie groupoid of a manifold, however principality is required in a weaker sense. Also, principal 2-bundles as in Definition <ref> are studied in [15].

[Ordinary principal bundles]
Let $P\to X$ be a usual principal $G$-bundle over a manifold. Then, viewing manifolds as unit groupoids yields a principal 2-bundle $\mathbb{P}\curvearrowleft \GG\to \mathbb{X}$.

[Principal 2-bundles over manifolds]

A principal 2-bundle over a manifold is a principal 2-bundle $\mathbb{P}\curvearrowleft \GG \to (X\rr X)$ where the base space is the unit groupoid of a manifold $X$. A more general notion of principal 2-bundle over a manifold was introduced in [63], where principality is defined in a weak sense, i.e. the map (<ref>) is a Morita map rather than an isomorphism.

[Principal bundles over Lie groupoids]
Let $\pi_0:P_0\to X_0$ be a principal $G$-bundle  with a left action of a Lie groupoid $\mathbb{X}=(X_1\rightrightarrows
X_0)$ along $\pi_0$ which commutes with the right action of $G$. The transformation groupoid $\p:=X_1\ltimes P_0\rightrightarrows P_0$ is a principal $\GG$-2-bundle over $\mathbb{X}$ with $\GG=(G\rr G)$ defined as the unit groupoid of $G$. These kind of principal bundles are studied in [38].

[Homogeneous Lie groupoids]
Let $\mathbb{G}$ be a Lie 2-group and $\mathbb{H}$ a Lie 2-subgroup such that $H_i\subseteq G_i$ is a closed subgroup for $i=0,1$. Then $\mathbb{G}$ is a principal $\mathbb{H}$-2-bundle over the Lie groupoid, $G_1/H_1\rightrightarrows G_0/H_0$.

Let $\phi:\mathbb{Y}\to \mathbb{X}$ be a Lie groupoid morphism. If $\mathbb{P}\curvearrowleft \GG\to \mathbb{X}$ is a principal 2-bundle, then the pullback bundles $\phi^*_1P_1\to Y_1$ and $\phi^*_0P_0\to X_0$ define a principal 2-bundle $\phi^*\mathbb{P}\curvearrowleft \GG\to \mathbb{Y}$ called the pullback 2-bundle of $\mathbb{P}$ by $\phi:\mathbb{Y}\to \mathbb{X}$.

Every surjective submersion $q:Y\to X$ defines a Lie groupoid $Y\times_{X}Y\to Y$ called the submersion groupoid. The nerve of the submersion groupoid yields a simplicial manifold $(Y_{n})_n$ whose face maps are denoted by $d_i:Y_n\to Y_{n-1}, i=0,1,...,n$. An $\mathbb{S}^1$-gerbe over a manifold $X$ is given by a triple $\mathcal{G}=(Y,P,\mu)$ where $q:Y\to X$ is a surjective submersion, $\pi:P\to Y\times_{X}Y$ is a principal $\mathbb{S}^1$-bundle and $\mu:d^*_0P\otimes d^*_2P\to d^*_1P$ is a bundle isomorphism over $Y_{3}$ called the gerbe multiplication, satisfying

$$d^*_1\mu\circ(d^*_3\mu \otimes \id)=d^*_2\mu \circ (\id \otimes d^*_0\mu),$$

over $Y_{4}$. One can see that an $\mathbb{S}^1$-gerbe determines a principal 2-bundle with 2-group $\mathbb{S}^1\rr \{*\}$. Indeed, there is a natural Lie groupoid $P\rr Y$ with source and target maps $s:=d_0\circ \pi$, $t:=d_1\circ \pi$ and multiplication $m:P\times_YP\to P$ defined by

$$\mu((x,y,z),p_1\otimes p_2):=((x,y,z),m(p_1,p_2)).$$

One can see that the action of $\mathbb{S}^1$ on $P$ defines an action of the Lie 2-group $\mathbb{S}^1\rr \{*\}$ on $P\rr Y$, making $\p:=(P\rr Y)$ a principal 2-bundle over the submersion groupoid $Y\times_{X}Y\to Y$. We refer to $\p\to Y\times_{X}Y$ as the principal 2-bundle underlying the $\mathbb{S}^1$-gerbe $\mathcal{G}$.

§.§ Equivariant VB-groupoids and LA-groupoids

Let $\mathbb{X}=(X_1\rr X_0)$ be a Lie groupoid. A VB-groupoid $\pi:\mathbb{E}\to \mathbb{X}$ is a Lie groupoid $\mathbb{E}=(E_1\rr E_0)$ where $E_1\to X_1$ and $E_0\to X_0$ are vector bundles and all the structural maps of $\mathbb{E}$ are vector bundle morphisms over the corresponding structural maps of $\mathbb{X}$. An LA-groupoid is a VB-groupoid $\pi:\mathbb{E}\to \mathbb{X}$ where $E_1\to X_1$ and $E_0\to X_0$ are Lie algebroids and all the structural maps are Lie algebroid morphisms. See [43] for more details.

Let $\GG$ be a Lie 2-group. A VB-groupoid $\pi:\mathbb{E}\to \mathbb{X}$ is called $\GG$-equivariant if there is a left action $\GG\curvearrowright \mathbb{E}$ by vector bundle automorphisms. As a consequence, $E_1\to X_1$ and $E_0\to X_0$ are $G_1$-equivariant and $G_0$-equivariant, respectively. It follows from Remark <ref> that $G_0\curvearrowright \mathbb{E}$ is an action by VB-groupoid automorphisms. In a similar manner, one can introduce $\GG$-equivariant LA-groupoids and in this case $G_0\curvearrowright \mathbb{E}$ is an action by LA-groupoid automorphisms.

Let $\GG \curvearrowright \mathbb{X}$ be a left action. Then the tangent LA-groupoid $T\mathbb{X}\to \mathbb{X}$ is $\GG$-equivariant with the action defined in Example <ref>.

Let $\p\curvearrowleft \GG\to \mathbb{X}$ be a principal 2-bundle. Let $\mathbb{g}=(\mathfrak{g_1}\rr \mathfrak{g_0})$ be the Lie 2-algebra of $\GG$ equipped with the adjoint representation of Example <ref>. The trivial VB-groupoid $\p\times \mathbb{g}\to \p$ is $\GG$-equivariant. Viewing $P_1\times \mathfrak{g}_1\to P_1$ and $P_0\times \mathfrak{g}_0\to P_0$ as bundles of Lie algebras, then $\p\times \mathbb{g}\to \p$ is actually a $\GG$-equivariant LA-groupoid.

Let $\pi:\mathbb{E}\to \mathbb{X}$ be a $\GG$-equivariant VB-groupoid. If $G_1\curvearrowright E_1$ and $G_0\curvearrowright E_0$ are principal actions, then $G_1\curvearrowright X_1$ and $G_0\curvearrowright X_0$ are principal as well and both $E_1/G_1\to X_1/G_1$ and $E_0/G_0\to X_0/G_0$ are vector bundles. It is easy to see that in this case, all groupoid structures also descend to the quotient, yielding a quotient VB-groupoid $\overline{\pi}:\mathbb{E}/\GG\to \mathbb{X}/\GG$. A similar result holds for $\GG$-equivariant LA-groupoids.

The following examples of quotients LA-groupoids associated to $\GG$-equivariant ones are particularly important in this work. See also [15], where both examples are discussed.

Let us consider the $\GG$-equivariant LA-groupoid $\p\times \mathbb{g}\to \p$ of Example <ref>. The corresponding quotient LA-groupoid is denoted $\Ad(\p)\to \mathbb{X}$ and referred to as the adjoint LA-groupoid of $\p$.

Let $\p\curvearrowleft \GG\to \mathbb{X}$ be a principal 2-bundle. Consider the tangent bundle $T\p\to \p$ equipped with the $\GG$-equivariant LA-groupoid structure of Example <ref>. The corresponding quotient LA-groupoid is denoted $\At(\p)\to \mathbb{X}$ and called the Atiyah LA-groupoid of $\p$.

As shown in [49], every VB-groupoid has an associated 2-vector space of sections, which in the case of an LA-groupoid it is also a Lie 2-algebra. We proceed now to recall the definition of the Lie 2-algebra of sections of an LA-groupoid following [49].

Let $\pi:\mathbb{E}\to \mathbb{X}$ be a VB-groupoid. A multiplicative section of $\mathbb{E}$ is a Lie groupoid morphism $e:\mathbb{X}\to \mathbb{E}$ satisfying $\pi\circ e=\id_{\mathbb{X}}$. The space of multiplicative sections is denoted by $\Gamma_{mult}(\mathbb{E})$. Multiplicative sections form a category in which morphisms are natural transformations $\tau:e\Rightarrow e'$ satisfying $1_{\pi} \bullet \tau=1_{\id_{\mathbb{X}}}$, where $\bullet$ denotes the horizontal composition of natural transformations:

\[\begin{tikzcd}
\mathbb{X} \arrow[rr, bend left=50,""{name=U, below}, "{e'}" description]
\arrow[rr, bend right=50,""{name=D}, "{e}" description]
\mathbb{E}\arrow[Rightarrow, from=D,to=U,"\tau"]\arrow[rr, bend left=50,""{name=D, below}, "{\pi}" description]
\arrow[rr, bend right=50,""{name=U}, "{\pi}" description]&&\mathbb{X} \arrow[Rightarrow,"1_{\pi}", from=U, to=D]&=& \mathbb{X}\arrow[rr, bend left=50, "{\id_{\mathbb{X}}}" description,""{name=U, below}]
\arrow[rr, bend right=50, "{\id_{\mathbb{X}}}" description,""{name=D}]&&\mathbb{X}.\arrow[Rightarrow,"1_{\id_{\mathbb{X}}}", from=D, to=U]
\end{tikzcd}
\]

The core bundle of $\mathbb{E}$ is the vector bundle over $X_0$ defined as the pullback $C:=\ker(s:E_1\to X_1)|_{X_0}$. The core complex is the 2-term complex of vector bundles $\partial:C\to E_0$ with $\partial(c)=t(c)$ where $t:\mathbb{E}\to \mathbb{X}$ is the target map.

The complex of multiplicative sections is the 2-term complex $C^{\bullet}_{mult}(\mathbb{E})$ defined by

\begin{equation}\label{eq:complexmultiplicative}
\delta:\Gamma(C)\to \Gamma_{mult}(\mathbb{E}); c\mapsto c^r-c^l,
\end{equation}

where $c^r(x):=c_{t(x)}\cdot 0_x$ and $c^l(x):=-0_x\cdot (c_{s(x)})^{-1}$ for every $x\in X_1$. Here $0:\mathbb{X}\to \mathbb{E}$ denotes the zero section and $\cdot$ the groupoid multiplication in $\mathbb{E}$. As shown in [49], the category of multiplicative sections of $\mathbb{E}$ is isomorphic to the 2-vector space $\Gamma(C)\oplus \Gamma_{mult}(\mathbb{E})\rr \Gamma_{mult}(\mathbb{E})$ defined by the complex of multiplicative sections.

Let us consider now an LA-groupoid $\pi:\mathbb{E}\to \mathbb{X}$. In this case, the core bundle $C\to X_0$ is naturally a Lie algebroid and the complex of multiplicative sections is part of a crossed module of Lie algebras $\Gamma(C)\to \Gamma_{mult}(\mathbb{E})\to \mathrm{Der}(\Gamma(C))$ where the last map $e\mapsto D_e$ is given by

\begin{equation}\label{eq:multiplicativecrossedmodule}
\end{equation}

In other words, the category of multiplicative sections of $\mathbb{E}$ inherits a natural structure of strict Lie 2-algebra which we denote by $\mathbb{L(E)}$. For more details see [49].

Let $\mathbb{X}$ be a Lie groupoid with Lie algebroid $A_{\mathbb{X}}$. As shown in [49] (see also [8]), the category of multiplicative sections of the tangent LA-groupoid $T\mathbb{X}\to \mathbb{X}$ is given by the crossed module $\Gamma(A_{\mathbb{X}})\to \mathfrak{X}_{mult}(\mathbb{X})\to \mathrm{Der}(\Gamma(A_{\mathbb{X}}))$, where $\mathfrak{X}_{mult}(\mathbb{X})$ denotes the Lie algebra of multiplicative vector fields on $\mathbb{X}$ of Mackenzie and Xu [42].

Let $\mathbb{X}=(X_1\rr X_0)$ be a Lie groupoid and $\mathbb{V}:=V_{-1}\oplus V_0$ the 2-vector defined by a 2-term cochain complex $\partial:V_{-1}\to V_0$. The trivial VB-groupoid $\mathbb{X}\times \mathbb{V}\to \mathbb{X}$ has core bundle $C=X_0\times V_{-1}\to X_0$, the space of multiplicative sections is $\mathrm{Hom}_{gpd}(\mathbb{X},\mathbb{V})$ and the complex of multiplicative sections is given by

$$C^{\infty}(X_0,V_{-1})\to \mathrm{Hom}_{gpd}(\mathbb{X},\mathbb{V}); f\mapsto (t^*f-s^*f)\oplus s^*(\partial\circ f).$$

The corresponding 2-vector space is denoted $\mathbb{F(X,V)}$ and referred to as the 2-vector space of $\mathbb{V}$-valued multiplicative functions on $\mathbb{X}$.

Let us consider now a $\GG$-equivariant LA-groupoid $\pi:\mathbb{E}\to \mathbb{X}$ with core bundle $C\to X_0$. Since $G_0$ acts on $\pi:\mathbb{E}\to \mathbb{X}$ by LA-groupoid automorphisms, the core $C\to X_0$ inherits the structure of a $G_0$-equivariant Lie algebroid. Also, since $t:\mathbb{E}\to \mathbb{X}$ is $\GG$-equivariant (Remark <ref>), the core complex $\partial:C\to E_0$ is a 2-term complex of $G_0$-equivariant vector bundles, i.e. $\partial$ is $G_0$-equivariant. It is straightforward to check the following.

Let $\pi:\mathbb{E}\to \mathbb{X}$ be a $\GG$-equivariant LA-groupoid with core $C\to X_0$. If the action is principal, then $G_0\curvearrowright C$ is also principal and quotient LA-groupoid $\mathbb{E}/\GG\to \mathbb{X}/\GG$ has core bundle $C/G_0\to X_0/G_0$ with core anchor map $\overline{\partial}:C/G_0\to E_0/G_0; [c]\mapsto [\partial(c)]$. Additionally, the space of multiplicative sections of $\mathbb{E}/\GG$ identifies with $\Gamma_{mult}(\mathbb{E})^\GG$ and the corresponding crossed module of multiplicative sections is isomorphic to

$$\Gamma(C)^{G_0}\to \Gamma_{mult}(\mathbb{E})^\GG \to \mathrm{Der}(\Gamma(C)^{G_0}),$$

where the maps are respectively given the restrictions of (<ref>) and (<ref>) to invariant sections.

This allows to describe the crossed module of multiplicative sections of both the adjoint LA-groupoid and the Atiyah LA-groupoid of a principal 2-bundle.

Let $\p\curvearrowleft \GG\to \mathbb{X}$ be a a principal 2-bundle and $\Ad(\p)\to \mathbb{X}$ be the adjoint LA-groupoid of Example <ref>. If $\mathbb{g}=(\mathfrak{g}_1\rr \mathfrak{g}_0)$ is the Lie 2-algebra of $\GG$, let $\partial:\mathfrak{h}\to \mathfrak{g}_0$ be the 2-term complex defined by the crossed module associated to $\mathbb{g}$. The core bundle of $\Ad(\p)$ is  the Lie algebra bundle $P_{0}\times_{G_0} \mathfrak{h}\to X_0$ hence sections of the core bundle are elements in $C^{\infty}(P_0,\mathfrak{h})^{G_0}$. The space of multiplicative sections of $\Ad(\p)$ is given by

$$\Gamma_{mult}(\Ad(\p))\cong \Hom_{gpd}(\p,\mathbb{g})^{\GG},$$

where the right hand side denotes the space of $\GG$-equivariant groupoid morphisms $F:\p\to \mathbb{g}$. By $\GG$-equivariant we mean

$$F_1(pg)=\Ad_{g^{-1}}^{G_1}f_1(p),\quad F_0(p_0g_0)=\Ad_{g^{-1}_0}^{G_0}f_0(p_0),$$

for every $p\in P_1, p_0\in P_0$ and $g\in G_1,g_0\in G_0$. As a consequence of Proposition <ref> and Example <ref>, the complex of multiplicative sections of $\Ad(\p)$ is

$$C^{\infty}(P_0,\mathfrak{h})^{G_0}\to \Hom_{gpd}(\p,\mathbb{g})^{\GG}; f\mapsto (t^*f-s^*f,s^*(\partial \circ f)),$$

where we have used the decomposition $\mathbb{g}=\mathfrak{h}\oplus \mathfrak{g}_1$ in terms of crossed module components.

Let $\At(\p)\to \mathbb{X}$ be the Atiyah LA-groupoid of a principal 2-bundle $\p\curvearrowleft \GG\to \mathbb{X}$. In this case, the core bundle of $T\p\to \p$ is the Lie algebroid $A_{\p}$ of $\p$, hence the core bundle of $\At(\p)$ is $A_{\p}/G_0$. The space of multiplicative sections of the Atiyah LA-groupoid is given by

$$\Gamma_{mult}(\At(\p))\cong \mathfrak{X}_{mult}(\p)^{\GG},$$

where the right hand side is the space of $\GG$-invariant multiplicative vector fields on $\p$. It follows from Proposition <ref> and Example <ref> that the complex of multiplicative sections of $\At(\p)$ is given by

$$\Gamma(A_{\p})^{G_0}\to \mathfrak{X}_{mult}(\p)^{\GG}; a\mapsto a^r-a^l.$$

Just as a usual principal bundle has an associated Atiyah sequence, for a  given principal 2-bundle the adjoint LA-groupoid and the Atiyah LA-groupoid fit into an exact sequence in the category of LA-groupoids [15].

Let $\pi:\p\curvearrowleft \GG\to \mathbb{X}$ be a principal 2-bundle. The Atiyah sequence of $\p$ is the exact sequence of LA-groupoids

As shown in [49], every LA-groupoid morphism covering the identity induces a Lie 2-algebra morphism between the corresponding categories of multiplicative sections. In particular, the Atiyah sequence of $\p$ determines an extension of Lie 2-algebras

\begin{equation}\label{eq:Lie2atiyah}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

referred to as the Atiyah extension of $\pi:\p\curvearrowleft \GG\to \mathbb{X}$ and denoted by $\mathbb{At}(\p)$.

§.§ Connections on principal 2-bundles

Let $\mathbb{X}=(X_1\rr X_0)$ be a Lie groupoid and consider its nerve $N\mathbb{X}=(X_{n})$ with its usual structure of simplicial manifold. Here $X_n$ denotes the space of $n$-simplices, which are strings of arrows $(x_1,...,x_n)$ with $s(x_i)=t(x_{i+1}), i=1,...,n-1$ and face maps $d_i:X_{n+1}\to X_n; i=1,...,n+1$. The simplicial structure gives rise to a complex $\delta:\Omega^{\bullet}(X_n)\to \Omega^{\bullet}(X_{n+1})$ defined as the alternating sum of pullbacks by the face maps. This differential does not change the form degree, but it increases the simplicial degree by +1. For instance, $\delta\eta=t^*\eta-s^*\eta$ for every $\eta\in \Omega^{k}(X_0)$ and $\delta\omega=pr^*_1\omega-m^*\omega+pr^*_2\omega$ if $\omega\in\Omega^{k}(X_1)$, where $m,pr_1,pr_2:X_1\times_{X_0}X_1\to X_1$ denote the groupoid multiplication and the canonical projections, respectively.

