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\begin{document}

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\section*{infinity-Lie algebroid-valued differential form}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{curvature_characteristics}{Curvature characteristics}\dotfill \pageref*{curvature_characteristics} \linebreak
\noindent\hyperlink{InfGaugeTrafo}{1-Morphisms: integration of infinitesimal gauge transformations}\dotfill \pageref*{InfGaugeTrafo} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{lie_algebra_valued_1forms}{Lie algebra valued 1-forms}\dotfill \pageref*{lie_algebra_valued_1forms} \linebreak
\noindent\hyperlink{lie_2algebra_valued_forms}{Lie 2-algebra valued forms}\dotfill \pageref*{lie_2algebra_valued_forms} \linebreak
\noindent\hyperlink{ordinary_forms_and_the_de_rham_complex}{Ordinary $n$-forms and the de Rham complex}\dotfill \pageref*{ordinary_forms_and_the_de_rham_complex} \linebreak
\noindent\hyperlink{supergravity_fields}{Supergravity fields}\dotfill \pageref*{supergravity_fields} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $\mathfrak{g}$ an [[∞-Lie algebra]] (or more generally [[∞-Lie algebroid]]), the \textbf{$\infty$-groupoid of $\mathfrak{g}$-valued forms} is the [[∞-groupoid]] whose

\begin{itemize}%
\item [[object]]s are [[differential forms]] with values in $\mathfrak{g}$;


\item [[morphism]]s are [[gauge transformation]]s between these;


\item [[k-morphisms]] are order $k$ higher gauge transformations.



\end{itemize}
This naturally refines to a non-[[concrete sheaf|concrete]] [[∞-Lie groupoid]] whose $U$-parameterized smooth families of objects are [[∞-Lie algebroid]]-valued [[differential form]]s on $Z$.

A [[cocycle]] with coefficients in this is a [[connection on an ∞-bundle]].

For an introduction see the section  at [[∞-Chern-Weil theory introduction]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For $X$ a [[smooth manifold]] and $\mathfrak{g}$ an [[∞-Lie algebra]] or more generally an [[∞-Lie algebroid]], a \textbf{$\infty$-Lie algebroid valued differential form} on $X$ is a morphism of [[dg-algebra]]s

\begin{displaymath}
\Omega^\bullet(X)
  \leftarrow
  W(\mathfrak{g})
  : 
  A
\end{displaymath}
from the [[Weil algebra]] of $\mathfrak{g}$ to the [[de Rham complex]] of $X$. Dually this is a morphism of [[∞-Lie algebroid]]s

\begin{displaymath}
A : T X \to inn(\mathfrak{g})
\end{displaymath}
from the [[tangent Lie algebroid]] to the [[Weil algebra|inner automorphism ∞-Lie algebra]].

Its [[curvature]] is the composite of morphisms of [[graded vector space]]s

\begin{displaymath}
\Omega^\bullet(X) \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{F_{(-)}}{\leftarrow}
  \wedge^1 \mathfrak{g}^*
  : 
  F_{A}
  \,.
\end{displaymath}
Precisely if the curvatures vanish does the morphism factor through the [[Chevalley-Eilenberg algebra]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

\begin{displaymath}
(F_A = 0) 
  \;\;\Leftrightarrow
  \;\;
  \left(
  \itexarray{
     && CE(\mathfrak{g})
     \\
     & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow
     \\
     \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g})     
  }
  \right)
\end{displaymath}
in which case we call $A$ \textbf{flat}.

The [[curvature characteristic forms]] of $A$ are the composite

\begin{displaymath}
\Omega^\bullet(X)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{\langle F_{(-)} \rangle}{\leftarrow}
  inv(\mathfrak{g})
  :
  \langle F_A\rangle
  \,,
\end{displaymath}
where $inv(\mathfrak{g}) \to W(\mathfrak{g})$ is the inclusion of the [[invariant polynomial]]s.

\begin{defn}
\label{}\hypertarget{}{}
For $U$ a [[smooth manifold]], the \textbf{$\infty$-groupoid of $\mathfrak{g}$-valued forms is the [[Kan complex]]}

\begin{displaymath}
\exp(\mathfrak{g})_{conn}(U)
  :
  [k]
  \mapsto
  \left\{
     \Omega^\bullet(U \times \Delta^k)
      \stackrel{A}{\leftarrow}
     W(\mathfrak{g})
     \;\;
     |
     \;\;
       \forall v \in \Gamma(T \Delta^k) : \iota_v F_A = 0
  \right\}
\end{displaymath}
whose [[k-morphism]]s are $\mathfrak{g}$-valued forms $A$ on $U \times \Delta^k$ with sitting instants, and with the property that their [[curvature]] vanishes on vertical vectors.

