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\begin{document}

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\section*{2-groupoid of Lie 2-algebra valued forms}

\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{for_strict_lie_2algebras}{For strict Lie 2-algebras}\dotfill \pageref*{for_strict_lie_2algebras} \linebreak
\noindent\hyperlink{ForGeneralLie2Algebras}{For general Lie 2-algebras}\dotfill \pageref*{ForGeneralLie2Algebras} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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 $\mathfrak{g}$ a [[Lie 2-algebra]] the \textbf{2-groupoid of $\mathfrak{g}$-valued forms} is the [[2-groupoid]] whose objects are [[differential form]]s with values in $\mathfrak{g}$, whose morphisms are [[gauge transformation]]s between these, and whose 2-morphisms are \emph{higher order gauge transformations} of those.

This naturally refines to a non-[[concrete sheaf|concrete]] [[Lie 2-groupoid]] is the 2-[[truncated]] [[∞-Lie groupoid]] whose $U$-parameterized smooth families of objects are smooth [[differential form]]s with values in a [[Lie 2-algebra]], and whose morphisms are gauge transformations of these.

This is the [[higher category theory|higher category]] generalization of the [[groupoid of Lie-algebra valued forms]].

A [[cocycle]] with coefficients in this 2-groupoid is a [[connection on a 2-bundle]].

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

\hypertarget{for_strict_lie_2algebras}{}\subsubsection*{{For strict Lie 2-algebras}}\label{for_strict_lie_2algebras}

Consider a Lie [[strict 2-group]] $G$ corresponding to a Lie [[crossed module]] $(G_2 \stackrel{\delta}{\to} G_1)$ with action $\alpha : G_1 \to Aut(G_2)$. Write $\mathbf{B}G$ for the corresponding [[delooping]] 2-groupoid, the one coming from the [[crossed complex]]

\begin{displaymath}
[\mathbf{B}G]
  =
  (G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *)
  \,.
\end{displaymath}
Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding [[differential crossed module]] with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$

\begin{udef}
The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack

\begin{displaymath}
\bar \mathbf{B}G : CartSp{}^{op} \to 2Grpd
\end{displaymath}
which assigns to $U \in CartSp$ the following 2-groupoid:

\begin{itemize}%
\item An [[object]] is a pair

\begin{displaymath}
A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\;
  B \in \Omega^2(U,\mathfrak{g}_2)
  \,.
\end{displaymath}

\item A 1-[[morphism]] $(g,a) : (A,B) \to (A',B')$ is a pair

\begin{displaymath}
g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)
\end{displaymath}
such that

\begin{displaymath}
A' = Ad_{g^{-1}}\left( A + \delta_* a \right) + g^{-1} d g
\end{displaymath}
and

\begin{displaymath}
B' = \alpha_{g^{-1}}(
    B + d a + [a \wedge a] + \alpha_*(A \wedge a)
  )
  \,.
\end{displaymath}
The composite of two 1-morphisms

\begin{displaymath}
(A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to}
  (A'', B'')
\end{displaymath}
is given by the pair

\begin{displaymath}
(g_1 g_2, a_1 + (\alpha_{g_2})_* a_2)
  \,.
\end{displaymath}

\item a [[2-morphism]] $f : (g,a) \Rightarrow (g', a'):(A,B)\to (A',B')$ is a function

\begin{displaymath}
f \in C^\infty(U,G_2)
\end{displaymath}
such that

\begin{displaymath}
g' = \delta(f)^{-1} \cdot g
\end{displaymath}
and

\begin{displaymath}
a' = Ad_{f^{-1}} \left(a  + (r_f^{-1} \circ \alpha_f)_*(A)\right) + f^{-1} d f
\end{displaymath}


\end{itemize}
and composition is defined as follows: vertical composition is given by pointwise multiplication ([[David Roberts|DR]]: the order still needs sorting out!) and horizontal composition is given as horizontal composition in the one-object 2-groupoid $\mathbf{B}G)$.

\end{udef}
\hypertarget{ForGeneralLie2Algebras}{}\subsubsection*{{For general Lie 2-algebras}}\label{ForGeneralLie2Algebras}

We consider now $\mathfrak{g}$ a general [[Lie 2-algebra]].

Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let

\begin{displaymath}
\{t^a\} \,, \;\;\; \{b^i\}
\end{displaymath}
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general [[Chevalley-Eilenberg algebra]]

\begin{displaymath}
CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}
\end{displaymath}
with these generators.

We thus have

\begin{displaymath}
d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i
\end{displaymath}
\begin{displaymath}
d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j  
    - 
    r_{a b c} t^a \wedge t^b \wedge t^c
  \,,
\end{displaymath}
for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the [[action]] of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the ``Jacobiator'' for the bracket on $\mathfrak{g}_0$).

These constants are subject to constraints (the weak [[Jacobi identity]] and its higher [[coherence law]]s) which are precisely equivalent to the condition

\begin{displaymath}
(d_{CE(\mathfrak{g})})^2 = 0
  \,.
\end{displaymath}
Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism

\begin{displaymath}
\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) 
   : 
  (A,B)
\end{displaymath}
from the [[Weil algebra]] $W(\mathfrak{g})$.