The Bott-Shulman-Stasheff complex of $\mathbb{X}$ is the double complex $\Omega^{\bullet}(X_{\bullet})$ with vertical differential given by the usual de Rham differential of forms and horizontal differential defined by the simplicial structure. The de Rham complex of $\mathbb{X}$ is the total complex:

with differential $D\omega:=d\omega+(-1)^{j+1}\delta\omega$ for every $\omega \in \Omega^{i}(X_j)$.

A multiplicative $k$-form on  $\mathbb{X}$ is a $k$-form $\omega\in \Omega^k(X_1)$ which satisfies $\delta\omega=0$. That is,

\begin{equation}\label{eq:multiplicativeform}
\end{equation}

As shown in [11, 12], a $k$-form $\omega\in \Omega^k(X_1)$ is multiplicative if and only if the induced bundle map $\omega:\oplus^{k-1}T\mathbb{X}\to T^*\mathbb{X}$ is a Lie groupoid morphism.

Let $\eta\in \Omega^{k+1}_{cl}(X_0)$ be a closed form. We say that a multiplicative form $\omega\in \Omega^k(X_1)$ is closed with respect to $\eta$ if $d\omega=t^*\eta-s^*\eta$. In other words, the element $(\omega+\eta)\in C^{k+1}_{dR}(\mathbb{X})$ is a cocycle in the de Rham complex of $\mathbb{X}$.

A multiplicative $k$-form on $\mathbb{X}$ should be thought of as a $k$-form of degree 1 on the quotient stack $[X_0/X_1]$ since it yields a morphism of complexes $\bigoplus^{k-1} T_{\mathbb{X}}\to T^*_{\mathbb{X}}[1]$, where $T_{\mathbb{X}}:=(A_{\mathbb{X}}\to TX_0)$ is the tangent complex and $T^*_{\mathbb{X}}:=(T^*X_0\to A^*_{\mathbb{X}})$ is the cotangent complex. Similarly, a multiplicative $k$-form on $\mathbb{X}$ which is closed with respect to $\eta\in \Omega^{k+1}_{cl}(X_0)$ has to be thought of as a closed $k$-form on the quotient stack $[X_0/X_1]$. The special case of 2-forms arises in the context of Dirac structures [10] as well as in shifted symplectic geometry [51], see also [14].

A more general notion of multiplicative form with values in a VB-groupoid is considered in  [24]. We are interested in the particular case of $\mathbb{g}$-valued multiplicative $k$-forms on a Lie groupoid $\mathbb{X}$ where $\mathbb{g}=(\mathfrak{g}_1\rr \mathfrak{g}_0)$ is a Lie 2-algebra.

A $\mathbb{g}$-valued multiplicative $k$-form on a Lie groupoid $\mathbb{X}=(X_1\rr X_0)$ is a $k$-form $\omega_1\in \Omega^k(X_1,\mathfrak{g}_1)$ such that $\omega_1:\oplus^kTX_1\to \mathfrak{g}_1\times X_1$ is a Lie groupoid morphism covering $\omega_0:\oplus^kTX_0\to \mathfrak{g}_0\times X_0$ where $\omega_0\in \Omega^k(X_0,\mathfrak{g}_0)$.

If $\partial:\mathfrak{h}\to \mathfrak{g}_0$ is the cochain complex associated with $\mathbb{g}$, then a $\mathbb{g}$-valued multiplicative $k$-form on $\mathbb{X}$ is given by a pair of $k$-forms $\bar\omega\in \Omega^k(X_1,\mathfrak{h})$ and $\omega_0\in \Omega^k(X_0,\mathfrak{g}_0)$ with

\begin{equation}\label{eq:splitmultiplicativeform}
\omega_1=\bar\omega + s^*\omega_0,
\end{equation}

and satisfying the following conditions:

\begin{align}\label{eq:conditionssplitforms}
\partial \circ \bar\omega=&t^*\omega_0-s^*\omega_0. \nonumber
\end{align}

For more details, see [24].

Let $\mathbb{X}=(X_1\rr X_0)$ be a Lie groupoid and $\eta\in\Omega^{k+1}_{cl}(X_0,\mathfrak{h})$ be a closed form. We say that a $\mathbb{g}$-valued multiplicative $k$-form $\omega_1=(\bar\omega,s^*\omega_0)$ is closed with respect to $\eta$ if the following conditions hold:

\begin{align}\label{eq:gvaluedclosed}
d\omega_0=&\partial\circ \eta\\
\end{align}

A $\mathbb{g}$-valued multiplicative $k$-form on $\mathbb{X}$ should be thought of as a $k$-form of degree 1 on the stack $[X_0/X_1]$ with values in the stacky Lie algebra $[\mathfrak{g}_0/\mathfrak{g}_1]$. A similar interpretation holds for $\mathbb{g}$-valued multiplicative $k$-forms which are closed with respect to $\eta\in\Omega^{k+1}_{cl}(X_0,\mathfrak{h})$.

Just as multiplicative $k$-forms with respect to closed $(k+1)$-form are $(k+1)$-cocycles in the de Rham complex of a Lie groupoid, closed $\mathbb{g}$-valued multiplicative $k$-form are cocycles in the differential graded Lie algebra given by the de Rham complex of a Lie groupoid with values in a Lie 2-algebra [61]. This will be discussed below.

Let $\mathbb{g}$ be a Lie 2-algebra with associated 2-term cochain complex $\partial:\mathfrak{h}\to \mathfrak{g}_0$. Due to notation purposes, it is useful to see this 2-term complex as $\partial_k:\mathfrak{v}_k\to \mathfrak{v}_{k+1}$ where $\partial_{-1}:=\partial:\mathfrak{h}\to \mathfrak{g}_0$ and $\mathfrak{v}_k=0$ for every $k\neq -1,0$. Let $\mathbb{X}=(X_1\rr X_0)$ be a Lie groupoid and consider the simplicial manifold given by the nerve $N\mathbb{X}=(X_{n})$. There is a total complex

with differential $C^p(\mathbb{X},\mathbb{g})\to C^{p+1}(\mathbb{X},\mathbb{g})$ given by $d+(-1)^i\delta+(-1)^{i+j}\partial_k$ on each component $\Omega^i(X_j,\mathfrak{v}_k)$. Here $d$ denotes the usual de Rham differential of forms and  $\delta$ is the simplicial differential on forms. As explained in [61], if one considers elements in $C^p(\mathbb{X},\mathbb{g})$ whose form degree is at least $p$, then the total differential does not preserve such an elements. However, one can co-restrict the total differential yielding a map $\mathbb{D}:C^p_{dR}(\mathbb{X},\mathbb{g})\to C^{p+1}_{dR}(\mathbb{X},\mathbb{g})$ which actually defines a differential. An element $\Phi\in C^p_{dR}(\mathbb{X},\mathbb{g})$ is given by a triple of differential forms:

$$\Phi^a\in \Omega^p(X_0,\mathfrak{g}_0); \quad \Phi^b\in \Omega^p(X_1,\mathfrak{h}); \quad \Phi^c\in \Omega^{p+1}(X_0,\mathfrak{h}),$$

such that

\begin{align}\label{eq:waldrofsplitforms}
\partial \circ \Phi^b=&t^*\Phi^a-s^*\Phi^a. \nonumber
\end{align}

The component $\Phi^c$ controls the differential of $\Phi$. More concretely, the differential $\mathbb{D}\Phi\in C^{p+1}_{dR}(\mathbb{X},\mathbb{g})$ is given by the triple $((\mathbb{D}\Phi)^a,(\mathbb{D}\Phi)^b,(\mathbb{D}\Phi)^c)$ with

\begin{align}\label{eq:waldorfsplitdifferential}
(\mathbb{D}\Phi)^a=&d\Phi^a-(-1)^p\partial\circ \Phi^c\\
(\mathbb{D}\Phi)^b=&d\Phi^b-(-1)^p\delta_0\circ \Phi^c\\
\end{align}

Note that $\Phi\in C^p_{dR}(\mathbb{X},\mathbb{g})$ if and only if $\Phi^b+s^*\Phi^a$ is a $\mathbb{g}$-valued multiplicative form on $\mathbb{X}$ together with conditions on $\Phi^c$ (<ref>) which encodes information about derivatives. In [61], elements in $C^{\bullet}_{dR}(\mathbb{X},\mathbb{g})$ are called $\mathbb{g}$-valued differential forms on $\mathbb{X}$. It is important to observe that in this paper we will always consider multiplicative $\mathbb{g}$-valued forms in the sense of Definition <ref>, which is a particular case of [61]. Also, note that $\Phi=(\Phi^a,\Phi^b,\Phi^c)\in C^p_{dR}(\mathbb{X},\mathbb{g})$ being a cocycle, i.e. $\mathbb{D}\Phi=0$, is equivalent to saying that $\Phi^b+s^*\Phi^a$ is a $\mathbb{g}$-valued multiplicative $p$-form on $\mathbb{X}$ which is closed with respect to $\Phi^c$, see Definition <ref>.

We are particularly concerned with the case of $\mathbb{g}$-valued multiplicative connection 1-forms.

Let $\p\curvearrowleft \GG\to \mathbb{X}$ be a principal 2-bundle and let $\mathbb{g}$ be the Lie 2-algebra of $\GG$. A $\mathbb{g}$-valued multiplicative 1-form $\theta$ on $\p$ is called a 2-connection if it is a connection on both morphisms and objects.

In other words, a 2-connection is given by a pair $\theta=(\theta_1,\theta_0)$ of usual principal connections $\theta_i\in \Omega^1(P_i,\mathfrak{g}_i), i=1,2$ such that $\theta_1:TP_1\to \mathfrak{g}_1\times P_1$ is a Lie groupoid morphism covering $\theta_0:TP_0\to \mathfrak{g}_0 \times P_0$. As observed in (<ref>),  one can write $\theta_1=\bar\theta+s^*\theta_0$ where $\bar\theta\in\Omega^1(P_1,\mathfrak{h})$ satisfies the identities in (<ref>). From now on a 2-connection will be written $\theta=\bar\theta+s^*\theta_0$.

Connections on principal 2-bundles were introduced in [61] as a triple $(\theta^a,\theta^b,\theta^c)$ of forms $\theta^a\in \Omega^1(P_0,\mathfrak{g}_0), \theta^b\in \Omega^1(P_1,\mathfrak{h})$ and $\theta^c\in\Omega^2(P_0,\mathfrak{h})$ with $\theta^a$ and $\theta^b$ satisfying identities (<ref>) and (<ref>) together with a suitable notion of invariance. Every 2-connection $\theta=\bar\theta+s^*\theta_0$ in our sense defines such a triple as $(\theta_0,\bar\theta,0)$, however the notion of invariance is different from that of [61]. Regarding the term $\theta^c$, as shown in [61] it only plays a role when introducing the curvature of a connection, once connections in [61] are defined in order to have a well-defined parallel transport not only along paths but also along surfaces called bigons. In this paper, 2-connections are defined as a model for connections on principal bundles over differentiable stacks. We also should mention that the notion of 2-connection of our work coincides with the one in [15] .

Let us see some examples of principal 2-bundles with 2-connections. For more examples, see [61].

Every usual principal $G$-bundle $P\to X$ can be seen as a principal 2-bundle $\p=(P\rr P)\to \mathbb{X}=(X\rr X)$ as in Example <ref>. Every connection 1-form $\theta_0$ on $P$ can be seen as a 2-connection $\bar\theta + s^*\theta_0$ on $\p$ with $\bar\theta=0$.

Let $\GG=(G_1\rr G_0)$ be the Lie 2-group defined by a crossed module $H\to G_0$. The Maurer-Cartan form $\theta_{MC}\in\Omega^1(\GG,\mathbb{g})$ defines a 2-connection on $\GG$ seen as a principal 2-bundle $\GG\curvearrowleft \GG\to \{*\}$. One has $\theta_{MC}=(\bar\theta, \theta_0)$ where $\theta_0$ is the Maurer-Cartan form on $G_0$ and $\bar\theta\in\Omega^1(H\ltimes G, \mathfrak{h})$ is defined by

where $\theta^{H}_{MC}$ is the Maurer-Cartan form on $H$. See [61] for details.

Let $\mathcal{G}=(Y,P,\mu)$ be an $\mathbb{S}^1$-gerbe over $X$, see Example <ref>. A connection on $\mathcal{G}$ is a connection 1-form $\theta$ on the principal $\mathbb{S}^1$-bundle $P\to Y\times_{X}Y$ which is invariant by the gerbe multiplication. That is,

\begin{equation}\label{eq:gerbeconnectioninvariance}
\mu^*(\hat{d_1}^*\theta)=(\hat{d_2}^*\theta)\otimes (\hat{d_0}^*\theta).
\end{equation}

Here $\hat{d}_i:d^*_iP\to P$ is the natural map covering the face map $d_i$ on $Y_{3}$. One easily observes that condition (<ref>) is equivalent to $\theta$ being a multiplicative 1-form on the Lie groupoid $P\rr Y$. Hence, connections on $\mathbb{S}^1$-gerbes can be regarded as 2-connections on its underlying principal 2-bundle. For more details see [35].

It is well-known that connections on a usual principal bundle are in one-to-one correspondence with splittings of its Atiyah sequence. One immediately observes that the proof can carry over verbatim in the case of a principal 2-bundle. Hence, there is a one-to-one correspondence between 2-connections on a principal 2-bundle $\p\curvearrowleft \GG\to \mathbb{X}$  and splittings
of the Atiyah sequence of Definition <ref>. By a splitting we mean a VB-groupoid morphism $h:T\mathbb{X}\to \At(\mathbb{P})$ with $\tilde{d\pi}\circ h=\id_{T\mathbb{X}}$. Such a splitting will be referred to as a multiplicative horizontal lift of $\At(\mathbb{P})$.

§.§ The curvature of a 2-connection

Our main goal now is to define the curvature of a 2-connection. For that, we start by fixing the notation in the case of ordinary principal bundles.

Let $P\curvearrowleft G \to X$ be a principal bundle with connection $\theta\in\Omega^1(P,\mathfrak{g})$. Let $\Omega(P,V)$ be the space of $V$-valued differential forms on $P$ where $V$ is a vector space. The covariant derivative of $\omega\in \Omega(P,V)$  is defined as $\nabla^{\theta}\omega:=(d\omega)^h$, the horizontal part of the usual exterior derivative. The curvature of a connection $\theta$ is the 2-form $\Omega:=\nabla^{\theta}\theta\in \Omega^2(P,\mathfrak{g})$. The curvature 2-form can be written as

\begin{equation}\label{eq:MCcurvature}
\Omega=d\theta+\frac{1}{2}[\theta,\theta].
\end{equation}

If $\rho:G\to GL(V)$ a representation, we say that $\omega\in \Omega^k(P,V)$ is basic if it is horizontal and invariant in the sense that $R^*_g\omega=\rho_{g^{-1}}\omega$ for every $g\in G$. We denote by $\Omega^k_{\rho}(P,V)$ the space of basic $k$-forms on $P$ with values in $V$. The space $\Omega^k_{\rho}(P,V)$ identifies canonically with $\Omega^k(X,P\times_GV)$, the space of $k$-forms on $X$ with values in the associated bundle $P\times_GV\to X$. The curvature of a connection is basic with respect to the adjoint representation, hence it can be seen as a 2-form on $X$ with values in the adjoint bundle $\Ad(P)$.

For basic forms $\omega\in \Omega^k_{\rho}(P,V)$ the covariant derivative has the following explicit formula:

\begin{equation}\label{eq:covariantbasic}
\nabla^{\theta}\omega=d\omega + \rho_{*}(\theta\wedge \omega).
\end{equation}

Here $\rho_{*}(\theta\wedge \omega)\in \Omega^{k+1}(P,V)$ is defined as

$$\rho_{*}(\theta\wedge \omega)(U_1,...,U_{k+1})=\sum_{\sigma \in \Sigma_{k+1}}(-1)^{\sigma}\rho_{*}(\theta(U_{\sigma(1)}))\omega(U_{\sigma(2)},...,U_{\sigma(k+1)}),$$

where $\rho_{*}:\mathfrak{g}\to \mathfrak{gl}(V)$ is the induced Lie algebra representation.

We proceed now to define the curvature of a 2-connection on a principal 2-bundle.

Let $\theta=\bar\theta+s^*\theta_0\in \Omega^1(\p,\mathbb{g})$ be a 2-connection on a principal 2-bundle $\p\curvearrowleft \GG\to \mathbb{X}$. The curvature of $\theta$ is the 2-form $\Omega:=d\theta+\frac{1}{2}[\theta,\theta]\in \Omega^2(\p,\mathbb{g})$.

It is reasonable to define the curvature of a 2-connection $\theta=\bar\theta+s^*\theta_0$ by looking at the expression (<ref>) of each of its components $\bar\theta$ and $\theta_0$. However, the 2-form $d\bar\theta+\frac{1}{2}[\bar\theta,\bar\theta]\in \Omega^2(P_1,\mathfrak{h})$ fails to be multiplicative, hence it does not provide a good notion of curvature in the setting of principal bundles over Lie groupoids.

Note that the curvature $\Omega\in \Omega^2(\p,\mathbb{g})$ as in Definition <ref> is a multiplicative 2-form. This follows immediately from the expression $\Omega=(d\theta)^h$ combined with the fact that $d\theta$ is multiplicative and the horizontal projection $TP\to H:=\Ker(\theta)$ is a VB-groupoid morphism. As a consequence, one can write the curvature as

\begin{equation}\label{eq:2curvaturecomponents}
\Omega=\bar\Omega+s^*\Omega_0,
\end{equation}

where $\bar\Omega\in\Omega^2(P_1,\mathfrak{h})$ and $\Omega_0\in\Omega^2(P_0,\mathfrak{g}_0)$. More concretely, it follows from Definition <ref> that $\Omega_0=d\theta_0+\frac{1}{2}[\theta_0,\theta_0]$ is just the curvature of $\theta_0$, and

Now we describe $\bar\Omega$  in (<ref>) in terms of the covariant derivative. For that, recall that the source map $P_1\to P_0$ is a morphism of principal bundles with connections, hence the corresponding covariant derivatives are related by $\nabla^{\theta}\circ s^*=s^*\circ \nabla^{\theta_0}$. Since $\Omega=\nabla^{\theta}\theta$ and $\theta=\bar\theta+s^*\theta_0$, one has

\begin{align*}
\Omega=&\nabla^{\theta}\bar\theta+s^*\nabla^{\theta_0}\theta_0\\
=&\nabla^{\theta}\bar\theta + s^*\Omega_0.
\end{align*}

The uniqueness of the decomposition (<ref>) implies $\bar\Omega:=\nabla^{\theta}\bar\theta$. In particular, as a consequence of Bianchi identity $\nabla^{\theta}\Omega=0$, one has the following Bianchi identity for $\bar\Omega$:

\begin{equation}\label{eq:Bianchibaromega}
\nabla^{\theta}\bar\Omega=0.
\end{equation}

Let $\Omega^2(\p,\mathbb{g})$ be the curvature of a 2-connection $\theta\in \Omega^1(\p,\mathbb{g})$. Since $\Omega=\bar\Omega+ s^*\Omega_0$ is basic with respect to $\pi:\p\curvearrowleft \GG\to \mathbb{X}$, there exists a unique multiplicative 2-form $\omega\in \Omega^2(\mathbb{X},\Ad(\p))$ with $\pi^*\omega=\Omega$. As shown in [24], a VB-groupoid valued form is completely determined by its core and side components, which in the case of $\omega\in \Omega^2(\mathbb{X},\Ad(\p))$ such a components are $\bar\omega\in\Omega^2(X_1,t^*(P_0\times_{G_0} \mathfrak{h}))$ and $\omega_0\in \Omega^2(X_0,P_0\times_{G_0}\mathfrak{g}_0)$, and

Since $\pi^*\omega=\Omega$, then $\pi^*_0\omega_0=\Omega_0$ and $\pi^*_1\bar\omega=\bar\Omega$. Also, due to Bianchi identity (<ref>) $\nabla^{\theta}\bar\Omega=0$, one has that $\nabla_0\bar\omega=0$ where $\nabla_0$ is a covariant exterior derivative defined by the induced connection on $t^*(P_0\times_{G_0} \mathfrak{h})\to X_1$.