The canonical morphism

\begin{displaymath}
\exp(\mathfrak{g})_{conn} \to \exp(\mathfrak{g})
\end{displaymath}
to the untruncated [[Lie integration]] of $\mathfrak{g}$ is given by restriction of $A$ to [[vertical differential form]]s (see below).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Here we are thinking of $U \times \Delta^k \to U$ as a trivial [[bundle]].

The \emph{first} [[Ehresmann connection|Ehresmann condition]] will be identified with the conditions on lifts $\nabla$ in [[∞-anafunctor]]s

\begin{displaymath}
\itexarray{
    && \exp(\mathfrak{g})_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    C(U) &\stackrel{g}{\to}& \exp(\mathfrak{g})
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
\end{displaymath}
that define [[connections on ∞-bundles]]. More on this in the \hyperlink{Properties}{Properties}-section below.

\end{remark}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{curvature_characteristics}{}\subsubsection*{{Curvature characteristics}}\label{curvature_characteristics}

\begin{prop}
\label{}\hypertarget{}{}
For $A \in \exp(\mathfrak{g})_{conn}(U,[k])$ a $\mathfrak{g}$-valued form on $U \times \Delta^k$ and for $\langle - \rangle \in W(\mathfrak{g})$ any [[invariant polynomial]], the corresponding [[curvature characteristic form]] $\langle F_A \rangle \in \Omega^\bullet(U \times \Delta^k)$ descends down to $U$.

\end{prop}
\begin{proof}
It is sufficient to show that for all $v \in \Gamma(T \Delta^k)$ we have

\begin{enumerate}%
\item $\iota_v \langle F_A \rangle = 0$;


\item $\mathcal{L}_v \langle F_A \rangle = 0$.



\end{enumerate}
The first condition is evidently satisfied if already $\iota_v F_A = 0$. The second condition follows with [[Cartan calculus]] and using that $d_{dR} \langle F_A\rangle = 0$:

\begin{displaymath}
\mathcal{L}_v \langle F_A \rangle = 
  d \iota_v \langle F_A \rangle
  + 
  \iota_v d \langle F_A \rangle
  = 0
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
For a general $\infty$-Lie algebra $\mathfrak{g}$ the curvature forms $F_A$ themselves are not closed, hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian $\infty$-Lie algebras: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.

\end{remark}
It is useful to organize the $\mathfrak{g}$-valued form $A$, together with its restriction $A_{vert}$ to [[vertical differential form]]s and with its [[curvature characteristic form]]s in the [[commuting diagram]] (following \emph{\href{Weil+algebra#CharacterizationInSmoothTopos}{Weil algebra -- Characterization in the smooth infinity-topos}})

\begin{displaymath}
\itexarray{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      &&&
      gauge\;transformation
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
      &&&
      \mathfrak{g}-valued\;form
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
      &&&
      curvature\;characteristic\;forms
    }
\end{displaymath}
in [[dgAlg]].

The commutativity of this diagram is implied by $\iota_v F_A = 0$.

\begin{defn}
\label{}\hypertarget{}{}
Write $\exp(\mathfrak{g})_{CW}(U)$ for the $\infty$-groupoid of $\mathfrak{g}$-valued forms fitting into such diagrams.

\begin{displaymath}
\exp(\mathfrak{g})_{CW}(U)
  :
  [k]
  \mapsto
  \left\{
    \itexarray{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
    }
  \right\}
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If we just consider the top horizontal morphism in this diagram we obtain the object

\begin{displaymath}
\exp(\mathfrak{g})(U)
  :
  [k]
  \mapsto
  \left\{
    \itexarray{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
     }
  \right\}
\end{displaymath}
discussed in detail at [[Lie integration]]. If we consider the top square of the diagram we obtain the object

\begin{displaymath}
\exp(\mathfrak{g})_{diff}(U)
  :
  [k]
  \mapsto
  \left\{
    \itexarray{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
   }
  \right\}
  \,.
\end{displaymath}
This forms a [[resolution]] of $\exp(\mathfrak{g})$ and may be thought of as the $\infty$-groupoid of [[pseudo-connection]]s.