This is given by a 1-form

\begin{displaymath}
A \in \Omega^1(U, \mathfrak{g}_0)
\end{displaymath}
and a 2-form

\begin{displaymath}
B \in \Omega^2(U, \mathfrak{g}_1)
  \,.
\end{displaymath}
The [[curvature]] of this is $(\beta, H)$, where the 2-form component (``fake curvature'') is

\begin{displaymath}
\beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c
  + r^a{}_i B^i
\end{displaymath}
and whose 3-form component is

\begin{displaymath}
H^i
     =
    d_{dR} B^i 
    +
   \alpha^i{}_{a j} A^a \wedge B^j
   +
   t_{a b c} A^a \wedge A^b \wedge A^c
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{udef}
\textbf{(flat Lie 2-algebra valued forms)}

The full sub-2-groupoid on \emph{flat} Lie 2-algebra valued forms, i.e. those pairs $(A,B)$ for which the 2-form [[curvature]]

\begin{displaymath}
\delta_* B - d A + [A \wedge A] = 0
\end{displaymath}
and the 3-form [[curvature]]

\begin{displaymath}
d B + [A \wedge B] = 0
\end{displaymath}
vanishes is a resolution of the underlying discrete Lie 2-groupoid $\mathbf{\flat} \mathbf{B}G$ of the Lie 2-groupoid $\mathbf{B}G$.

\end{udef}
This is discussed at [[∞-Lie groupoid]] in the section .

\begin{udef}
Let $\mathbf{\Pi}_2 : CartSp \to 2LieGrpd$ be the smooth 2-[[fundamental groupoid]] functor and let $P_2 : CartSp \to 2LieGrpd$ be the [[path n-groupoid|path 2-groupoid]] functor, taking values in the 2-catgeory $2Grpd(Difeol)$ of 2-groupoids [[internal category|internalization]] to [[diffeological space]]s. Then

\begin{itemize}%
\item the 2-groupoid of Lie 2-algebra valued forms for which both 2- and 3-form curvature vanish is canonically equivalent to

\begin{displaymath}
Hom_{2Grpd(Diffeol)}(\Pi_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd
  \,;
\end{displaymath}

\item the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to

\begin{displaymath}
Hom_{2Grpd(Diffeol)}(P_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd
  \,;
\end{displaymath}


\end{itemize}
\end{udef}
The equivalence is given by 2-dimensional [[parallel transport]]. A proof is in \href{http://arxiv.org/abs/0802.0663}{SchrWalII}.

The following proposition asserts that the Lie 2-groupoid of Lie 2-algebra valued forms is the coefficient object for for \emph{differential nonabelian cohomology} in degree 2, namely for \emph{connections} on [[principal 2-bundle]]s and in particular on [[gerbe]]s.

\begin{udef}
\textbf{(2-bundles with connection)}

For $X$ a [[paracompact space|paracompact]] [[smooth manifold]] and $\{U_i \to X\}$ a [[good open cover]] the 2-groupoid, let $X \stackrel{\simeq}{\leftarrow} C(\{U_i\})$ be the corresponding [[Cech nerve]] smooth 2-groupoid. Then

\begin{displaymath}
Hom_{2Grpd(Diffeol)}( C(\{U_i\}), \bar \mathbf{B}G)
\end{displaymath}
is equivalent to the [[2-groupoid]] of $G$-[[principal 2-bundle]]s with [[connection on a 2-bundle|2-connection]].

\end{udef}
This is discussed and proven in \href{http://arxiv.org/abs/0802.0663}{SchrWalII} for the case where the 2-form curvature is restricted to vanish. In this case the above can be written as

\begin{displaymath}
Hom( C(\{U_i\}), Hom(P_2(-), \mathbf{B}G))
  \simeq
  Hom(P_2(C(\{\U_i\})), \mathbf{B}G)
  \,,
\end{displaymath}
where $P_2(C(\{\U_i\}) \in 2LieGrpd$ is a resolution of the [[path 2-groupoid]] of $X$.

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

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


\item \textbf{2-groupoid of Lie 2-algebra valued forms}

\begin{itemize}%
\item [[nonabelian Stokes theorem]]

\end{itemize}

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


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



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

The 2-groupoid of Lie 2-algebra valued forms described in \href{http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.0663v3.pdf#page=27}{definition 2.11} of

\begin{itemize}%
\item Schreiber, Waldorf, \emph{Smooth functors versus differential forms} ().

\end{itemize}
There are many possible conventions. The one reproduced above is supposed to describe the \emph{bidual} [[opposite 2-category]] of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.

See also

.

[[!redirects Lie 2-algebra valued 2-form]]

[[!redirects Lie 2-algebra valued forms]] [[!redirects Lie 2-algebra valued differential forms]] [[!redirects Lie 2-algebra valued form]]



\end{document}