It is useful to consider 2-connections with prescribed curvature. This is formalized by introducing the notion of a curving 2-form. Let $G_0\to H$ be the crossed module associated to the Lie 2-group $\GG$. One of the components of the crossed module is a Lie group morphism $\alpha:G_0\to \Aut(H)$ which yields a Lie group representation $\tau:G_0\to GL(\mathfrak{h})$. In particular, one can look at basic forms $\Omega_{\tau}(P_0,\mathfrak{h})$ with respect to the representation $\tau$.

Let $\theta=\bar\theta+s^*\theta_0$ be a 2-connection on a principal 2-bundle $\p\curvearrowleft\GG\to \mathbb{X}$ with curvature $\Omega=\bar\Omega + s^*\Omega_0$.  A curving for $\theta$ is a basic 2-form $B\in\Omega^2_{\tau}(P_0,\mathfrak{h})$ with:

* $\Omega_0+\partial\circ B=0$, and

* $\bar\Omega+(t^*-s^*)(B)=0$.

A connective structure on a principal 2-bundle $\p\curvearrowleft\GG\to \mathbb{X}$ is a pair $(\theta,B)$ where $\theta$ is a 2-connection on $\p$ and $B$ is a curving for $\theta$. The curvature 3-form of a connective structure $(\theta,B)$ is the 3-form $\nabla^c_{\theta_0}B\in \Omega^3(P_0,\mathfrak{h})$. Due to (<ref>), the curvature 3-form has the following expression

\begin{equation}\label{eq:curvature3form}
\nabla^c_{\theta_0}B=dB+\tau_{*}(\theta_0\wedge B),
\end{equation}

where $\tau_{*}:\mathfrak{g}_0\to \mathfrak{gl}(\mathfrak{h})$ is the Lie algebra representation induced by $\tau:G_0\to GL(\mathfrak{h})$.

The terminology is motivated by the following example.

Let $\mathcal{G}=(Y,P,\mu)$ be an $\mathbb{S}^1$-gerbe over a manifold $X$. A connective structure $(\theta,B)$ on the underlying principal 2-bundle associated to $\mathcal{G}$ is just a connection 1-form on $P\to Y\times_XY$ and $B\in \Omega^2(Y)$ such that $\Omega=-(d^*_1-d^*_0)B$, where $\Omega$ is the curvature of $\theta$. This recovers the notion of connective structure on an $\mathbb{S}^1$-gerbe, [35]. In this case, the curvature 3-form is $dB\in \Omega^3(Y)$. Since $dB$ is basic with respect to $q:Y\to X$, there exists a unique closed 3-form $\chi\in\Omega^3_{cl}(X)$ with $dB=q^*\chi$. The 3-form $\chi$ is called the 3-curvature of the connective structure $(\theta,B)$ on the gerbe $\mathcal{G}=(Y,P,\mu)$.

As observed in Example <ref>, the curvature 3-form of a connective structure on a gerbe $\mathcal{G}=(Y,P,\mu)$ is a closed 3-form on the manifold $X$, which is the base of the submersion $q:Y\to X$. Regarding $X$ as the quotient stack of of the submersion groupoid $Y\times_XY\rr Y$, the curvature 3-form can be seen as a 3-form on the stack $X=[Y/Y\times_XY]$. For a general principal 2-bundle $\p\curvearrowleft \GG\to \mathbb{X}$, the curvature 3-form of a connective structure is an element in $\Omega^3(P_0,\mathfrak{h})$. However, one can see the curvature 3-form as a suitable multiplicative 3-form on $\mathbb{X}$ with values in the VB-groupoid $\Ad(\p)$, which in the case of a regular groupoid $\mathbb{X}$ yields a 3-form on the quotient stack $[X_0/X_1]$ with values in  $\Ad(\p)$. We will explain this below.

As shown in [24], $k$-forms on a Lie groupoid $\mathbb{Y}$ with values in a $VB$-groupoid $\mathbb{E}\to \mathbb{Y}$ determine a cochain complex $\delta_{\mathbb{E}}:C^{\bullet,k}(\mathbb{Y},\mathbb{E})\to C^{\bullet,k}(\mathbb{Y},\mathbb{E})$ whose 1-cocycles are precisely multiplicative $k$-forms. If $C\to Y_0$ is the core bundle of $\mathbb{E}\to \mathbb{Y}$, then $C^{0,k}(\mathbb{Y},\mathbb{E}):=\Omega^k(Y_0,C)$. A degree zero cocycle is called $\mathbb{Y}$-invariant.

In the case of a principal 2-bundle $\p\curvearrowleft \GG \to \mathbb{X}$ and the trivial VB-groupoid $\p\times \mathbb{g}$, one has the complex $\delta_{\p\times\mathbb{g}}:C^{p,2}(\p,\mathbb{g})\to C^{p+1,2}(\p,\mathbb{g})$ of $\mathbb{g}$-valued 2-forms on $\p$ whose 1-cocycles are $\mathbb{g}$-valued multiplicative $2$-forms on $\p$.  A connective structure $\tilde\theta:=(\theta,B)$ on $\p$ allows us to write the curvature

$$\Omega=-((t^*B-s^*B+s^*(\partial\circ B))=\delta_{\p\times \mathbb{g}} B$$

as a multiplicatively exact 2-form with values in $\mathbb{g}$. The groupoid morphism $\pi:\p\to \mathbb{X}$ defines a cochain map $\pi^*:C^{\bullet,2}(\mathbb{X},\Ad(\p))\to C^{\bullet,2}(\p,\mathbb{g})$, hence the multiplicative 2-form $\omega\in\Omega^2(\mathbb{X},\Ad(\p))$ described in Remark <ref> is also multiplicatively exact. Indeed,  $\omega=\delta_{\Ad(\p)}B_0$ where $B_0\in \Omega^2(X_0,P_0\times_{G_0}\mathfrak{h})$ is the unique 2-form with $\pi^*_0B_0=B$.

Let $\nabla^c_{0}:\Omega(X_0,P_0\times_{G_0}\mathfrak{h})\to \Omega(X_0,P_0\times_{G_0}\mathfrak{h})$ the covariant exterior derivative defined by the induced connection on $P_0\times_{G_0}\mathfrak{h}\to X_0$. We claim that $\nabla^c_{0}B_0\in \Omega^3(X_0,P_0\times_{G_0}\mathfrak{h})$ is $\mathbb{X}$-invariant, that is, it is a zero cocycle in the complex $C^{\bullet,3}(\mathbb{X},\Ad(\p))$. To see this, it suffices to check that $\pi^*(\delta_{\Ad(\p)}\nabla^c_{0}B_0)=0$ for the cochain map $\pi^*:C^{\bullet,3}(\mathbb{X},\Ad(\p))\to C^{\bullet,3}(\p,\mathbb{g})$. Indeed,

\begin{align*}
\pi^*(\delta_{\Ad(\p)}\nabla^c_{0}B_0)=&\delta_{\p\times \mathbb{g}}(\pi^*_0\nabla^c_{0}B_0)\\
=& \delta_{\p\times \mathbb{g}}(\nabla^c_{\theta_{0}}\pi^*_0B_0)\\
=& \delta_{\p\times \mathbb{g}}(\nabla^c_{\theta_{0}} B)
\end{align*}

where $\nabla^c_{\theta_{0}}$ is the induced covariant exterior derivative on $P_0\times \mathfrak{h}$. But $\delta_{\p\times \mathbb{g}}(\nabla^c_{\theta_{0}} B)=\nabla (\delta_{\p\times \mathbb{g}}B)=0$ because of Bianchi identity and $\delta_{\p\times \mathbb{g}}B=\Omega$. The corresponding cohomology class $[\nabla^c_{0}B_0]\in H^{0,3}(\mathbb{X},\Ad(\p))$ is an invariant of principal 2-bundles with connective structures.

[3-curvature of a gerbe]

Let $(\theta,B)$ a connective structure on the underlying principal 2-bundle associated to an $\mathbb{S}^1$-gerbe $\mathcal{G}$ over $X$. In this case, the covariant exterior derivative $\nabla^c_0B=dB$ is the ordinary exterior derivative and being a cocycle $\delta_{\Ad(\p)}\nabla^c_{0}B=0$ is equivalent to saying that $dB\in \Omega^3(Y)$ is basic with respect to the surjective submersion $q:Y\to X$. Hence there exists a unique closed 3-form $\chi\in \Omega^3(X)$ with $q^*\chi=dB$. The 3-form $\chi$ is called the 3-curvature of the connective structure $(\theta,B)$ on $\mathcal{G}$.

§.§.§ Flatness

A connective structure $(\theta,B)$ on $\p\curvearrowleft \GG\to \mathbb{X}$ is called weakly flat if its curvature 3-form vanishes, i.e. $\nabla^c_{\theta_0}B=0$, where $\nabla^{c}_{\theta_0}:\Omega(P_0,\mathfrak{h})\to \Omega(P_0,\mathfrak{h})$ is the induced a covariant derivative. Since $B\in \Omega^2(P_0,\mathfrak{h})$ is basic, it follows from (<ref>) that being weakly flat is equivalent to

\begin{equation}\label{eq:flatcurving}
dB+\tau_{*}(\theta_0\wedge B)=0,
\end{equation}

where $\tau_{*}:\mathfrak{g}_0\to \mathfrak{gl}(\mathfrak{h})$ is the Lie algebra representation induced by $\tau:G_0\to GL(\mathfrak{h})$. In the special case $B=0$, we say that $\theta$ is a flat 2-connection on $\p$. In this case, conditions (1) and (2) in Definition <ref> imply that the curvature $\Omega=\bar{\Omega}+s^*\Omega_0$ vanishes.

In what follows, we give a geometric interpretation of these notions of flatness in terms of horizontal lifts. For that, let $\p\curvearrowleft \GG\to \mathbb{X}$ be a principal 2-bundle and consider the corresponding Atiyah sequence

As observed in (<ref>), a multiplicative horizontal lift $h:T\mathbb{X}\to \At(\mathbb{\p})$ defines a splitting of the Atiyah extension

\begin{equation}\label{eq:atiyahextension}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

We spell this out in terms of the corresponding crossed modules of multiplicative sections. Since by definition $h:T\mathbb{X}\to \At(\mathbb{\p})$ is a VB-groupoid map, it induces a vector bundle morphism $h_c:A_{\mathbb{X}}\to A_{\p}/G_0$ between the core bundles as well as a map between multiplicative sections $h_m:\mathfrak{X}_{mult}(\mathbb{X})\to \Gamma_{mult}(\At(\mathbb{P})); (\xi,u)\mapsto(h\circ \xi,h_0\circ u),$ where $h_0:TX_0\to \At(P_0)$ is the map induced by $h$ on the units. The fact that $h:T\mathbb{X}\to \At(\mathbb{\p})$ is a VB-groupoid morphism implies that it induces a cochain map between the underlying complexes of multiplicative sections, i.e. $\delta\circ h_c=h_m\circ\delta$.

Using the explicit description of the crossed modules of multiplicative sections as in Examples <ref> and <ref>, a multiplicative horizontal lift yields to a diagram

\begin{equation}\label{eq:splittingAtiyah2extension}
\begin{tikzcd}
0\arrow[r]&C^{\infty}(P_0,\mathfrak{h})^{G_0}\arrow[d]\arrow[r,"\iota_c"]&\Gamma(A_{\mathbb{P}}/G_0)\arrow[d]\arrow[r,"\pi_c"]& \Gamma(A_{\mathbb{X}})\arrow[l,bend right,dashed,"h_c"']\arrow[d]\arrow[r]&0\\
0\arrow[r]& \mathrm{Hom}_{gpd}(\p,\mathbb{g})^{\GG}\arrow[r,"\iota_m"]&  \mathfrak{X}_{mult}(\mathbb{P})^{\GG}\arrow[r,"\pi_m"]&\mathfrak{X}_{mult}(\mathbb{X})\arrow[l,bend right,dashed,"h_m"']\arrow[
\end{tikzcd}
\end{equation}

where $\pi_c:\Gamma(A_{\p}/G_0)\to \Gamma(A_{\mathbb{X}})$ and $\pi_m:\mathfrak{X}_{mult}(\p)^{\GG}\to \mathfrak{X}_{mult}(\mathbb{X})$ are maps induced by the VB-groupoid morphism $\overline{T\pi}:\At(\p)\to T\mathbb{X}$ between core and multiplicative sections, respectively.

The induced map $h:\mathbb{L}(T\mathbb{X})\to \mathbb{L}(\At(\p))$ is not necessarily a Lie 2-algebra morphism. Being a weak Lie 2-algebra morphism amounts to the existence of a map $\Omega_m:\wedge^2\mathfrak{X}_m(\mathbb{X})\to C^{\infty}(P_0,\mathfrak{h})^{G_0}$ such that for every $\xi_1,\xi_2,\xi_3\in \mathfrak{X}_{mult}(\mathbb{X})$ and $a\in\Gamma(A_{\mathbb{X}})$, the following conditions are satisfied:

WF1) $h_m[\xi_1,\xi_2]-[h_m(\xi_1),h_m(\xi_2)]=\delta\Omega_m(\xi_1,\xi_2)$,

WF2) $h_{c}(D_{\xi_1} a)-D_{h_m(\xi_1)}(h_c(a))=\Omega_m(\xi_1,\delta(a))$,

WF3) $\big{(}\Omega_m([\xi_1,\xi_2],\xi_3)-D_{h_m(\xi_1)}(\Omega_m(\xi_2,\xi_

where $c.p.$ in the last condition stands for cyclic permutations of $\xi_1,\xi_2,\xi_3$. Here $D_{\xi}(a):=[\xi,a^r]|_{X_0}$ denotes the action of $\xi\in\mathfrak{X}_{mult}(\mathbb{X})$
by derivations of $\Gamma(A_{\mathbb{X}})$ defined by the crossed module of multiplicative vector fields, see (<ref>).

The next result establishes a relation between connective structures which are weakly flat and weak morphisms of Lie 2-algebras.

Let $(\theta, B)$ be a connective structure on a principal 2-bundle $\p\curvearrowleft \GG\to \mathbb{X}$. If $(\theta,B)$ is weakly flat (resp. flat), then the multiplicative horizontal lift $h:\mathbb{L}(T\mathbb{X})\to \mathbb{L}(\At(\mathbb{\p}))$ defined by the 2-connection $\theta$ is a weak (resp. strict) morphism of 2-term $L_{\infty}$-algebras.

Let $(\xi_1,u_1),(\xi_2,u_2)$ be multiplicative vector fields on $\mathbb{X}$ and $u^h_1,u^h_2\in \mathfrak{X}(P_0)$ the corresponding horizontal lifts with respect to the connection $\theta_0$. Let us consider

Since the curving $B$ is invariant, one has a well defined map $\Omega_m:\bigwedge^2\mathfrak{X}_{mult}(\mathbb{X})\to C^{\infty}(P_0,\mathfrak{h})^{G_0}$ and we will show that $\Omega_m$ satisfies conditions WF1), WF2) and WF3). As observed in Example <ref>, the Lie 2-algebra of multiplicative sections of the adjoint 2-bundle is given by  the crossed module

where $D$ is defined by (<ref>) and $\delta(f)=(t^*f-s^*f,s^*(\partial\circ f))$. Recall that $B$ being a weakly flat curving amounts to the next three equations:

\begin{align}
\Omega_0+\partial \circ B=&0\label{eq:curving1};\\
\bar\Omega+(t^*-s^*)B=&0\label{eq:curving2};\\
dB+\tau_{*}(\theta_0\wedge B)=&0,\label{eq:curvingflat}
\end{align}

where $\Omega=\bar\Omega+s^*\Omega_0$ is the curvature of $\theta$.

For every $\xi_1,\xi_2\in\mathfrak{X}_{mult}(\mathbb{X})$ covering $u_1,u_2\in\mathfrak{X}(X_0)$, respectively, one has

which combined with equations (<ref>) and (<ref>) and the fact that $\xi^h_i$ is both $t$-related and $s$-related with $u^h_i, i=1,2$, imply

\begin{align*}
=&\delta(\Omega_m(\xi_1,\xi_2))\in \mathrm{Hom}_{gpd}(\p,\mathbb{g})^{\GG},
\end{align*}

showing   WF1). To see WF2), let $a \in \Gamma(A_{\mathbb{X}})$ and $\xi\in\mathfrak{X}_{mult}(\mathbb{X})$ and note that

\begin{align}\label{eq:eq1thmflatconnectivestructure}
h_c(D_{\xi}a)-D_{h_m(\xi)}(h_c(a))=&h\circ [\xi,a^r]|_{X_0}-[\xi^h,h_c(a)^r]|_{P_0}\\ \nonumber
=&([\xi,a^r]^h-[\xi^h,h_c(a)^r])|_{P_0}\\ \nonumber
=&\Omega(\xi^h,h_c(a)^r)|_{P_0}\\ \nonumber
=&\bar\Omega(\xi^h,h_c(a)^r)+s^*\Omega_0(\xi^h,h_c(a)^r).\\ \nonumber
\end{align}

On the right hand side of the previous identity, one observes that the second term vanishes once $h_c(a)^r$ is right-invariant, hence $s$-related to zero.  Regarding the first term, due to (<ref>), one has

\begin{align}\label{eq:eq2thmflatconnectivestructure}
\bar\Omega(\xi^h,h_c(a)^r)|_{X_0}=&-(t^*-s^*)(B)(\xi^h,h_c(a)^r)|_{X_0}\\ \nonumber
\end{align}

Since $h_m\circ \delta=\delta\circ h_c$, the multiplicative vector field  $\delta(a)^h=\delta(h_c(a))$ covers $\rho(h_c(a))$, hence $\Omega_m(\xi,\delta(a))=-B(u^h,\rho(h_c(a)))$. Combining this with (<ref>) and (<ref>), one concludes

which is WF2).