We have an evident sequence of morphisms

\begin{displaymath}
\itexarray{
    \exp(\mathfrak{g})_{conn} &&& genuine\;connections
    \\
    \downarrow
    \\
    \exp(\mathfrak{g})_{CW} &&& pseudo-connection\;with\;global\;curvature\;characteristics 
    \\
    \downarrow
    \\
    \exp(\mathfrak{g})_{diff} &&& pseudo-connections
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \exp(\mathfrak{g}) &&& bare bundles
  }
  \,,
\end{displaymath}
where we label the objects by the structures they classify, as discussed at [[∞-Chern-Weil theory]].

Here the botton morphism is a weak equivalence and the others are [[monomorphism]]s.

Notice that in full [[∞-Chern-Weil theory]] the fundamental object of interest is really $\exp(\mathfrak{g})_{diff}$ -- the object of [[pseudo-connection]]s. The other objects only serve the purpose of picking particularly nice representatives:

the object $\exp(\mathfrak{g})_{CW}$ may contain pseudo-connections, those for which at least the associated [[circle n-bundles with connection]] given by the $\infty$-Chern Weil homomorphism are genuine connections.

This distinction is important: over objects $X \in$ [[?LieGrpd]] that are not [[smooth manifold]]s but for instance [[orbifold]]s, the genuine connections for higher Lie algebras do \emph{not} exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative in the usual sense.

\end{remark}
\hypertarget{InfGaugeTrafo}{}\subsubsection*{{1-Morphisms: integration of infinitesimal gauge transformations}}\label{InfGaugeTrafo}

The 1-[[morphism]]s in $\exp(\mathfrak{g})(U)$ may be thought of as [[gauge transformation]]s between $\mathfrak{g}$-valued forms. We unwind what these look like concretely.

\begin{defn}
\label{}\hypertarget{}{}
Given a 1-morphism in $\exp(\mathfrak{g})(X)$, represented by $\mathfrak{g}$-valued forms

\begin{displaymath}
\Omega^\bullet(U \times \Delta^1) 
  \leftarrow
  W(\mathfrak{g})
  : 
  A
\end{displaymath}
consider the unique decomposition

\begin{displaymath}
A = A_U + ( A_{vert} := \lambda \wedge d t)  \; \; 
  \,,
\end{displaymath}
with $A_U$ the horizonal differential form component and $t : \Delta^1 = [0,1] \to \mathbb{R}$ the canonical coordinate.

We call $\lambda$ the \textbf{gauge parameter} . This is a function on $\Delta^1$ with values in 0-forms on $U$ for $\mathfrak{g}$ an ordinary [[Lie algebra]], plus 1-forms on $U$ for $\mathfrak{g}$ a [[Lie 2-algebra]], plus 2-forms for a Lie 3-algebra, and so forth.

\end{defn}
We describe now how this enccodes a gauge transformation

\begin{displaymath}
A_0(s=1) \stackrel{\lambda}{\to} A_U(s = 1)
  \,.
\end{displaymath}
\begin{remark}
\label{}\hypertarget{}{}
By the nature of the [[Weil algebra]] we have

\begin{displaymath}
\frac{d}{d s} A_U
  =
  d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots
  + 
  \iota_s F_A
  \,,
\end{displaymath}
where the sum is over all higher brackets of the [[∞-Lie algebra]] $\mathfrak{g}$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
In [[Cartan calculus]] for $\mathfrak{g}$ an ordinary Lie algebra may write (see \href{Ehresmann+connection#FormulationInTermsOfCurvatureForm}{here}) the corresponding [[Ehresmann connection|second Ehresmann condition]] $\iota_{\partial_s} F_A = 0$ equivalently

\begin{displaymath}
\mathcal{L}_{\partial_s} A = ad_\lambda A
  \,.
\end{displaymath}
\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
Define the \textbf{[[covariant derivative]] of the gauge parameter} to be

\begin{displaymath}
\nabla \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots
  \,.
\end{displaymath}
\end{defn}
In this notation we have

\begin{itemize}%
\item the general identity

\begin{equation}
\frac{d}{d s} A_U = \nabla \lambda + (F_A)_s
\label{ShiftedGaugeTrafo}\end{equation}

\item the \textbf{horizontality} or \textbf{[[Ehresmann connection|second Ehresmann condition]]} (or ``strict \href{D%27Auria-Fre+formulation+of+supergravity#Rheonomy}{rheonomy}'') $\iota_{\partial_s} F_A = 0$, the [[differential equation]]

\begin{equation}
\frac{d}{d s} A_U = \nabla \lambda
  
  \,.
\label{GaugeTrafo}\end{equation}


\end{itemize}
This is known as the equation for \textbf{infinitesimal [[gauge transformation]]s} of an $\infty$-Lie algebra valued form.

\begin{remark}
\label{}\hypertarget{}{}
By [[Lie integration]] we have that $A_{vert}$ -- and hence $\lambda$ -- defines an element $\exp(\lambda)$ in the [[∞-Lie group]] that integrates $\mathfrak{g}$.

The unique solution $A_U(s = 1)$ of the above [[differential equation]] at $s = 1$ for the initial values $A_U(s = 0)$ we may think of as the result of acting on $A_U(0)$ with the [[gauge transformation]] $\exp(\lambda)$.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{lie_algebra_valued_1forms}{}\subsubsection*{{Lie algebra valued 1-forms}}\label{lie_algebra_valued_1forms}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(connections on ordinary bundles)}

For $\mathfrak{g}$ an ordinary [[Lie algebra]] with simply connected [[Lie group]] $G$ and for $\mathbf{B}G_{conn}$ the [[groupoid of Lie algebra-valued forms]] we have an [[isomorphism]]

\begin{displaymath}
\tau_1 \exp(\mathfrak{g})_{conn} = \mathbf{B}G_{conn}
\end{displaymath}
\end{prop}
\begin{proof}
To see this, first note that the sheaves of objects on both sides are manifestly isomorphic, both are the sheaf of $\Omega^1(-,\mathfrak{g})$. For morphisms, observe that for a form $\Omega^\bullet(U \times \Delta^1) \leftarow W(\mathfrak{g}) : A$ which we may decompose into a horizontal and a verical pice as $A = A_U + \lamnda \wedge d t$ the condition $\iota_{\partial_t} F_A = 0$ is equivalent to the [[differential equation]]

\begin{displaymath}
\frac{\partial}{\partial t} A
  =
  d_U \lambda + [\lambda, A]
  \,.
\end{displaymath}
For any initial value $A(0)$ this has the unique solution

\begin{displaymath}
A(t) = g(t)^{-1} (A + d_{U}) g(t)
  \,,
\end{displaymath}
where $g : [0,1] \to G$ is the [[parallel transport]] of $\lambda$:

\begin{displaymath}
\begin{aligned}
    &
    \frac{\partial}{\partial t}
    \left(
       g_(t)^{-1} (A + d_{U}) g(t)
     \right)
     \\
     = &
     g(t)^{-1} (A + d_{U}) \lambda g(t)
     -
     g(t)^{-1} \lambda (A + d_{U}) g(t)    
   \end{aligned}
\end{displaymath}
(where for ease of notaton we write actions as if $G$ were a [[matrix Lie group]]).

In particular this implies that the endpoints of the path of $\mathfrak{g}$-valued 1-forms are related by the usual cocycle condition in $\mathbf{B}G_{conn}$

\begin{displaymath}
A(1) = g(1)^{-1} (A + d_U) g(1)
  \,.
\end{displaymath}
In the same fashion one sees that given 2-cell in $\exp(\mathfrak{g})(U)$ and any 1-form on $U$ at one vertex, there is a unique lift to a 2-cell in $\exp(\mathfrak{g})_{conn}$, obtained by parallel transporting the form around. The claim then follows from the previous statement of [[Lie integration]] that $\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G$.

\end{proof}
\hypertarget{lie_2algebra_valued_forms}{}\subsubsection*{{Lie 2-algebra valued forms}}\label{lie_2algebra_valued_forms}

\begin{itemize}%
\item For $\mathfrak{g}$ [[Lie 2-algebra]], a $\mathfrak{g}$-valued differential form in the sense described here is precisely an [[Lie 2-algebra valued form]].

\end{itemize}
\hypertarget{ordinary_forms_and_the_de_rham_complex}{}\subsubsection*{{Ordinary $n$-forms and the de Rham complex}}\label{ordinary_forms_and_the_de_rham_complex}

For $n \in \mathbb{N}$, $n \geq 1$ we have that $b^{n-1}\mathbb{R}$-valued differential forms are in natural bijection to ordinary closed [[differential form]]s in degree $n$

\begin{remark}
\label{}\hypertarget{}{}
Notice that under addition of differential forms, $\exp(b^{n-1}\mathbb{R})_{conn}$ is over each $U \in CartSp$ an abelian [[simplicial group]].