Finally, we show WF3). For $i=1,2,3$ let $\xi_i\in \mathfrak{X}_{mult}(\mathbb{X})$ be a multiplicative vector field covering $u_i\in\mathfrak{X}(X_0)$. It follows from equation (<ref>)  that

\begin{align}\label{eq:eq3thmflatconnectivestructure}
0=&(dB+\tau_{*}(\theta_0\wedge B))(u^h_1,u^h_2,u^h_3)\\ \nonumber
=&\big{(}u^h_1(B(u^h_2,u^h_3))-B([u^h_1,u^h_2],u^h_3)+c.p.\big{)},\\ \nonumber
\end{align}

since $\alpha_*(\theta_0\wedge B)(u^h_1,u^h_2,u^h_3)=0$ once $\theta_0$ vanishes on horizontal vector fields. Also,   $B([u^h_1,u^h_2],u^h_3)=B([u_1,u_2]^h,u^h_3)$ since $B$ is a horizontal form.  Replacing this in (<ref>) yields

\begin{align}
=&\big{(}\Omega_m([\xi_1,\xi_2],\xi_3)-D_{\xi^h_1}(\Omega_m(\xi_2,\xi_3))\big{)}+ c.p..
\end{align}

This shows WF3) and therefore $(h_c,h_m,\Omega_m)$ is a weak morphism of Lie 2-algebras, as claimed.

Let $\mathcal{G}=(Y,P,\mu)$ be an $\mathbb{S}^1$-gerbe over a manifold $X$. A weakly flat connective structure $(\theta,B)$ on the underlying principal 2-bundle associated to $\mathcal{G}$ is just a connection 1-form on $P\to Y\times_XY$ with curving $B\in \Omega^2_{cl}(Y)$  a closed 2-form. This says that the 3-curvature $q^*\chi:=dB \in\Omega^3(X)$ of the connective structure vanishes. It follows from Theorem <ref> that the corresponding horizontal lift is a weak morphism of Lie 2-algebras. This recovers a result by Krepski and Vaughan [35] on $\mathbb{S}^1$-gerbes with flat 3-curvature.

§.§ Chern-Weil-Lecomte map

In what follows, we will state and prove the main result of this section, namely the construction of the Chern-Weil morphism for principal 2-bundles with 2-connections. This relies on the general set up of Theorem <ref> which gives rise to characteristic classes of extensions of $L_{\infty}$-algebras together with a representation up to homotopy.

Let $\p\curvearrowleft \GG\to \mathbb{X}$ be a principal 2-bundle with associated Atiyah extension:

\begin{equation}\label{eq:atiyahextension2}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

Consider the multiplicative horizontal lift $h:T\mathbb{X}\to \At(\p)$ defined by a 2-connection and look at the associated splitting $h:\mathbb{L}(T\mathbb{X})\to \mathbb{L}(\At(\p))$ of (<ref>). In order to apply Theorem <ref> to the extension <ref>, it remains to define a representation up to homotopy of $\mathbb{L}(T\mathbb{X})$ on a cochain complex. We will see that such a representation up to homotopy can be defined in a canonical way. For that, consider the 2-vector space $\mathbb{F}(\mathbb{X})$ of real valued multiplicative functions on $\mathbb{X}$ defined by the 2-term cochain complex $\delta:C^{\infty}(X_0)\to C^{\infty}_{mult}(\mathbb{X}); f\mapsto t^*f-s^*f$, see Example <ref>. As observed in Example <ref>, the Lie 2-algebra of multiplicative vector fields $\mathbb{L}(T\mathbb{X})$ is defined by the crossed module $\Gamma(A_{\mathbb{X}})\to \mathfrak{X}_{mult}(\mathbb{X}); a\mapsto a^r-a^l$, where $A_{\mathbb{X}}$ is the Lie algebroid of $\mathbb{X}$. Elements in $\mathbb{L}(T\mathbb{X})$ are pairs $(a,(\xi,u))$ with $a\in\Gamma(A_{\mathbb{X}})$ and $(\xi,u)$ a multiplicative vector field on $\mathbb{X}$. Similarly, elements in $\mathbb{F(X)}$ are pairs $(f,F)$ where $f\in C^{\infty}(X_0)$ and $F\in C^{\infty}_{mult}(\mathbb{X})$.

There is a canonical degree zero map $\rho:\mathbb{L}(T\mathbb{X})\otimes \mathbb{F(X)}\to \mathbb{F(X)}$ defined by

\begin{equation}\label{eq:canonicalruthderivations}
\rho_{\mathbb{X}}((a,(\xi,u))\otimes (f,F)):=(\Lie_uf+(\Lie_{a^r}F)|_{X_0},\Lie_{\xi}F)
\end{equation}

It is possible to check that (<ref>) is a representation up to homotopy of $\mathbb{L}(T\mathbb{X})$ on the cochain complex $C_m(\mathbb{X})$ of multiplicative functions. More details will appear in [29].

The representation up to homotopy (<ref>) together with the splitting $h:\mathbb{L}(T\mathbb{X})\to \mathbb{L}(\At(\p))$ of (<ref>), allows to apply Theorem <ref>, yielding the following result.

Let $\p\curvearrowleft\GG\to \mathbb{X}$ be a principal 2-bundle equipped with a 2-connection. Let $h:\mathbb{L}(T\mathbb{X})\to \mathbb{L}(\At(\p))$ the induced splitting of the Atiyah extension of Lie 2-algebras <ref>. For every $k\geq 0$ there exists a natural map

\begin{align}\label{eq:CWL2bundles}
cw:\underline{\Hom}^{\bullet}(\wedge^k\mathbb{L}(\Ad(\p))&[1],\mathbb{F(X)})^{\mathbb{L}(\mathrm{At}(\mathbb{P}))}\to H^{\bullet+k}(\mathbb{L}(T\mathbb{X}),\mathbb{F(X)})\\ \nonumber
&f\mapsto [f\circ K^k_h],
\end{align}

where $K^k_h=K_h\wedge_{\id}\cdots \wedge_{\id}K_h$ and $K_h\in C^1(\mathbb{L}(T\mathbb{X}),\mathbb{L}(\Ad(\p)))[1])$ is the curvature of $h$. The map $cw$ does not depend of the 2-connection.

One observes that since <ref> is an extension of strict Lie 2-algebras and the horizontal lift $h:T\mathbb{X}\to \mathrm{At}(\p)$ is a VB-groupoid morphism, then Remark <ref> implies that the curvature of the induced splitting $h:\mathbb{L}(T\mathbb{X})\to \mathbb{L}(\mathrm{At}(\p))$ has only two possibly non-zero components, namely:

* $K_h:\Gamma(A_{\mathbb{X}})\times \mathfrak{X}_{mult}(\mathbb{X})\to C^{\infty}(P_0,\mathfrak{h})^{G_0}$, with

$$K_h(a,\xi)= h_{c}([\xi,a^r]|_{X_0})-[h_m(\xi),(h_c(a))^r]|_{P_0}.$$

* $K_h:\wedge^2\mathfrak{X}_{mult}(\mathbb{X})\to \Hom_{gpd}(\p,\mathbb{g})^{\GG}$, where

$$K_h(\xi_1.\xi_2)= \Omega(\xi_1,\xi_2),$$

where $\Omega=\bar{\Omega}+s^*\Omega_0$ is the curvature of the 2-connection associated to $h:T\mathbb{X}\to \At(\p)$.

Note that in this case, the component $K_h:\wedge^3\mathfrak{g}_0\to \mathfrak{n}_{-1}$ (see (2) in <ref>) is always zero since all the brackets of order 3 are zero in a strict Lie 2-algebra.

Suppose that $B\in \Omega^2_{\tau}(P_0,\mathfrak{h})$ is a curving for a 2-connection $\theta\in \Omega^1(\p,\mathbb{\mathfrak{g}})$. Then, the second component of the curvature $K_h:\wedge^2\mathfrak{X}_{mult}(\mathbb{X})\to \Hom_{gpd}(\p,\mathbb{g})^{\GG}$ is given by

$$K_h=-(t^*B-s^*B)-\partial\circ B.$$

The curvature 3-form $\nabla^c_{\theta_0}B\in \Omega^3(P_0,\mathfrak{h})$ is not necessarily zero unless the horizontal lift $h:\mathbb{L}(T\mathbb{X})\to \At(\mathbb{P})$ is a weak $L_{\infty}$-algebra morphism (see Theorem <ref>). In particular, since the component $K_h:\wedge^3\mathfrak{g}_0\to \mathfrak{n}_{-1}$ always vanishes, one cannot recover the curvature 3-form as a component of the curvature $K_h\in C^1(T\mathbb{X},\mathbb{F}(\mathbb{X}))$.

Let us discuss some examples.

[Ordinary principal bundles]

Let $P\to X$ be an ordinary principal G-bundle viewed as a 2-bundle as in Example <ref>. In this case, the map (<ref>) recovers the ordinary Chern-Weil map of a principal bundle with connection. See [39].

Let $\mathcal{G}=(Y,P,\mu)$ be an $\mathbb{S}^1$-gerbe over a manifold $X$ equipped with a connective structure $(\theta,B)$. Let $H\in \Omega^3_{cl}(X)$ be the corresponding curvature 3-form. One would expect to recover the cohomology class $[H]\in H^3_{dR}(X)$ from the Chern-Weil-Lecomte map defined by the induced connection on the principal 2-bundle $\mathbb{P}\to Y\times_XY\rr Y$ underlying the gerbe $\mathcal{G}$.  However, as observed in Remark <ref>, this is not possible because each of the Lie 2-algebras involved in the Atiyah sequence is strict. In spite of this, one can still describe the curvature 3-form via the Chern-Weil-Lecomte map associated to a suitable exact Courant algebroid. Indeed, as shown in [30, 31], the choice of a connection and curving on an $\mathbb{S}^1$-gerbe over $X$ with curvature 3-form $H\in\Omega^3_{cl}(X)$ defines an exact Courant algebroid $E\to X$ with a splitting whose Ševera class is $[H]\in H^3_{dR}(X)$. The class $[H]$ can be recovered as a characteristic class via Chern-Weil-Lecomte as in Example <ref>. Alternatively, as shown in [23, 35], every $\mathbb{S}^1$-gerbe over $X$ equipped with a connective structure and 3-curvature $H\in \Omega^3_{cl}(X)$ has an associated Lie 2-algebra of infinitesimal symmetries which is quasi-isomorphic to the Poisson-Lie 2-algebra of $(X,H)$. Hence the class $[H]$ can also be described as in Example <ref>.

§.§.§ Morita invariance

We finish this section by describing how does the Chern-Weil-Lecomte map behave under pullbacks by Morita maps. For that, we briefly recall some facts about projectable sections of VB-groupoids, see [49] for more details. Let $\Phi:E\to E'$ be a vector bundle morphism covering $\phi:M\to M'$. Consider the set $\Gamma(E)^{\Phi}$ of $\Phi$-projectable sections, that is, sections $e\in\Gamma(E)$ for which there exists $e'\in\Gamma(E')$ with $\Phi\circ e=e'\circ \phi$. The section $e'$ is not necessarily unique, but uniqueness holds whenever $\phi:M\to M'$ is surjective. In this case, there are maps

\begin{equation}\label{eq:projectablesections}
\begin{tikzcd}
\Gamma(\mathbb{E}) &\arrow[l,hook',"\iota"']\Gamma(\mathbb{E})^{\Phi}\arrow[r,"\Phi_*"]&\Gamma(\mathbb{E}'),
\end{tikzcd}
\end{equation}

where $\Phi_{*}:\Gamma(E)^{\Phi}\to \Gamma(E'); e\mapsto e'$ and $\Gamma(E)^{\Phi}\hookrightarrow \Gamma(E)$ is the canonical inclusion.

Let $\Phi:\mathbb{E}\to \mathbb{E}'$ be a VB-groupoid morphism covering $\phi:\mathbb{X}\to \mathbb{X}'$ and inducing $\Phi_c:C\to C'$ between the core bundles.  The complex of multiplicative projectable sections $C^{\bullet}_{mult}(\mathbb{E})^{\Phi}$ is the subcomplex of $C^{\bullet}_{mult}(\mathbb{E})$ ((<ref>)) given by

$$\delta:\Gamma(C)^{\Phi_c}\to \Gamma_{mult}(\mathbb{E})^{\Phi}; c\mapsto c^r-c^l.$$

If $\Phi:\mathbb{E}\to \mathbb{E}'$ is surjective on objects, then diagram (<ref>) can be upgraded to the level of complexes of multiplicative sections, yielding

\begin{equation}\label{eq:multiplicativeprojectablesections}
\begin{tikzcd}
C^{\bullet}_{mult}(\mathbb{E}) &\arrow[l,hook',"\iota"']C^{\bullet}_{mult}(\mathbb{E})^{\Phi}\arrow[r,"\Phi_*"]&C^{\bullet}_{mult}(\mathbb{E}').
\end{tikzcd}
\end{equation}

It was proven in [49] that if $\Phi:\mathbb{E}\to \mathbb{E}'$ is a VB-Morita map which is also surjective on objects, then the maps in (<ref>) are quasi-isomorphisms. Additionally, if $\mathbb{E},\mathbb{E}'$ are LA-groupoids and $\Phi:\mathbb{E}\to \mathbb{E}'$ is an LA-groupoid morphism covering $\phi:\mathbb{X}\to \mathbb{X}'$, then $\Gamma(C)^{\Phi_c}\to \Gamma_{mult}(\mathbb{E})^{\Phi}$ is actually a crossed sub-module of the crossed module of multiplicative sections. In particular, there is a Lie 2-subalgebra $\mathbb{L}(\mathbb{E})^{\Phi}\subseteq \mathbb{L}(\mathbb{E})$ of $\Phi$-projectable sections. Moreover, if $\Phi:\mathbb{E}\to \mathbb{E}'$ is an LA-Morita map which is surjective on objects, then the maps in (<ref>) are quasi-isomorphisms of crossed modules, see Theorem 7.3. in [49]. In particular, one has quasi-isomorphisms of Lie 2-algebras

\begin{equation}\label{eq:quasiLie2algebras}
\begin{tikzcd}
\mathbb{L}(\mathbb{E}) &\arrow[l,hook',"\iota"']\mathbb{L}(\mathbb{E})^{\Phi}\arrow[r,"\phi_*"]&\mathbb{L}(\mathbb{E}').
\end{tikzcd}
\end{equation}

As a consequence, an LA-Morita equivalence $\begin{tikzcd}
\mathbb{E} &\arrow[l,"\Phi"']\tilde{\mathbb{E}}\arrow[r,"\Psi"]&\mathbb{E}'
\end{tikzcd}$ induces a zig-zag of Lie 2-algebra quasi-isomorphisms

\begin{equation}
\begin{tikzcd}
\mathbb{L}(\mathbb{E})&\mathbb{L}(\tilde{\mathbb{E}})^{\Phi}\arrow[l,"\Phi_*"']\arrow[r,hook]&\mathbb{L}(\tilde{\mathbb{E}})&\mathbb{L}(\tilde{\mathbb{E}})^{\Psi}\arrow[l,hook']\arrow[r,"\Psi_*"]&\mathbb{L}(\mathbb{E}').
\end{tikzcd}
\end{equation}

In what follows, we specialize the previous discussion in the case of the VB-groupoid map $T\phi:T\mathbb{X}\to T\mathbb{X}'$ associated with a Lie groupoid morphism  $\phi:\mathbb{X}\to \mathbb{X}'$. Our main goal is to show that the cohomology $H(\mathbb{L}(T\mathbb{X}),\mathbb{F}(\mathbb{X}))$ associated with the ruth $\rho_{\mathbb{X}}:\mathbb{L}(\mathbb{X})\otimes \mathbb{F}(\mathbb{X})\to \mathbb{F}(\mathbb{X})$ as in (<ref>), is invariant by Morita equivalence. This follows from the invariance of the $L_{\infty}$-cohomology in the derived category of ruths, see Corollary <ref>.

Indeed, let $\phi:\mathbb{X}\to \mathbb{X}'$ be a Morita map which is surjective on objects. Then, the LA-Morita map $T\phi:T\mathbb{X}\to T\mathbb{X}'$ is surjective on objects, hence there are quasi-isomorphisms of Lie 2-algebras

\begin{equation*}
\begin{tikzcd}
\mathbb{L}(T\mathbb{X}) &\arrow[l,hook',"\iota"']\mathbb{L}(T\mathbb{X})^{T\phi}\arrow[r,"(T\phi)_*"]&\mathbb{L}(T\mathbb{X}').
\end{tikzcd}
\end{equation*}

The next result is proven in [29] using the language of differential graded modules.

Let $\mathbb{X}, \mathbb{X}'$ be Morita equivalent Lie groupoids. Then the representations up to homotopy $(\mathbb{F(X)},\rho_{\mathbb{X}})$ and $(\mathbb{F(X')},\rho_{\mathbb{X}'})$ are quasi-isomorphic.

We actually need the following quasi-isomorphism. First, it is well-known that the pullback of functions defines a quasi-isomorphism of complexes $\phi^*:\mathbb{F}(\mathbb{X}')\to \mathbb{F}(\mathbb{X})$. Also, one can see that the map $((T\phi)_{*},\phi^*):(\mathbb{F}(\mathbb{X}),\iota^*\rho_{\mathbb{X}})\to (\mathbb{F}(\mathbb{X}'),\rho_{\mathbb{X}'})$ is a quasi-isomorphism of ruths, more details will appear in [29]. As a consequence of Corollary <ref> one has

\begin{equation}\label{eq:Moritainvariancecohomologyfunctions}
H(\mathbb{L}(\mathbb{X}),\mathbb{F}(\mathbb{X}))\cong H(\mathbb{L}(\mathbb{X}'),\mathbb{F}(\mathbb{X}')).
\end{equation}

The following result establishes the Morita invariance of the $L_{\infty}$-cohomology of the Lie 2-algebra of multiplicative vector fields on a Lie groupoid.

Let $\mathbb{X},\mathbb{X}'$ be Morita equivalent Lie groupoids. Then, there is an isomorphism

$$H(\mathbb{L}(T\mathbb{X}),\mathbb{F}(\mathbb{X}))\cong H(\mathbb{L}(T\mathbb{X}'),\mathbb{F}(\mathbb{X}')).$$

Let us consider Morita maps $\begin{tikzcd}\mathbb{X} &\arrow[l,"\Phi"']\tilde{\mathbb{Y}}\arrow[r,"\Psi"]&\mathbb{X}'\end{tikzcd}$ which are surjective on objects. This induces an LA-Morita equivalence $\begin{tikzcd}T\mathbb{X} &\arrow[l,"T\Phi"']T\tilde{\mathbb{Y}}\arrow[r,"T\Psi"]&T\mathbb{X}'\end{tikzcd}$ which is surjective on objects as well. The argument used to conclude (<ref>) can be applied to each map of this Morita equivalence, giving rise to isomorphisms:

$$H(\mathbb{L}(T\mathbb{X}),\mathbb{F}(\mathbb{X}))\cong H(\mathbb{L}(T\tilde{\mathbb{Y}}),\mathbb{F}(\tilde{\mathbb{Y}}))\cong H(\mathbb{L}(T\mathbb{X}'),\mathbb{F}(\mathbb{X}')),$$

which finishes the proof.

The $L_{\infty}$-cohomology $H(\mathbb{L}(T\mathbb{X}),\mathbb{F}(\mathbb{X}))$ is the target space of the Chern-Weil-Lecomte (<ref>) associated with a principal 2-bundle over $\mathbb{X}$. Due to Theorem <ref>, one is naturally led to study the behavior of the Chern-Weil-Lecomte map under pullbacks by Morita maps, which is explained below.

Let $\phi:\mathbb{X}\to \mathbb{X}'$ be a Lie groupoid morphism which is surjective on objects and let $\p\curvearrowleft \GG \to \mathbb{X}'$ be a principal 2-bundle. The pullback 2-bundle $(\phi^*\p)\curvearrowleft \GG \to \mathbb{X}$ is given by $\phi^*\p=(\phi^*_1P_1\rr \phi^*_0P_0)$. The natural map $\Phi:\phi^*\p\to \p$ is both a Lie groupoid morphism and a bundle map covering $\phi:\mathbb{X}\to \mathbb{X}'$. In other words, the map $\Phi:\phi^*\p\to \p$ is a principal 2-bundle morphism over $\phi:\mathbb{X}\to \mathbb{X}'$. Note that if $\phi:\mathbb{X}\to \mathbb{X}'$ is a Morita map, so is $\Phi:\phi^*\p\to \p$.

We proceed now to relate the Atiyah extensions $\mathbb{At}(\p)$ and $\mathbb{At}(\phi^{*}\p)$ of $\p$ and $\phi^{*}\p$, respectively. The Atiyah LA-groupoid of $\phi^*\p \to \mathbb{X}$ is given by the pullback Lie algebroid $\phi^{!}\At(\p)\to \mathbb{X}$, which turns out to be an LA-groupoid since $\phi:\mathbb{X}\to \mathbb{X}'$ is a Lie groupoid morphism. Similarly, the adjoint LA-groupoid of $\phi^*\p \to \mathbb{X}$ is $\Ad(\phi^*\p)=\phi^*\p\to \mathbb{X}$ the usual pullback of VB-groupoids. The principal 2-bundle map $\Phi:\phi^{!}\p\to \p$ induces natural LA-groupoid morphisms

$$\At(\Phi):\phi^{!}\At(\p)\to \At(\p); \quad \Ad(\Phi):\phi^*\Ad(\p)\to \Ad(\p),$$

which are surjective on objects. Hence, it follows from (<ref>) that there are morphisms of Lie 2-algebras

\begin{equation*}
\begin{tikzcd}
\mathbb{L}(\phi^{*}\Ad(\p)) &\arrow[l,hook',""']\mathbb{L}(\phi^{*}\Ad(\p))^{\Ad(\Phi)}\arrow[r,"\Ad(\Phi)_*"]&\mathbb{L}(\Ad(\p)),
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\mathbb{L}(\phi^{!}\At(\p)) &\arrow[l,hook',""']\mathbb{L}(\phi^{!}\At(\p))^{\At(\Phi)}\arrow[r,"\At(\Phi)_*"]&\mathbb{L}(\At(\p)),
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\mathbb{L}(T\mathbb{X}) &\arrow[l,hook',""']\mathbb{L}(T\mathbb{X})^{T\phi}\arrow[r,"(T\phi)_*"]&\mathbb{L}(T\mathbb{X}'),
\end{tikzcd}
\end{equation*}

which determine morphisms of Lie 2-algebra extensions

\begin{equation}\label{eq:Moritaequivalenceextensions}
\begin{tikzcd}
\mathbb{At}(\phi^*\p) &\arrow[l,hook',""']\mathbb{At}(\phi^{*}\p)^{\mathbb{At}(\Phi)}\arrow[r,"\mathbb{At}(\Phi)_*"]&\mathbb{At}(\p),
\end{tikzcd}
\end{equation}

where $\mathbb{At}(\phi^{*}\p)^{\mathbb{At}(\Phi)}$ is the extension of Lie 2-algebras

$$\mathbb{L}(\phi^{*}\Ad(\p))^{\Ad(\Phi)} \to \mathbb{L}(\phi^{!}\At(\p))^{\At(\Phi)} \to \mathbb{L}(T\mathbb{X})^{T\phi}$$

defined by projectable sections.

One can see that  $\mathbb{At}(\phi^{*}\p)^{\mathbb{At}(\Phi)}$ is the pullback extension of $\mathbb{At}(\p)$ by the Lie 2-algebra morphism $(T\phi)_{*}:\mathbb{L}(T\mathbb{X})^{T\phi}\to \mathbb{L}(T\mathbb{X}')$. In particular, one has $\mathbb{L}(\phi^{*}\Ad(\p))^{\Ad(\Phi)}\cong \mathbb{L}(\Ad(\p))$.

Note that every multiplicative horizontal lift $h:T\mathbb{X}'\to \At(\p)$ induces a multiplicative horizontal lift $h^{!}:T\mathbb{X}\to \phi^!(\At(\p)); v\mapsto (v,(h\circ T\phi)(v))$. This is the horizontal lift corresponding to the pullback 2-connection $\Phi^*\theta$ where $\theta\in \Omega^1(\p,\mathbb{g})$ is the 2-connection associated with $h:T\mathbb{X}'\to \At(\p)$.

The horizontal lift $h:T\mathbb{X}'\to \At(\p)$, induces a horizontal lift of the pullback extension $\mathbb{At}(\phi^{*}\p)^{\mathbb{At}(\Phi)}$, which coincides with the restriction of $h^{!}:T\mathbb{X}\to \phi^!(\At(\p))$ to projectable sections. The latter is well defined once $h^{!}$ preserves projectable sections. Hence one can see (<ref>) as morphisms of extensions of Lie 2-algebras with horizontal lifts.

The inclusion in (<ref>) gives rise to the following commutative diagram relating the associated Chern-Weil-Lecomte maps

\begin{equation}
\begin{tikzcd}
\underline{\Hom}^{\bullet}(\wedge^k\mathbb{L}(\phi^*\Ad(\p))\left[1\right],\mathbb{F(X)})^{\mathbb{L}(\phi^{!}\At(\p))}\arrow[r,"cw"]\arrow[d,""]&H^{k+\bullet}(\mathbb{L}(T\mathbb{X}),\mathbb{F(X)})\arrow[d,""]\\
\underline{\Hom}^{\bullet}(\wedge^k\mathbb{L}(\Ad(\p))\left[1\right],\mathbb{F(X)})^{\mathbb{L}(\phi^{!}\At(\p))^{\At(\Phi)}}\arrow[r,"cw"] &H^{k+\bullet}(\mathbb{L}(T\mathbb{X})^{T\phi},\mathbb{F(X)}).
\end{tikzcd}
\end{equation}

Additionally, it follows from Theorem <ref> that

\begin{equation}
\begin{tikzcd}
\underline{\Hom}^{\bullet}(\wedge^k\mathbb{L}(\Ad(\p))\left[1\right],\mathbb{F(X')})^{\mathbb{L}(\At(\p))}\arrow[r,"cw"]\arrow[d,""]&H^{k+\bullet}(\mathbb{L}(T\mathbb{X}'),\mathbb{F(X')})\arrow[d,""]\\
\underline{\Hom}^{\bullet}(\wedge^k\mathbb{L}(\Ad(\p))\left[1\right],\mathbb{F(X)})^{\mathbb{L}(\phi^{!}\At(\p))^{\At(\Phi)}}\arrow[r,"cw"] &H^{k+\bullet}(\mathbb{L}(T\mathbb{X})^{T\phi},\mathbb{F(X)}).
\end{tikzcd}
\end{equation}

As a result, we have the following.

Let $(\mathbb{P}\curvearrowleft \mathbb{G}\to \mathbb{X}')$ be a principal 2-bundle equipped with a 2-connection. If $\phi:\mathbb{X}\to \mathbb{X}'$ is a  Lie groupoid morphism, then the following the diagram

\begin{equation}\label{eq:MoritaCWL2bundles}
\begin{tikzcd}
\Hom^{\bullet}(\wedge^k \mathbb{L}(\phi^{*}\mathrm{Ad}(\mathbb{P}))[1],\mathbb{F}(\mathbb{X}))^{\mathbb{L}(\phi^{!}\mathrm{At}(\mathbb{P}))}\arrow[r,"cw"]\arrow[d]&H^{\bullet+ k}(\mathbb{L}(T\mathbb{X}),\mathbb{F}(\mathbb{X}))\arrow[d]\\
\Hom^{\bullet}(\wedge^k \mathbb{L}(\mathrm{Ad}(\mathbb{P}))[1],\mathbb{F}(\mathbb{X}))^{\mathbb{L}(\phi^{!}\At(\p))^{\At(\Phi)}}\arrow[r,"cw"]&H^{\bullet+ k}(\mathbb{L}(T\mathbb{X})^{\phi},\mathbb{F}(\mathbb{X}))\\
\Hom^{\bullet}(\wedge^k \mathbb{L}(\mathrm{Ad}(\mathbb{P}))[1],\mathbb{F}(\mathbb{X}'))^{\mathbb{L}(\mathrm{At}(\mathbb{P}))}\arrow[r,"cw"]\arrow[u]&H^{\bullet+ k}(\mathbb{L}(T\mathbb{X}'),\mathbb{F}(\mathbb{X}')),\arrow[u]\\
\end{tikzcd}
\end{equation}

commutes. Additionally, if $\phi:\mathbb{X}\to \mathbb{X}'$ is a Morita map which is surjective on objects, then the vertical arrows on the right hand side of the diagram are isomorphisms.

Since $\mathbb{L}(\phi^{*}\Ad(\p))^{\Ad(\Phi)}\cong \mathbb{L}(\Ad(\p))$, see Remark <ref>, the commutative diagram on top of (<ref>) says that given a ruth morphism $f:\wedge^k\mathbb{L}(\phi^{*}\Ad(\mathbb{P}))[1]\to \mathbb{F}(\mathbb{X})$, then:

where $\iota_{\mathbb{L}(T\mathbb{X})^{\phi}}:\mathbb{L}(T\mathbb{X})^{\phi}\to \mathbb{L}(T\mathbb{X})$ is the inclusion. In other words, the Chern-Weil-Lecomte map is compatible with projectable complexes. The restriction to projectable complex does not change the space coefficients $\mathbb{F(X)}$ where the cohomology class $cw(f)$ takes values. However, the second commutative diagram in (<ref>) comes from the compatibility between the Chern-Weil-Lecomte map and pullbacks of extensions <ref>, hence the space of coefficients $\mathbb{F(X)}$ is replaced by $\mathbb{F(X')}$. If $g:\wedge^k\mathbb{L}(\Ad(\p))\to \mathbb{F(X')}$, then

$$\big{(}(T\phi)_*\big{)}^{*}cw(g)=cw(g\circ \phi^{*}).$$

[1] A. Alekseev, E. Meinrenken, Lie theory and the Chern-Weil homomorphism, Ann. Sci. 'École Normale Supérieure, Serie 4 38 (2005), no. 2, 303–338.

[2] C. Arias Abad, M. Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. (Crelles) 663 (2012) 91–126.

[3] C. Arias Abad, S. Pineda Montoya, A. Quintero Velez, Chern-Weil theory for $\infty$-local systems, arXiv:2105.00461 (2021)

[4] J. Baez, A. Crans,  Higher-dimensional algebra VI: Lie 2-algebras, Theory and Applications of Categories, 12 (2004), no. 15, 492–528.

[5] J. Baez, U. Schreiber, Higher gauge theory, Categories in algebra, geometry and mathematical physics, Contemporary Mathematics, 431 (2007), 7–30.

[7] K. Behrend, P. Xu, Differentiable stacks and gerbes, Journal of Symplectic Geometry 9 (2011), no. 3, 285–341.

[8] D. Berwick-Evans, E. Lerman, Lie 2-algebras of vector fields, Pacific Journal of Mathematics 309 (2020), no. 1, 1–34.

[9] J-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics vol.107 Birkhaüser Boston, 1993.

[10] H. Bursztyn, M. Crainic, A. Weinstein, C. Zhu, Integration of twisted Dirac brackets, Duke Math. Journal 123 (2004), 549–607.

[11] H. Bursztyn, A. Cabrera, Multiplicative forms at the infinitesimal level, Math. Annalen 353 (2012), 663–705.

[12] H. Bursztyn, A. Cabrera, C. Ortiz, Multiplicative and linear 2-forms, Lett. Math. Phys. 90 (2009), 59–83.

[13] H. Bursztyn, F. Noseda, C. Zhu, Principal actions of stacky Lie groupoids, International Mathematics Research Notices 2020 (2020), no. 16, 5055–5125.

[14] D. Calaque, Derived stacks in symplectic geometry, arXiv:1802.09643 (2018)

[15] S. Chatterjee, A. Chaudhuri, P. Koushik, Atiyah sequence and qauge transformations of a principal 2-bundle over a Lie groupoid, Journal of Geometry and Physics, 176 (2022), 104509.

[16]  Z. Chen, H. Lang, Z. Liu, The weak Lie 2-algebra of multiplicative forms on a quasi-Poisson groupoid, arXiv:2302:01294 (2023)

[18] J. Chuang, A. Lazarev, Combinatorics and formal geometry of the Maurer-Cartan equation, Letters in Mathematical Physics 103 (2013), no.1, 79–112.

[19] M. Crainic, Cyclic cohomology of Hopf algebras, Journal of Pure and Applied Algebra, 166 (2022), no. 1-2, 29–66.

[20] B. Collier, Infinitesimal symmetries of Dixmier-Douady gerbes, arXiv:1108.1525 (2011).

[21] B. Collier, Infinitesimal symmetries of Dixmier-Douady gerbes, PhD thesis, University
of Texas at Austin (2012).

[22] M. Dehling, Shifted $L_{\infty}$-bialgebras, Master Thesis, Göttingen University (2011).

[23] D. Djounvouna, D. Krepski, Infinitesimal symmetries of bundles gerbes and Courant algebroids, arXiv:2304.05347 (2023)

[24] T. Drummond, L. Egea, Differential forms with values in VB-groupoids, Journal of Geometry and Physics 135 (2019), 42–69.

[25] D. Fiorenza, C. Rogers, U. Schreiber, $L_{\infty}$-algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, 16 (2014) no. 2, 107–142.

[26] A. Gracia-Saz, R. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Advances in Mathematics 223 (2010), no. 4, 1236–1275.

[28] J.S. Herrera-Carmona, The Chern-Weil-Lecomte morphism for $L_{\infty}$-algebras, Ph.D. Thesis, University of Sao Paulo (2022).

[29] J.S. Herrera-Carmona, C. Ortiz, J. Waldron, Vector fields and derivations on differentiable stacks, in preparation.

[30] N. Hitchin, Generalized Calabi–Yau manifolds, Quarterly Journal of Mathematics,
54 (2003), no. 3, 281–308.

[31] N. Hitchin, Brackets, forms and invariant functionals, Asian Journal of Mathematics,
10 (2006), no. 3, 541–560.

[32] H. Kajiura, Noncommutative homotopy algebras associated with open strings, Reviews in Mathematical Physics 19 (2007), no. 01, 1–99.

[33] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66, 157–216 (2003)

[34] A. Kotov, T. Strobl, Characteristic classes associated to $Q$-bundles, International Journal of Geometric Methods in Modern Physics,  12 (2015), no. 1, 1550006.

[35] D. Krepski, J. Vaughan, Multiplicative vector fields on bundle gerbes, Differential Geometry and its Applications, 84 (2022) 101931.

[36] T. Lada, M. Markl, Strongly homotopy Lie algebras. Communications in Algebra, 23 (1996), no. 6, 2147–2161.

[37] T. Lada, J. Stasheff, Introduction to SH Lie algebras for physicists, International Journal of Theoretical Physics 32 (1993), 1087–1103.

[38] C. Laurent-Gengoux, J.-L. Tu, P. Xu, Chern-Weil map for principal bundles over groupoids, Mathematische Zeitschrift, 255 (2007), no. 3, 451–491.

[39] P. Lecomte, Sur la suite exacte canonique associée un fibré principal, Bull. Soc. Math. Fr. 13 (1985), 259-271.

[41] Z-J. Liu, A. Weinstein, P. Xu, Manin triples for Lie bialgebroids, Journal of Differential Geometry 45 (1997), 547–574.

[42] K.C.H. Mackenzie, P. Xu, Classical lifting properties and multiplicative vector fields, Quart. J. Math. Oxford
49 (1998), no. 2, 59–85.

[43] K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series 213, Cambridge University Press, 2005.

[44] M. Manetti, Lectures on deformations of complex manifolds, arXiv:math/0507286 (2005).

[47] R. Mehta, M. Zambon, $L_{\infty}$-algebra actions, Differential Geometry and its Applications, 30 (2012), no. 6, 576–587.

[49] C. Ortiz, J. Waldron, On the Lie 2-algebra of sections of an $\mathcal{LA}$-groupoid, Journal of Geometry and Physics, 145 (2019) 103474.

[50] M. Penkava, $L_{\infty}$-algebras and their cohomology, arXiv:q-alg/9512014 (1995).

[51] T. Pantev, B. Toën, M. Vaquié, G. Vezzosi, Shifted symplectic structures, Publications Mathématiques de l’IHES, 117 (2013), no. 1, 271–328.

[52] B. Reinhold, $L_{\infty}$-algebras and their cohomology, Emergent Scientist, 3 (2019), no. 4.

[53] C. Rogers, 2-plectic geometry, Courant algebroids and categorified prequantization, Journal of Symplectic Geometry 11 (2013), no. 1, 53–91.

[54] C. Rogers, An explicit model for the homotopy theory of finite-type Lie $n$-algebras, Algebraic and Geometric Topology, 20 (2020), no. 3, 1371–1429.

[55] D. Roytenberg, A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys. 46 (1998), no. 1, 81–93.

[56] L. Ryvkin, T. Burzwacher, An invitation to multisymplectic geometry, Journal of Geometry and Physics, 142 (2019), 9–36.

[59] B. Vallette, Homotopy theory of homotopy algebras, Annales de l'Institut Fourier 70 (2020), no. 2, 683-738.

[61] K. Waldorf, A global perspective to connections on principal 2-bundles, Forum Math. 30 (2017), no. 4, 809–843.

[62] C.A. Weibel, An introduction to homological algebra, Cambridge University Press 38 (1994).

[63] C. Wockel, Principal 2-bundles and their gauge 2-groups, Forum Math.  23 (2011), 565–610.

,every to/.append style=
edge node=node [sloped, allow upside down, auto=false]$#1$

Sebastián Herrera
Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP 05508-090, Brasil

Cristian Ortiz
Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP 05508-090, Brasil

Herrera was supported by a CAPES Ph.D. Fellowship. Ortiz was partially supported by CNPq Bolsa de Produtividade em Pesquisa. This project received financial support from Fapesp Grant 2019/13204-0.

The Chern-Weil-Lecomte characteristic map for L_∞-algebras

In this paper we extend the Chern-Weil-Lecomte characteristic map to the setting of $L_{\infty}$-algebras. In this general framework, the target of the Chern-Weil-Lecomte map is the cohomology of an $L_{\infty}$-algebra with values in a representation up to homotopy. As applications, we introduce a Chern-Weil morphism for principal 2-bundles over Lie groupoids.