Under the [[Dold-Kan correspondence]] $Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Xi}{\to}} sAb$ we may therefore identify $\exp(b^{n-1}\mathbb{R})_{conn}$ with a presheaf $N \exp(b^{n-1}\mathbb{R})_{conn}$ of chain complexes.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The degreewise [[fiber integration]] of differential forms over simplices constitutes a morphism

\begin{displaymath}
\int_{\Delta^\bullet} :
  N\exp(b^{n-1}\mathbb{R})_{conn}
  \to 
  \left(
    C^\infty(-, \mathbb{R})
    \stackrel{d_{dR}}{\to}
    \Omega^1(-)
    \stackrel{d_{dR}}{\to}
    \cdots
    \stackrel{d_{dR}}{\to}
    \Omega^n(-)
  \right)
  \,.
\end{displaymath}
that is a weak equivalence.

\end{prop}
This is shown at  based on the discussion at .

\hypertarget{supergravity_fields}{}\subsubsection*{{Supergravity fields}}\label{supergravity_fields}

What is called an ``extended soft group manifold'' in the literature on the [[D'Auria-Fre formulation of supergravity]] is really precisely a collection of $\infty$-Lie algebroid valued forms with values in a super $\infty$-Lie algebra such as the [[supergravity Lie 3-algebra]] (for 11-dimensional [[supergravity]]). The way [[curvature]] and [[Bianchi identity]] are read off from ``extded soft group manifolds'' in this literature is -- apart from this difference in terminology -- precisely what is described above.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[groupoid of Lie-algebra valued forms]]


\item [[2-groupoid of Lie 2-algebra valued forms]]


\item [[3-groupoid of Lie 3-algebra valued forms]]


\item \textbf{$\infty$-groupoid of ∞-Lie-algebra valued forms}



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The (obvious but conceptually important) observation that [[Lie algebra-valued 1-forms]] regarded as morphisms of [[graded vector space]]s $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ are equivalently morphisms of dg-algebras out of the [[Weil algebra]] $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ and that one may think of as the identity $W(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id$ as the \emph{universal $\mathfrak{g}$-connection} appears in early articles for instance highlighted on p. 15 of

\begin{itemize}%
\item Franz W. Kamber; Philippe Tondeur, \emph{Semisimplicial Weil algebras and characteristic classes for foliated bundles in ech cohomology} , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283--294. Amer. Math. Soc., Providence, R.I., (1975).

\end{itemize}
following [[Eli Cartan]]`s influential work (see [[Weil algebra]] for more references).

The (evident) generalization to Weil algebras of [[∞-Lie algebra]]s and [[∞-Lie algebroid]]s is considered explicitly in

\begin{itemize}%
\item Hisham Sati, Urs Schreiber, Jim Staasheff, \emph{$L_\infty$-algebra valued connections} ()

\end{itemize}
but -- somewhat implicitly -- this construction appears earlier, notably in the [[D'Auria-Fre formulation of supergravity]]. A collection of such precursors to these notions is collected at

\begin{itemize}%
\item [[schreiber:differential cohomology in an (∞,1)-topos -- references]]: 

\end{itemize}
The structure of the formula \eqref{GaugeTrafo} for infinitesimal gauge transformations of higher forms is widely known in the literature on [[supergravity]] and [[string theory]], if maybe not formalized in terms of $\infty$-Lie algebra theory as we do here. One exception is the remarkable book

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fre]], \emph{[[Supergravity and Superstrings - A Geometric Perspective]]} .

\end{itemize}
In this old book no $\infty$-Lie algebras are mentioned explicitly, but the [[dg-algebra]] computations that are considered are easily seen to be precisely their [[Chevalley-Eilenberg algebra]]-incarnations.

The authors use the term \emph{extended [[soft group manifold]]} for what here we identify as an $\infty$-Lie algebra valued form $T X \to inn(\mathfrak{g})$.

With this terminological translation understood, and observing that all their constructions straightforwardly generalize to more general dg-algebras than these authors conisder explicitly, we find that

\begin{itemize}%
\item our equation \eqref{ShiftedGaugeTrafo} for the possibly shifted gauge transformation is their equation I.3.136;


\item our equation \eqref{GaugeTrafo} for the genuine gauge transfomation is their equation for \emph{horizontal} or [[rheonomy|rheonomic]] gauge transformations III.3.23 .



\end{itemize}
In fact their full rheonomy constraint III.3.32 is essentialy the same horizontality constraint, but applied not just to the 1-simplex $\Delta^1$, but to the [[supermanifold|super simplex]] $\Delta^{1|p}$.

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\end{document}