§ INTRODUCTION

Main ideas:

- motivation: why ordinary CW is not the right approach?
- ordinary CWL main steps; L infty version
- principal 2-bundles

- state the main results

§ BACKGROUND MATERIAL

In this section we briefly recall the definition of $L_{\infty}$-algebras and their morphisms. We follow [52] closely.

§.§ Graded vector spaces

Let $\mathbb{V}=\bigoplus_{n\in \mathbb{Z}}V_n$ be a graded vector space. An element $v\in V_n$ is called homogeneous of degree $n$ and its degree is denoted by $|v|$. Given a second graded vector space $\mathbb{W}$, the space $\underline{\Hom}(\mathbb{V},\mathbb{W})$ is the graded vector space whose homogeneous component of degree $p$ is defined by

$$\underline{\Hom}(\mathbb{V},\mathbb{W})_p:=\bigoplus_{n\in \Z}\mathrm{Hom}(V_n,W_{n+p}).$$

A linear map $f\in \underline{\Hom}(\mathbb{V},\mathbb{W})_p$ is called homogeneous of degree $|f|=p$. A linear map $f:\mathbb{V}\to \mathbb{W}$ is called degree preserving if it is homogeneous of degree zero. If $\mathbb{W}=\mathbb{V}$ we write $\underline{\End}(\mathbb{V}):=\underline{\Hom}(\mathbb{V},\mathbb{V})$.

The tensor product $\mathbb{V}\otimes\mathbb{W}$ is the graded vector space with grading given by

$$(\mathbb{V}\otimes\mathbb{W})_n:=\bigoplus_{i+j=n}V_i\otimes W_j,$$

for all $n\in \mathbb{Z}$. In particular, for every $n\geq 0$ one has a graded vector space $\mathbb{V}^{\otimes n}:= \bigotimes_{i=1}^{n} \mathbb{V}$.

One can also define the tensor product of two homogeneous linear maps $f\in \underline{\Hom}(\mathbb{V},\mathbb{W})_p$ and $g\in \underline{\Hom}(\mathbb{V}' ,\mathbb{W}')_q$ as the homogeneous linear map $f\otimes g\in \underline{\Hom}(\mathbb{V}\otimes\mathbb{V}',\mathbb{W}\otimes\mathbb{W}')_{p+q}$ defined by

\begin{equation}\label{eq:tensormaps}
(f\otimes g)(v\otimes v'):=(-1)^{|g||v|}f(v)\otimes g(v'),
\end{equation}

for any homogeneous elements $v \in \mathbb{V}$ and $v'\in \mathbb{V}'$. The tensor product of composable linear maps is composable and one has the following interchange law

\begin{equation}\label{eq:interchangelaw}
(f_1\otimes f_2)\circ (g_1\otimes g_2)=(-1)^{|f_2||g_1|}(f_1\circ g_1)\otimes (f_{2}\circ g_2),
\end{equation}

where $f_1,g_1$ and  $f_2,g_2$ are couples of composable homogeneous linear maps.

The graded vector space $\mathbb{V}^{\otimes n}$ carries a natural $\Sigma_n$-action by degree preserving maps $\hat{\epsilon}:\Sigma_n\to \underline{\End}(\mathbb{V}^{\otimes n})$, where $\Sigma_n$ denotes the symmetric group in $n$-letters. For every transposition $\sigma_i=(i, i+1), 1\leq i \leq n$ the action is defined by

$$\hat{\epsilon}(\sigma_i)(v_1\otimes\dots\otimes v_n)=(-1)^{|v_i||v_{i+1}|}v_1\otimes\cdots \otimes v_{i+1}\otimes v_i\otimes \cdots\otimes v_{n}$$

where $v_i\in \mathbb{V}$ are homogeneous elements with $1\leq i \leq n$. The action of an arbitrary permutation $\sigma\in \Sigma_n$ is defined as

\begin{equation}\label{eq:koszulsign}
\hat{\epsilon}(\sigma)(v_1\otimes \cdots \otimes v_n):=\epsilon(\sigma)v_{\sigma(1)}\otimes \cdots v_{\sigma(n)}
\end{equation}

where $\epsilon(\sigma)$ is the Koszul sign. Although the Koszul sign also depends on $v_1,...,v_n$ we write $\epsilon(\sigma)$ to make our notation simpler. This action naturally extends to the tensor algebra $T(\mathbb{V}):=\oplus_{n\geq 0}\mathbb{V}^{\otimes n}$, which is equipped with the graded product given by concatenation. For every $n\geq 0$ there is a graded subspace $I_{\epsilon}\subset \mathbb{V}^{\otimes n}$ defined by

$$I_{\epsilon}:=\mathrm{span}\{ v_1\otimes\cdots \otimes v_n - \epsilon(\sigma)v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(n)}; \sigma\in \Sigma_n \}.$$

The $n$-th symmetric power of $\mathbb{V}$ is defined by $S^{n}(\mathbb{V}):=\mathbb{V}^{\otimes n}/I_{\epsilon}$. The symmetric algebra of $\mathbb{V}$ is the free unital graded commutative algebra given by

$$S(\mathbb{V})=\bigoplus_{n\geq 0}S^{n}(\mathbb{V}):=\bigoplus_{n\geq 0} \mathbb{V}^{\otimes n}/I_{\epsilon}.$$

The canonical projection $\pi_S:T(\mathbb{V})\to S(\mathbb{V})$ is a surjective linear degree preserving map, whose restriction to the fixed points $\mathrm{Sym}^n(\mathbb{V})\subset \mathbb{V}^{\otimes n}$ of the action $\hat{\epsilon}:S_n\to \underline{\mathrm{End}}(\mathbb{V})^{\otimes n}$ yields an isomorphism $\mathrm{Sym}^n(\mathbb{V})\cong S^n(\mathbb{V})$. For $v_1\otimes\cdots \otimes v_n\in \mathbb{V}^n$ we write $v_1\vee \cdots \vee v_n:=\pi_S(v_1\otimes\cdots \otimes v_n)$ and the graded product in the symmetric algebra is denoted $\mu_S:S(\mathbb{V})\otimes S(\mathbb{V})\to S(\mathbb{V})$. For a homogeneous element $v=v_1\vee\cdots \vee v_n$ in $S^n(\mathbb{V})$ its  weight is the integer $n$ and its degree is $|v|=|v_1|+\cdots+|v_n|$.

The exterior algebra of $\mathbb{V}$ is defined in a similar manner by

$$\wedge \mathbb{V}=\bigoplus_{0\leq n}\wedge^n(\mathbb{V}),$$

where $\wedge^n(\mathbb{V}):=\mathbb{V}^{\otimes n}/I_{\chi}$ for every $n\geq 0$. Here $I_{\chi}\subset \mathbb{V}^{\otimes n}$ is defined similarly by considering the skew-symmetric $\Sigma_n$-action on $\mathbb{V}^{\otimes n}$ defined on transpositions by

$$\hat{\chi}(\sigma_i)(v_1\otimes\dots\otimes v_n)=-(-1)^{|v_i||v_{i+1}|}v_1\otimes\cdots \otimes v_{i+1}\otimes v_i\otimes \cdots\otimes v_{n},$$

The tensor algebra $T(\mathbb{V})$ carries the structure of a coassociative, cocommutative coalgebra with coproduct given by deconcatenation. It turns out that the symmetric algebra $S(\mathbb{V})$ inherits a coalgebra structure which we recall now. A permutation $\sigma \in \Sigma_{n}$ is called a $(k,n-k)$-unshuffle with $0\leq k\leq n$, if $\sigma(1)<\cdots<\sigma(k)$ and $\sigma(k+1)<\cdots<\sigma(n)$. The set of $(k,n-k)$-unshuffles is denoted by $Sh(k,n-k)$. The coproduct $\Delta_S:S(\mathbb{V})\to S(\mathbb{V})\otimes S(\mathbb{V})$ is defined by the symmetric deconcatenation

\begin{equation}\label{eq:symmetriccoproduct}
\Delta_S(v_1\vee\cdots\vee v_n)=\sum_{k=0}^{n}\sum_{\sigma \in Sh(k,n-k)}\epsilon(\sigma)(v_{\sigma(1)}\vee\cdots\vee v_{\sigma(k)})\otimes (v_{\sigma(k+1)}\vee\cdots\vee v_{\sigma(n)})
\end{equation}

This makes $S(\mathbb{V})$ a coassociative and cocommutative coalgebra, which is also coaugmented with $S(\mathbb{V})=\mathbb{R}1+\bar{S}(\mathbb{V})$ where $\bar{S}(\mathbb{V}):=\oplus_{n\geq 1}S^n(\mathbb{V})$.  The induced coproduct $\bar{\Delta}_S:\bar{S}(\mathbb{V})\to \bar{S}(\mathbb{V})\otimes \bar{S}(\mathbb{V})$ is defined as $\Delta_S=1\otimes \id + \bar{\Delta}_S+ \id\otimes 1$ and it is referred to as the reduced coproduct.

§.§ $L_{\infty}$-algebras

Let $\mathfrak{g}=\bigoplus_{n\leq 0}\mathfrak{g}_n$ be a non-positively graded vector space. The structure of an $L_{\infty}$-algebra on $\mathfrak{g}$ can be seen in three equivalent ways, either as higher Lie brackets, higher symmetric functions or coderivations of the coalgebra $S(\mathbb{V})$, respectively. Each of these perspectives will be explained below. For more details, the reader can see [37, 36, 44] and references therein.

An $L_{\infty}$-structure on $\mathfrak{g}=\bigoplus_{n\leq 0}\mathfrak{g}_n$ is given by a family of linear maps $[\cdot]^k:\wedge^k\mathfrak{g}\to \mathfrak{g}; k\geq 1$ of degree $2-k$ called higher brackets, with vanishing graded Jacobiator:

\begin{equation}\label{eq:gradedJacobi}
\sum_{i+j=n+1}\sum_{\sigma \in Sh(i,n-i)}(-1)^{i(j-1)}\chi(\sigma)\left[\left[v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(i)}\right]^i\wedge v_{\sigma(i+1)}\wedge\cdots\wedge v_{\sigma(n)}\right]^{j}=0,
\end{equation}

for every $n\geq 1$.

An $L_{\infty}$-algebra is a pair $(\mathfrak{g},[\cdot]^{\bullet})$ given by a non-positively graded vector space equipped with an $L_{\infty}$-structure.

Let us see some examples of $L_{\infty}$-algebras.

Let $(\mathfrak{g},[\cdot]^{\bullet})$ be an $L_{\infty}$-algebra. The degree one map $\partial:=[\cdot]^1:\mathfrak{g}\to \mathfrak{g}$ satisfies $\partial^2=0$. In particular, an $L_{\infty}$-algebra in which $[\cdot]^k=0$ for every $k\geq 2$ is the same that a cochain complex of vector spaces.

Let $\mathfrak{g}=\bigoplus_{n\leq 0}\mathfrak{g}_n$ be an $L_{\infty}$-algebra in which $[\cdot]^k=0$ for every $k\geq 2$. This is equivalent to having a structure of a differential graded Lie algebra (dgla for short)  on the graded vector space $\mathfrak{g}$ with bracket $[\cdot,\cdot]:=[\cdot]^2:\mathfrak{g}\otimes \mathfrak{g}\to \mathfrak{g}$ and differential $d:=[\cdot]^1:\mathfrak{g}\to \mathfrak{g}$.

An $L_{\infty}$-algebra $\mathfrak{g}=\mathfrak{g}_{-1}\oplus \mathfrak{g}_0$ concentrated in degrees -1 and 0 is called 2-term $L_{\infty}$-algebra. In this case, the only not necessarily trivial brackets are $[\cdot]^i:\wedge^i\mathfrak{g}\to \mathfrak{g}$ with $i=1,2,3$. As shown in [4], the category of 2-term $L_{\infty}$-algebras is equivalent to that of semistrict Lie 2-algebras. If $[\cdot]^3=0$, one recovers the notion of a strict Lie 2-algebra, which is also equivalent to having a crossed module of Lie algebras. That is, a cochain complex of Lie algebras $\partial:\mathfrak{g}_{-1}\to \mathfrak{g}_0$ together with an action $D:\mathfrak{g}_{0}\to \mathrm{Der}(\mathfrak{g}_{-1})$ satisfying the Peiffer indentities:

* $D_{\partial(u)}v=\ad_uv$

* $\partial(D_xu)=\ad_x\partial(u)$

for every $u,v\in \mathfrak{g}_{-1}$ and $x\in \mathfrak{g}_{0}$.

The decálage isomorphism $S^n(\mathfrak{g}[1])\cong (\wedge^n\mathfrak{g})[n], n\geq 1$ (see [52]), yields an alternative description of an $L_{\infty}$-algebra structure on a graded vector space $\mathfrak{g}=\bigoplus_{n\leq 0}\mathfrak{g}_n$. Namely, as a family of linear maps $\lambda_k:S^{k}(\mathfrak{g}[1])\to \mathfrak{g}[1], k\geq 1$ of degree 1 satisfying

\begin{equation}\label{eq:symmetricJacobi}
\sum_{i=1}^n\sum_{\sigma\in Sh(i,n-i)}\epsilon(\sigma)\lambda_{n-i+1}(\lambda_{i}(v_{\sigma(1)}\vee\cdots\vee v_{\sigma(i)})\vee v_{\sigma(i+1)}\vee\cdots \vee v_{\sigma(n)})=0,
\end{equation}

for every $n\geq 1$.

The family $\lambda_k:S^k(\mathfrak{g}[1])\to \mathfrak{g}[1]; k\geq 1$ yields a linear map $\lambda:S(\mathfrak{g})\to \mathfrak{g}$ via $\lambda=\sum_k\lambda_k$. This data is equivalent to a degree 1 coderivation  $d_{\lambda}:S(\mathfrak{g}[1])\to S(\mathfrak{g}[1])$ defined by

\begin{equation}\label{eq:inducedcoderivation}
d_{\lambda}:=\mu_S\circ (\lambda\otimes\id)\circ \Delta_S.
\end{equation}

Also, the maps $\lambda_k:S^k(\mathfrak{g}[1])\to \mathfrak{g}[1]$ satisfy (<ref>) if and only if the induced coderivation squares to zero, i.e. $d_{\lambda}^2=0$.

The map $\underline{\mathrm{Hom}}(S(\mathfrak{g}),\mathfrak{g})\to \mathrm{Coder}(S(\mathfrak{g})); \lambda \mapsto d_{\lambda}$ as in (<ref>) is an isomorphism. In particular, an $L_{\infty}$-algebra structure on $\mathfrak{g}$ is equivalent to a degree one coderivation $d\in \mathrm{Coder}(S(\mathfrak{g}[1]))$ with $d(1)=0$ and $d^2=0$.

Accordingly, an $L_{\infty}$-algebra is denoted either by $(\mathfrak{g},[\cdot]^{\bullet}), (\mathfrak{g},\lambda)$ or $(S(\mathfrak{g}),d)$.

§.§ $L_{\infty}$-algebra morphisms
Viewing an $L_{\infty}$-structure on a graded vector space as a degree 1 coderivation on its symmetric coalgebra allows us to introduce $L_{\infty}$-algebra morphisms in a simple manner.

An $L_{\infty}$-morphism $F:(S(\mathfrak{g}),d_{\mathfrak{g}})\to (S(\mathfrak{h}),d_{\mathfrak{h}})$ is a coalgebra morphism which commutes with the differentials.

That is, $F:S(\mathfrak{g}[1])\to S(\mathfrak{h}[1])$ is a degree zero linear map with the following properties:

\begin{align}
F\otimes F\circ \Delta_{S}=&\Delta_{S}\circ F \label{eq:coalgebramap}\\
F\circ d_{\mathfrak{g}}=&d_{\mathfrak{h}}\circ F. \label{eq:Fcommutesdifferentials}
\end{align}

Any coalgebra morphism $F:S(\mathfrak{g}[1])\to S(\mathfrak{h}[1])$ is uniquely determined by its projection onto $\mathfrak{h}[1]$, therefore equivalent to a collection of degree zero linear maps

$$ F^1_n:S^n(\mathfrak{g}
[1])\to \mathfrak{h}[1]; \quad n\geq 1.$$

An $L_{\infty}$-morphism $F:(S(\mathfrak{g}),d_{\mathfrak{g}})\to (S(\mathfrak{h}),d_{\mathfrak{h}})$ is called strict if $F_n^1=0$ for $n\geq 2$. This is equivalent to having $F=S(F^1_1)$ where

\begin{equation}\label{eq:symmetricextensionmap}
(SF^1_1)(v_1\vee\cdots\vee v_n)=F_1^1(v_1)\vee\cdots\vee F_1^1(v_n),
\end{equation}

is the canonical extension of $F^1_1:\mathfrak{g}[1]\to \mathfrak{h}[1]$. In this case, identity (<ref>) can be written in terms of symmetric brackets. More precisely, if $d_{\mathfrak{g}}$ and $d_{\mathfrak{h}}$ are respectively induced by $\lambda^{\mathfrak{g}}:S(\mathfrak{g}[1])\to \mathfrak{g}[1]$ and $\lambda^{\mathfrak{h}}:S(\mathfrak{h})\to \mathfrak{h}[1]$ as in (<ref>), then (<ref>) is equivalent to

\begin{equation}\label{eq:strictmorphism}
\lambda^{\mathfrak{h}}\circ S(F^1_1)= F^1_1\circ \lambda^{\mathfrak{g}}.
\end{equation}

As shown in Example <ref>, an $L_{\infty}$-algebra $(\mathfrak{g},[\cdot])^{\bullet}$ has an underlying complex of vector spaces with differential $\partial:\mathfrak{g}\to \mathfrak{g}$ where $\partial:=[\cdot ]^{1}$. If $F:(S(\mathfrak{g}),d_{\mathfrak{g}})\to (S(\mathfrak{h}),d_{\mathfrak{h}})$ is an $L_{\infty}$-algebra morphism, then $F^1_1:\mathfrak{g}\to \mathfrak{h}$ is a morphism between the underlying complexes. We say that $F$ is an   $L_{\infty}$-quasi-isomorphism if the induced map in cohomology $H(F_1^1):H(\mathfrak{g},[\cdot]^1)\to H(\mathfrak{h},[\cdot]^1)$ is an isomorphism.

Let us consider an extension of Lie algebras

0[r]   𝔫[r,"ι"]   𝔤̂[r,"π"]   𝔤[r]   0

Let 2-$\mathrm{Der}(\mathfrak{n})$ be the Lie 2-algebra corresponding to the crossed module of Lie algebras $\ad:\mathfrak{n}\to \mathrm{Der}(\mathfrak{n})$. The choice of a splitting $\sigma:\mathfrak{g}\to \hat{\mathfrak{g}}$ yields an $L_{\infty}$-morphism $F:\mathfrak{g}\to 2\text{-}\mathrm{Der}(\mathfrak{n})$ whose components are

$$F^1_0:=0:0\to \mathfrak{n}; \quad F^1_1:=\ad\circ \sigma:\mathfrak{g}\to \mathrm{Der}(\mathfrak{n}); \quad F^1_2:=K_{\sigma}: \wedge^2\mathfrak{g}\to \mathfrak{n},$$

where $K_{\sigma}(x,y):=[\sigma(x),\sigma(y)]-\sigma[x,y]$ is the curvature of $\sigma$. For details, see [47].

An important example of $L_{\infty}$-quasi-isomorphism is given by the minimal model of an $L_{\infty}$-algebra. An $L_{\infty}$-algebra $(\mathfrak{g},[\cdot]^{\bullet}$ is called minimal if $[\cdot]^1=0$. We say that it is contractible if $[\cdot]^k=0$ for every $k\geq 2$ and $H(\mathfrak{g},\partial)=0$ where $\partial:=[\cdot]^1:\mathfrak{g}\to \mathfrak{g}$ is the underlying cochain complex.

[Kontsevich decomposition]

Every $L_{\infty}$-algebra $\mathfrak{g}$ is isomorphic to a direct sum $\mathfrak{g}\cong\mathfrak{g}_m\oplus \mathfrak{g}_c$ of a minimal and a contractible $L_{\infty}$-algebra, see [33]. It turns out that $\mathfrak{g}$ is quasi-isomorphic to its minimal component $\mathfrak{g}_m$, which is unique up to quasi-isomorphism and referred to as the minimal model of $\mathfrak{g}$.

The minimal model of a strict 2-term $L_{\infty}$-algebra can be easily described in terms of crossed modules. This is explained in the next example.

[Minimal model of 2-term $L_{\infty}$-algebras]

As shown in [4], minimal 2-term $L_{\infty}$-algebras are in one-to-one correspondence with quadruples $(\mathfrak{g}_0,V,\rho,\theta)$ where $\rho:\mathfrak{g}_0\to \mathrm{End}(V)$ is a Lie algebra representation and $\theta\in Z^3_{CE}(\mathfrak{g}_0,V)$ is a 3-cocycle in the corresponding Chevalley-Eilenberg complex. The correspondence assigns to every minimal 2-term $L_{\infty}$-algebra $\mathfrak{g}=\mathfrak{g}_{-1}\oplus \mathfrak{g}_0$ the vector space $V:=\mathfrak{g}_{-1}$ together with the representation $\rho:\mathfrak{g}_0\otimes V\to V; (x,u)\mapsto [x,u]^2$ and 3-cocycle $\theta:=[\cdot]^3$. See Thm. 55 in [4]. Isomorphic minimal 2-term $L_{\infty}$-algebras give rise to an isomorphism between the corresponding quadruples. In particular, there is a one-to-one correspondence between quasi-isomorphism classes of 2-term $L_{\infty}$-algebras and isomorphisms of quadruples $(\mathfrak{g}_0,V,\rho,[\theta])$ with $[\theta]\in H^3_{CE}(\mathfrak{g}_0,V)$. The cohomology class $[\theta]\in H^3_{CE}(\mathfrak{g}_0,V)$ is called the characteristic class of a 2-term $L_{\infty}$-algebra. Now we describe the minimal model of a strict 2-term $L_{\infty}$-algebra $\mathfrak{g}=\mathfrak{g}_{-1}\oplus \mathfrak{g}_0$, hence providing an explicit description of its characteristic class. For that, let $\partial:\mathfrak{g}_{-1}\to \mathfrak{g}_{0}$ be the underlying crossed module of $\mathfrak{g}$ and consider $\mathfrak{h}_{-1}:=\ker(\partial)$ and $\mathfrak{h}_0:=\mathrm{coker}(\partial)$. The choice of a splitting $h:\mathfrak{h}_0\to \mathfrak{g}_0$ of the canonical projection $\pi:\mathfrak{g}_0\to \mathfrak{h}_0$ induces a quasi-isomorphism of complexes

\begin{equation}\label{eq:eq1ex:2termminimal}
(\iota,h):\big{(}0:\mathfrak{h}_{-1}\to \mathfrak{h}_0\big{)}\to \big{(}\partial:\mathfrak{g}_{-1}\to \mathfrak{g}_{0}\big{)}.
\end{equation}

This quasi-isomorphism of complexes is part of a weak quasi-isomorphism of $L_{\infty}$-algebras $F:\mathfrak{h}\to \mathfrak{g}$ with $F^1_1=(\iota,h)$ and $F^1_2:\wedge^2\mathfrak{h}_{0}\to \mathfrak{g}_{-1}$ defined as $F^1_2=\sigma^*K_h$ where $K_h:\wedge^2\mathfrak{h}_0\to \mathrm{im}(\partial)$ and $\sigma:\mathrm{im}(\partial)\to \mathfrak{g}_{-1}$ is a splitting of the exact sequence $\ker(\partial)\to \mathfrak{g}_{-1}\to \mathrm{im}(\partial)$. The $L_{\infty}$-structure on $\mathfrak{h}$ has bracket of order three given by

where $h^*D:\mathfrak{h}_0\times \mathfrak{h}_{-1}\to \mathfrak{h}_{-1}; (\bar x, u)\mapsto [h(\bar x),u]$ is the pullback via $h$ of the representation $\mathfrak{g}_{0}\to \mathrm{Der}(\mathfrak{g}_{-1})$. Hence, the characteristic class of the strict 2-term $L_{\infty}$-algebra $\mathfrak{g}$ is

$$[d_{h^*D}\sigma^*K_h]\in H^3_{CE}(\mathrm{coker}(\partial),\ker(\partial)).$$

§ $L_{\INFTY}$-ALGEBRA COHOMOLOGY

In this section we recall the notion of $L_{\infty}$-algebra cohomology with coefficients in a representation up to homotopy. For that we follow [52] closely. See also [36].

§.§ Representations up to homotopy

Let $(\mathfrak{g},\lambda)$ be an $L_{\infty}$-algebra with induced coderivation $d_{\lambda}:S(\mathfrak{g})\to S(\mathfrak{g})$ as in (<ref>). Given a graded vector space $\mathbb{V}$, the tensor product $S(\mathfrak{g}[1])\otimes \mathbb{V}$ has a natural structure of left $S(\mathfrak{g}[1])$-comodule  defined by

$$\Delta_{\mathbb{V}}:=\Delta_S\otimes \id_{\mathbb{V}}:S(\mathfrak{g}[1])\to S(\mathfrak{g}[1])\otimes \mathbb{V},$$

where $\Delta_S$ denotes the symmetric coproduct in $S(\mathfrak{g}[1])$ as in (<ref>).

A representation up to homotopy (or ruth) of $\mathfrak{g}$ on $\mathbb{V}$ is a degree one coderivation $D:S(\mathfrak{g}[1])\otimes \mathbb{V}\to S(\mathfrak{g}[1])\otimes \mathbb{V}$ satisfying:

i) $\Delta_{\mathbb{V}}\circ D=(d_{\lambda}\otimes \id_{S(\mathfrak{g})\otimes \mathbb{V}}+\id_{S(\mathfrak{g})}\otimes D)\circ \Delta_{\mathbb{V}}$

ii) $D^2=0$

The set of all ruths of $(\mathfrak{g},\lambda)$ is denoted by $\mathrm{Rep}^{\infty}(\mathfrak{g})$. An element in  $\mathrm{Rep}^{\infty}(\mathfrak{g})$ is written $(\mathbb{V},D)$ or simply $\mathbb{V}$. As shown in [52], ruths can be equivalently described as an action of $S(\mathfrak{g}[1])$ on a graded vector space $\mathbb{V}$. More precisely, a representation up to homotopy of $\mathfrak{g}$ on $\mathbb{V}$ is equivalent to a linear map

\begin{equation}\label{eq:ruthasmap}
\rho:S(\mathfrak{g}[1])\otimes \mathbb{V}\to \mathbb{V},
\end{equation}

of degree one satisfying the following condition

\begin{equation}\label{eq:equivariance}
\rho \circ (d_{\lambda}\otimes \id_{\mathbb{V}})+\rho\circ(\id_{S(\mathfrak{g}[1])}\otimes \rho)\circ(\Delta_S \otimes\id_{\mathbb{V}})=0.
\end{equation}

In agreement with this perspective, a representation up to homotopy of $\mathfrak{g}$ is denoted by $(\mathbb{V},\rho)$ or simply $\mathbb{V}$.

One can think of Definition <ref> as a coalgebra version of the well-known fact that a Lie algebra representation is fully characterized by its Chevalley-Eilenberg differential. Also, Definition <ref> resembles the notion of representation up to homotopy as in [2] and [26]. However, describing a representation up to homotopy as an action map $\rho:S(\mathfrak{g}[1])\otimes \mathbb{V}\to \mathbb{V}$ satisfying (<ref>) is more geometric and we follow this perspective.

[Adjoint representation]
Let $(\mathfrak{g},\lambda)$ be an $L_{\infty}$-algebra. The adjoint representation of $\mathfrak{g}$ is the representation up to homotopy defined by the higher brackets

$$\ad:S(\mathfrak{g}\left[1\right])\otimes \mathfrak{g}\left[1\right]\to \mathfrak{g}\left[1\right],\quad \ad(x\otimes y):=\lambda(x\vee y).$$

Just as with Lie algebras and their representations, looking at ruths of a given $L_{\infty}$-algebra allows us to define two algebraic objects. Namely, its category of representations up to homotopy and the Chevalley-Eilenberg cohomology with coefficients in a representation up to homotopy. The former is described in what follows.

Let $(\mathbb{V},\rho)$ and $(\mathbb{V}',\rho')$ be two representations up to homotopy of $\mathfrak{g}$ and $\mathfrak{g}'$, respectively. A ruth morphism $(F,f):(\mathbb{V},\rho)\to (\mathbb{V}',\rho')$ is given by a couple of maps $F:\mathfrak{g}\to \mathfrak{g}'$ which is an $L_{\infty}$-morphism and $f:\mathbb{V}'\to \mathbb{V}$ a degree-preserving map satisfying

\begin{equation}\label{eq:ruthmorphism}
f\circ \rho' \circ (F\otimes \id_{\mathbb{V}'})=\rho\circ(\id_{S(\mathfrak{g}[1])}\otimes f).
\end{equation}

In the particular case that $\mathfrak{g}=\mathfrak{g}'$ and $F=\id$, we simply say that the map $f:\mathbb{V}'\to \mathbb{V}$ is ruth map. One easily checks that if $\mathfrak{g}$ and $\mathfrak{g}'$ are Lie algebras, $\mathbb{V}$ and $\mathbb{V}'$ are usual representations, then Definition <ref> recovers the known notion of morphism of Lie algebra representations.

One observes that ruth morphisms can be composed according to the next proposition, whose proof is a straightforward computation.

Let $(\mathbb{V},\rho), (\mathbb{V}',\rho')$ and $(\mathbb{V}'',\rho'')$ be representations up to homotopy of $\mathfrak{g},\mathfrak{g}'$ and $\mathfrak{g}''$, respectively. If $(F,f):(\mathbb{V},\rho)\to (\mathbb{V}',\rho')$ and $(F',f'):(\mathbb{V}',\rho')\to (\mathbb{V}'',\rho'')$ are ruths morphisms, then

\[(F'\circ F, f\circ f'):(\mathbb{V},\rho)\to (\mathbb{V}'',\rho'').\]

is a ruths morphism.

Given an $L_{\infty}$-algebra $\mathfrak{g}$, its category of representations up to homotopy is the category $\mathcal{R}ep_{\infty}(\mathfrak{g})$ whose objects are ruths of $\mathfrak{g}$ and morphisms are defined as ruth maps.

One observes that an $L_{\infty}$-algebra morphism $F:\mathfrak{g}\to \mathfrak{g}'$ yields a pullback $F^*:\mathrm{Rep}_{\infty}(\mathfrak{g}')\to \mathrm{Rep}_{\infty}(\mathfrak{g})$. If $(\mathbb{V}',\rho')\in \mathrm{Rep}_{\infty}(\mathfrak{g}')$ is a ruth, the pullback ruth $(\mathbb{V}',F^*\rho')\in \mathrm{Rep}_{\infty}(\mathfrak{g})$ is defined as

\begin{equation}\label{eq:pullbackruth}
F^*\rho':S(\mathfrak{g}[1])\otimes \mathbb{V}'\to \mathbb{V}',\quad F^*\rho':=\rho'\circ (F\otimes  \id_{\mathbb{V}'}).
\end{equation}

Let $(\mathbb{V},\rho)$ and $(\mathbb{V}',\rho')$ be representations up to homotopy of $\mathfrak{g}$ and $\mathfrak{g}'$, respectively. Any ruth morphism $(F,f):(\mathbb{V},\rho)\to (\mathbb{V}',\rho')$ factors through a pair of ruth morphisms $(F,\id_{\mathbb{V}'}):(\mathbb{V}',F^*\rho')\to (\mathbb{V}',\rho')$ and $(\id_{\mathfrak{g}},f):(\mathbb{V},\rho)\to (\mathbb{V}',F^*\rho')$. That is, $(F,f)=(\id_{\mathfrak{g}},f)\circ (F,\id_{\mathbb{V}'})$.

We proceed now to recall the Chevalley-Eilenberg complex of an $L_{\infty}$-algebra together with a representation up to homotopy. For details, see [22] and [52] and the references therein. Let $(\mathfrak{g},\lambda)$ be an $L_{\infty}$-algebra and $\mathbb{V}$ a graded vector space. Consider the graded vector space

\begin{equation}\label{eq:cochains}
C(\mathfrak{g},\mathbb{V}):=\bigoplus_{p\geq 0} C^p(\mathfrak{g},\mathbb{V}),
\end{equation}

where $C^p(\mathfrak{g},\mathbb{V})=\underline{\Hom}(S(\mathfrak{g}\left[1\right]),\mathbb{V})_p$. In other words $C(\mathfrak{g},\mathbb{V}):=\underline{\Hom}(S(\mathfrak{g}\left[1\right]),\mathbb{V})$ is the graded vector space of graded linear maps on $S\mathfrak{g}[1]$ with values in $\mathbb{V}$. Recall that $\mathfrak{g}$ has an associated coproduct $\Delta_S:S(\mathfrak{g}[1])\to S(\mathfrak{g}[1])\otimes S(\mathfrak{g}[1])$ as in (<ref>) and induced degree 1 coderivation $d_{\lambda}:S(\mathfrak{g}[1])\to S(\mathfrak{g}[1])$ as in (<ref>).

A representation up to homotopy $(\mathbb{V},\rho)\in \mathrm{Rep}_{\infty}(\mathfrak{g})$ determines a cochain complex $(C(\mathfrak{g},\mathbb{V}),D_{\rho})$ where the differential $D_{\rho}:C(\mathfrak{g},\mathbb{V})\to C(\mathfrak{g},\mathbb{V})$ is given by

\begin{equation}\label{eq:CEdifferential}
D_{\rho}\alpha:=\rho\circ (\id_{S(\mathfrak{g}\left[1\right])}\otimes \alpha)\circ\Delta_S -(-1)^{p}\alpha\circ d_{\lambda}.
\end{equation}

for every homogeneous element $\alpha \in C^p(\mathfrak{g},\mathbb{V})$.

The cohomology $H(\mathfrak{g},\mathbb{V})$ of the complex $(C(\mathfrak{g},\mathbb{V}),D_{\rho})$ is called the $L_{\infty}$- algebra cohomology of $\mathfrak{g}$ with coefficients in $\mathbb{V}$.

If  $\mathfrak{g}$ is a Lie algebra and $\mathbb{V}$ is a usual representation, then the cochain complex $(C(\mathfrak{g},\mathbb{V}),D_{\rho})$ coincides with the usual Chevalley-Eilenberg complex associated to a Lie algebra representation. In particular, the Chevalley-Eilenberg cohomology of a Lie algebra is a particular instance of the $L_{\infty}$-algebra cohomology with coefficients in a ruth. For more details see [52].

The $L_{\infty}$-algebra cohomology with coefficients is functorial as stated in the next proposition.

Let $(F,f):(\mathbb{V},\rho)\to (\mathbb{V}',\rho')$ be a morphism between ruths of $\mathfrak{g}$ and $\mathfrak{g}'$. The pullback map $F^*:C(\mathfrak{g}',\mathbb{V}')\to C(\mathfrak{g},\mathbb{V}); \alpha\mapsto F^*\alpha:=f\circ \alpha \circ F$  is a cochain map and hence induces a well-defined map $F^*:H(\mathfrak{g}',\mathbb{V}')\to H(\mathfrak{g},\mathbb{V}')$.

Let us see that $F^* D_{\rho'}=D_{\rho}F^*$. Indeed, if $\alpha$ is a homogeneous element of degree $p$ in  $C(\mathfrak{h},\mathbb{W})$, then

\begin{align*}
F^*(D_{\rho'}\alpha)=&f\circ (\rho'\circ (\id_{S(\mathfrak{g}'\left[1\right])}\otimes \alpha)\circ\Delta_S -(-1)^{p}\alpha\circ d_{\lambda'})\circ F\\
=&f\rho'(\id_{S(\mathfrak{g}'[1])}\otimes \alpha)(F\otimes F)\Delta_S-(-1)^{p} f\circ \alpha \circ F\circ d\\
=&f\rho'(F\otimes \id_{\mathbb{V}})(\id_{S(\mathfrak{g}'[1])}\otimes \alpha)(\id_{S(\mathfrak{g}'[1])}\otimes F)\Delta_S-(-1)^{p}(F^*\alpha) d\\
=&\rho(\id_{S(\mathfrak{g}[1])}\otimes f)(\id_{S(\mathfrak{g}[1])}\otimes \alpha)(\id_{S(\mathfrak{g}[1])}\otimes F)\Delta_S-(-1)^p(F^*\alpha) d,\quad \text{by equation (\ref{eq:ruthmorphism}})\\
\end{align*}

§.§ Tensor products

Recall that given graded vector spaces $\mathbb{V}_1,\mathbb{V}_2$, the twisting map is defined by

\begin{equation}\label{eq:twistingmap}
T:\mathbb{V}_1\otimes \mathbb{V}_2\to \mathbb{V}_2\otimes \mathbb{V}_1; v\otimes w\mapsto (-1)^{|v||w|}w\otimes v.
\end{equation}

The following result shows that representations up to homotopy have a natural tensor product operation.

Let $\mathfrak{g}$ be  a $L_{\infty}$-algebra and  $(\mathbb{V}_1,\rho_1), (\mathbb{V}_2,\rho_2)\in \mathrm{Rep}_{\infty}(\mathfrak{g})$. The tensor product $\mathbb{V}_1\otimes\mathbb{V}_2$ inherits the structure of a representation up to homotopy given by

$$\rho_1\otimes \rho_2:S(\mathfrak{g}\left[1\right])\otimes (\mathbb{V}_1\otimes\mathbb{V}_2)\to \mathbb{V}_1\otimes\mathbb{V}_2,\quad \rho:=\rho_1\otimes\id_{\mathbb{V}_2}+(\id_{\mathbb{V}_1}\otimes\rho_2)\circ (T\otimes \id_{\mathbb{V}_2})$$

where $T:S(\mathfrak{g}\left[1\right])\otimes\mathbb{V}_1\to \mathbb{V}_1 \otimes S(\mathfrak{g}\left[1\right])$ is the twisting map. More concretely, on homogeneous elements $x \in S(\mathfrak{g}\left[1\right]), v\in \mathbb{V}_1$ and $w\in \mathbb{V}_2$ one has

\begin{equation}\label{eq:tensorruthhomogeneous}
\rho_1\otimes \rho_2(x\otimes v\otimes w)=\rho_1(x\otimes v)\otimes w+(-1)^{(|x|)|v|}v\otimes \rho_2(x\otimes w).
\end{equation}

The proof of Theorem <ref> is a straightforward but long computation which can be found in [65]. The tensor product ruth of $(\mathbb{V}_1,\rho_{1})$ and $(\mathbb{V}_2,\rho_2)$ is denoted by $(\mathbb{V}_1\otimes \mathbb{V}_2,\rho_1\otimes \rho_2)$.

In particular, if $n\geq 2$ and $(\mathbb{V},\rho)\in \mathrm{Rep}_{\infty}(\mathfrak{g})$ then the $n^{th}$-tensor power $\mathbb{V}^{\otimes n}$ inherits the structure of a ruth defined as follows:

\begin{equation}\label{eq:ntensorruth}
\rho^{\otimes n}:=\rho\otimes \id^{\otimes n-1}+\sum_{k=2}^{n}(\id^{\otimes k-1}\otimes\rho \otimes \id^{\otimes n-k})\circ T^k,
\end{equation}

where $T^k:\mathbb{V}^{\otimes n}\to \mathbb{V}^{\otimes n}; k\geq 2$, is the degree preserving map

\begin{equation}\label{eq:kshiftmap}
T^k:=(\id^{\otimes k-2}\otimes T\otimes \id^{\otimes n-k})\circ T^{k-1},
\end{equation}

and $T:\mathbb{V}\otimes \mathbb{V}\to \mathbb{V}\otimes\mathbb{V}, v\otimes w\mapsto (-1)^{|v||w|}w\otimes v$ is the twisting map. More concretely, on homogeneous elements $x\in S(\mathfrak{g}[1])$ and $v_1,...,v_n\in \mathbb{V}$ one has

$$T^k(v_1\otimes\cdots\otimes v_n)=(-1)^{|v_1|(\sum_{j=2}^{k}|v_j|)}v_2\otimes\cdots\otimes v_k\otimes v_1\otimes v_{k-1}\otimes \cdots\otimes v_n$$

\begin{align*}
\rho^{\otimes n}(x\otimes v_1\otimes&\cdots \otimes v_n)=\rho(x\otimes v_1)\otimes v_2\otimes\cdots\otimes v_n+\\
&+\sum_{k=2}^{n}(-1)^{(|x|+1)(\sum_{j=1}^{k-1}|v_j|)}v_1\otimes\cdots v_{k-1}\otimes \rho (x\otimes v_k)\otimes v_{k+1}\otimes\cdots\otimes v_n.
\end{align*}

§.§ Cochain products

Let $\mathfrak{g}$ be a $L_{\infty}$-algebra and $\mathbb{V}_i; i=1,2,3$ graded vector spaces. A degree preserving map $m:\mathbb{V}_1\otimes \mathbb{V}_2\to \mathbb{V}_3$ yields a degree zero linear map

\begin{equation}\label{eq:cochainmap}
\wedge_m:C(\mathfrak{g}.\mathbb{V}_1)\otimes C(\mathfrak{g},\mathbb{V}_2)\to C(\mathfrak{g}.\mathbb{V}_3),
\end{equation}

defined on homogeneous elements $\alpha \in C^p(\mathfrak{g},\mathbb{V}_1)$ and  $\beta\in C^q(\mathfrak{g},\mathbb{V}_2)$ by

\begin{equation}\label{eq:cochainproduct}
\alpha\wedge_{m}\beta:=m\circ \alpha\otimes \beta \circ \Delta_S.
\end{equation}

The element $\alpha\wedge_m\beta \in C^{p+q}(\mathfrak{g},\mathbb{V}_3)$ is called the cochain product between $\alpha$ and $\beta$ along $m$.

We say that $m:\mathbb{V}_1\otimes \mathbb{V}_2\to \mathbb{V}_3$ is symmetric if $m=m\circ T$. Similarly, if $m=-m\circ T$ one says that $m$ is skew-symmetric. The following can be verified by a straightforward computation.

Let $\mathfrak{g}$ be a $L_{\infty}$-algebra and $m:\mathbb{V}_1\otimes \mathbb{V}_2\to \mathbb{V}_3$ a degree zero map with induced cochain product $\wedge_m:C(\mathfrak{g}.\mathbb{V}_1)\otimes C(\mathfrak{g},\mathbb{V}_2)\to C(\mathfrak{g}.\mathbb{V}_3)$. Then $\wedge_m$ is symmetric (resp. skew-symmetric) if $m$ is symmetric (resp. skew-symmetric).

Assume now that $(\mathbb{V}_i,\rho_i)\in \mathrm{Rep}_{\infty}(\mathfrak{g}); i=1,2,3$ are representations up to homotopy. It is reasonable to require $m:\mathbb{V}_1\otimes \mathbb{V}_2\to \mathbb{V}_3$ to be an equivariant map, that is

\begin{equation}\label{eq:equivariantcochainproduct}
m\circ (\rho_1\otimes \rho_2)=\rho_3\circ (\id_{S(\mathfrak{g}[1])}\otimes m),
\end{equation}

where $\rho_1\otimes \rho_2$ denotes the tensor product ruth on $\mathbb{V}_1\otimes \mathbb{V}_2$ as in Theorem <ref>. In this case, the following Leibniz identity holds.

Let $\mathfrak{g}$ be an $L_{\infty}$-algebra and $(\mathbb{V}_i,\rho_i)\in\mathrm{Rep}_{\infty}(\mathfrak{g})$ ruths with induced differentials $D_{\rho_i}:C(\mathfrak{g},\mathbb{V}_i)\to C(\mathfrak{g},\mathbb{V}_i)$ as in (<ref>). If $m:\mathbb{V}_1\otimes \mathbb{V}_2\to \mathbb{V}_3$ is equivariant, then

\begin{equation}\label{eq:leibnizcochain}
D_{\rho_3}(\alpha \wedge_{m}\beta)=(D_{\rho_1}\alpha)\wedge_{m}\beta +(-1)^{|\alpha|}\alpha\wedge (D_{\rho_2}\beta),
\end{equation}

for every homogeneus elements $\alpha\in C(\mathfrak{g},\mathbb{V}_1)$ and $\beta\in C(\mathfrak{g},\mathbb{V}_2)$.

We start by computing the left hand side of (<ref>). If $\alpha\in C(\mathfrak{g},\mathbb{V}_1)$ and $\beta\in C(\mathfrak{g},\mathbb{V}_2)$ are homogeneous, then using the definition of $D_{\rho_{i}}$ together with the interchange law (<ref>), one has

\begin{align*}
D_{\rho_3}(\alpha\wedge_{m}\beta)=&\rho_3\circ (\id_{S(\mathfrak{g}[1])}\otimes \alpha\wedge_{m} \beta)\circ \Delta_S-(-1)^{|\alpha|+|\beta|}\alpha\wedge_{m}\beta\circ d\\
=&\rho_3\circ (\id_{S(\mathfrak{g}[1])}\otimes(m\circ \alpha\otimes\beta \circ \Delta_S))\circ \Delta_S-(-1)^{|\alpha|+|\beta|}m\circ \alpha\otimes \beta \circ \Delta_S\circ d\\
=&\rho_3\circ (\id_{S(\mathfrak{g}[1])}\otimes m)\circ (\id_{S(\mathfrak{g}[1])}\otimes\alpha\otimes\beta)\circ (\id_{S(\mathfrak{g}[1])}\otimes\Delta_S)\circ\Delta_S-\\
&(-1)^{|\alpha|+|\beta|}m\circ \alpha\otimes\beta \circ (\id_{S(\mathfrak{g}[1])}\otimes d+d\otimes\id_{S(\mathfrak{g}[1])})\circ \Delta_S\\
=&m\circ \rho_1\otimes \rho_2\circ (\id_{S(\mathfrak{g}[1])}\otimes \alpha\otimes \beta)\circ \Delta_S^2-(-1)^{|\alpha|+|\beta|}m\circ (\alpha\otimes(\beta\circ d))\circ \Delta_S+\\
&-(-1)^{|\alpha|+|\beta|}m\circ (-1)^{|\beta|}((\alpha\circ d)\otimes \beta) \circ \Delta_S)\\
=&A+ B\\
\end{align*}

\begin{align*}
A=&m\circ \big{(}(\rho_1\circ (\id_{S(\mathfrak{g}[1])}\otimes\alpha))\otimes \beta+(\id_{S(\mathfrak{g}[1])}\otimes \rho_2)\circ(T\otimes \id_{S(\mathfrak{g}[1])})\circ(\id_{S(\mathfrak{g}[1])}\otimes \alpha\otimes \beta)\big{)}\circ \Delta_S^2\\
B=&m\circ( -(-1)^{|\alpha|+|\beta|}\alpha\otimes(\beta\circ d) -(-1)^{|\alpha|} (\alpha\circ d)\otimes \beta) \circ \Delta_S.\\
\end{align*}

Since the coproduct is coassociative and cocommutative $\Delta_S^2=(\Delta_S\otimes\id_s)\circ \Delta_S=(\id_s\otimes\Delta_S)\circ \Delta_S$ and $(T\otimes\id_s)\circ\Delta_S^2=\Delta_S^2$, one concludes

\begin{align*}
A=&m\circ((\rho_1\circ(\id_s\otimes\alpha))\otimes \beta \circ (\Delta_S\otimes \id_s)\circ \Delta_S+(\id_s\otimes \rho_2)\circ (\alpha\otimes\id_s\otimes \beta)\circ (\id_s\otimes \Delta_S)\circ \Delta_S)\\
=&m\circ(((\rho_1\circ(\id_s\otimes\alpha)\circ \Delta_S)\otimes \beta) \circ  \Delta_S+(-1)^{|\alpha|}(\alpha \otimes (\rho_2\circ (\id_s\otimes \beta)\circ \Delta_S))\circ \Delta_S)\\
=&m\circ(((\rho_1\circ(\id_s\otimes\alpha)\circ \Delta_S)\otimes \beta) +(-1)^{|\alpha|}(\alpha \otimes (\rho_2\circ (\id_s\otimes \beta)\circ \Delta_S)))\circ \Delta_S.\\
\end{align*}

Now, one can easily see that

\begin{align*}
=&D_{\rho_1} \alpha\wedge_{m}\beta+(-1)^{|\alpha|}\alpha\wedge_{m}D_{\rho_2}\beta,
\end{align*}

as claimed.

It follows from Proposition <ref> that the product (<ref>) descends to a map in cohomology

\begin{align}\label{eq:cochainmapcohomology}
\wedge_m:H^{\bullet}(\mathfrak{g}.\mathbb{V}_1)\otimes &H^{\bullet}(\mathfrak{g},\mathbb{V}_2)\to H^{\bullet}(\mathfrak{g}.\mathbb{V}_3)\\
([\alpha],&[\beta])\mapsto [\alpha\wedge_m\beta]\nonumber
\end{align}

The following special instance of Proposition <ref> will be relevant in section <ref>. Let $\mathbb{V}$ be a graded algebra with product $m:\mathbb{V}\otimes \mathbb{V}\to \mathbb{V}$ and $\rho:S(\mathfrak{g}[1])\otimes \mathbb{V}\to \mathbb{V}$ is a representation up to homotopy. Using (<ref>) one has that $m:\mathbb{V}\otimes \mathbb{V}\to \mathbb{V}$ is a ruth map if and only if for every homogeneous elements $x\in S(\mathfrak{g}[1])$ and $v,w\in \mathbb{V}$ the following holds

\begin{equation}\label{eq:ruthderivations}
(\rho\otimes \rho)(x,m(v\otimes w))=m(\rho(x,v)\otimes w)+(-1)^{|x||v|}m(v\otimes \rho(x,w)).
\end{equation}

In other words, property (<ref>) is equivalent to saying that $\mathfrak{g}$ acts on $(\mathbb{V},m)$ by graded derivations.

Let  $\mathfrak{g}$ be an $L_{\infty}$-algebra and $(\mathbb{V},m)$ a graded algebra. If $\rho:S(\mathfrak{g}[1])\otimes \mathbb{V}\to \mathbb{V}$ is an action by derivations, then $(H^{\bullet}(\mathfrak{g},\mathbb{V}),\wedge_m)$ is a graded algebra where $\wedge_m$ is defined by (<ref>)

\begin{align}\label{eq:cochainmapgradedalgebra}
\wedge_m:H^{\bullet}(\mathfrak{g}.\mathbb{V})\otimes &H^{\bullet}(\mathfrak{g},\mathbb{V})\to H^{\bullet}(\mathfrak{g}.\mathbb{V})\\
([\alpha],&[\beta])\mapsto [\alpha\wedge_m\beta].\nonumber
\end{align}

Let us see some examples of induced cochain products.

Let $(\mathbb{V},\rho)$ be a representation up to homotopy of $\mathfrak{g}$ on $\mathbb{V}$. The identity map $\id_{\mathbb{V}\otimes\mathbb{V}}:\mathbb{V}\otimes\mathbb{V}\to \mathbb{V}\otimes\mathbb{V}$ determines a cochain product map

$$C(\mathfrak{g},\mathbb{V})\otimes C(\mathfrak{g},\mathbb{V})\to C(\mathfrak{g},\mathbb{V}\otimes \mathbb{V}).$$

In this case, the cochain product of $\alpha,\beta\in C(\mathfrak{g},\mathbb{V})$ is denoted by $\alpha\wedge_{\id}\beta$. Note that this cochain product is neither symmetric nor skew-symmetric.

Consider now a graded vector space $\mathbb{V}$. The evaluation map is given by

\begin{equation}\label{eq:evaluationmap}
ev:\underline{\mathrm{End}}(\mathbb{V})\otimes \mathbb{V}\to \mathbb{V}; T\otimes v\mapsto T(v)
\end{equation}

Clearly (<ref>) has degree zero, so it induces a product on cochains $\wedge_{ev}:C(\mathfrak{g},\underline{\mathrm{End}}(\mathbb{V}))\otimes C(\mathfrak{g},\mathbb{V})\to C(\mathfrak{g},\mathbb{V})$.

Let $(L,\left[\cdot,\cdot\right])$ be a graded Lie algebra. That is, $L=\bigoplus_{n\in \mathbb{Z}}L_n$ is a graded vector space and $\left[\cdot,\cdot\right]:L\otimes L\to L$ is a degree zero linear map which is skew-symmetric and satisfies the graded Jacobi identity:

If $\mathfrak{g}$ is an $L_{\infty}$-algebra, the bracket $\left[\cdot,\cdot\right]:L\otimes L\to L$  defines a cochain product $\wedge_{[\cdot,\cdot]}:C(\mathfrak{g},L)\otimes C(\mathfrak{g},L)\to C(\mathfrak{g},L)$ which is skew-symmetric.

In particular, if $(L,[\cdot,\cdot], d)$ is a dgla (see Example <ref>), that is, the linear map $d:L\to L$ has degree +1, it squares to zero $d^2=0$ and

then the cochain product $\wedge_{[\cdot,\cdot]}:C(\mathfrak{g},L)\otimes C(\mathfrak{g},L)\to C(\mathfrak{g},L)$  is well-defined too. As shown in Corollary <ref>, if the space of coefficients of a ruth is a graded algebra, then such an additional structure descends to cohomology. The next result shows that if the space of coefficients is a dgla, then the corresponding space of cochains inherits the structure of a dgla.

Let $(\mathfrak{g},\lambda)$ be an $L_{\infty}$-algebra with  induced degree 1 coderivation $d_{\lambda}:S(\mathfrak{g}[1])\to S(\mathfrak{g}[1])$ as in (<ref>). Assume that $(L, \left[\cdot,\cdot\right])$ is a graded Lie algebra, then $(C(\mathfrak{g},L),\wedge_{\left[\cdot,\cdot\right]})$ is a graded Lie algebra. Moreover, if $(L,\left[\cdot,\cdot\right],d)$ is a dgla, then $(C(\mathfrak{g},L),\wedge_{\left[\cdot,\cdot\right]},\textbf{d})$ is dgla with differential given by

\[\textbf{d}(\alpha):=d\circ \alpha-(-1)^{|\alpha|}\alpha\circ d_{\lambda}\]

for every homogeneous element $\alpha\in C(\mathfrak{g},L)$.

It follows from Proposition <ref> that the product $\wedge_{\left[\cdot,\cdot\right]}$ is skew-symmetric, we will see that it also satisfies the Jacobi identity. For that, let $\alpha,\beta,\gamma\in C(\mathfrak{g},L)$ be homogeneous elements, then:

\begin{align*}
\alpha\wedge_{\left[\cdot,\cdot\right]}(\beta\wedge_{\left[\cdot,\cdot\right]} \gamma)=&\left[\cdot,\cdot\right]\circ(\alpha \otimes \beta\wedge_{\left[\cdot,\cdot\right]} \gamma)\circ \Delta_S=\left[\cdot,\cdot\right]\circ (\alpha \otimes (\left[\cdot,\cdot\right]\circ \beta \otimes \gamma\circ \Delta))\circ \Delta\\
=&\left[\cdot,\cdot\right]\circ(\id\otimes \left[\cdot,\cdot\right])\circ\alpha\otimes\beta\otimes\gamma \circ(\id\otimes \Delta)\circ\Delta\\
=&\left[\cdot,\cdot\right]\circ(\left[\cdot,\cdot\right]\otimes\id)\circ \alpha\otimes\beta\otimes\gamma \circ \Delta^2+\left[\cdot,\cdot\right]\circ(\id\otimes \left[\cdot,\cdot\right])\circ(T\otimes\id)\circ\alpha\otimes\beta\otimes\gamma \circ \Delta^2\\
\end{align*}

Assume now that $(L,\left[\cdot,\cdot\right],d)$ is dgla. We will show that the map $\textbf{d}$ is derivation of the product $\wedge_{\left[\cdot,\cdot\right]}$ with $\textbf{d}^2=0.$ The derivation property follows by a direct computation on homogeneous elements $\alpha,\beta\in C(\mathfrak{g},L)$. In fact,

\begin{align*}
\textbf{d}(\alpha\wedge_{\left[\cdot,\cdot\right]}\beta)=&d\circ \alpha\wedge_{\left[\cdot,\cdot\right]}\beta-(-1)^{|\alpha|+|\beta|}\alpha\wedge_{\left[\cdot,\cdot\right]}\beta\circ d_{\lambda}\\
=&d\circ\left[\cdot,\cdot\right]\circ\alpha\otimes\beta\circ\Delta-(-1)^{|\alpha|+|\beta|}\left[\cdot,\cdot\right]\circ\alpha\otimes\beta \circ\Delta\circ d_{\lambda}\\
=&\left[\cdot,\cdot\right]\circ(d\otimes \id+\id\otimes d)\circ\alpha\otimes\beta\circ\Delta-(-1)^{|\alpha|+|\beta|}\left[\cdot,\cdot\right]\circ\alpha\otimes\beta \circ(d_{\lambda}\otimes\id+\id\otimes d_{\lambda})\circ\Delta\\