\documentclass{beamer}

\usepackage{beamerthemesplit}
\usepackage{amssymb}
\usepackage[all]{xy}

\setbeamercovered{transparent}

\usetheme{Antibes}
\usecolortheme{default}

\title{On String and Chern-Simons $n$-Transport}
\author{Urs Schreiber}
\institute{\emph{with} \\ Jim Stasheff 
\\
\hspace{1pt}
\\ \emph{based in parts on work with}\\
John Baez 
\\ Alissa Crans
\\ David Roberts 
\\ Danny Stevenson
\\ Todd Trimble
\\ Konrad Waldorf }
\date{\today}

\def\of #1{(#1)}
\def\path {\gamma}
\def\surface {S}
\def\transport {\mathrm{tra}}
\def\trivializationTransition {f}
\def\trivializationTransitionModification {f}


\def\qDGCA {qDGCA}
\def\wedgebullet {\mbox{$\bigwedge^\bullet$}}
\def\gg {\mathfrak{g}}
\def\ff {\mathfrak{f}}
\def\hh {\mathfrak{h}}

\begin{document}









\frame{\titlepage}



\frame{
  \frametitle{Table of Contents}


\begin{itemize}

  \item Introduction

    \begin{enumerate}

       \item \underline{\hyperlink{motivation}{Motivation}}

       \item \underline{\hyperlink{plan}{Plan}}

    \end{enumerate}

  
  \item Integral Picture: Parallel $G_{(n)}$-transport

    \begin{itemize}

      \item \underline{\hyperlink{parallel transport}{Parallel $n$-Transport }}

    \end{itemize}
  
  \item Differential Picture: $\gg_{(n)}$-connections

     \begin{enumerate}
     
    \item  \underline{\hyperlink{Lie n-algebra cohomology}{Lie $n$-algebra cohomology}}


    \item  \underline{\hyperlink{n-bundles with connection}{Bundles with Lie $n$-algebra connection}}

    \item  \underline{\hyperlink{examples of gn bundles}{Examples of $\gg_{(n)}$-bundles}}


     \end{enumerate}


  \item Epilogue

    \begin{enumerate}

        \item \underline{\hyperlink{conclusion}{Conclusion}}

        \item \underline{\hyperlink{questions}{Questions}}

    \end{enumerate}

  \item  \underline{\hyperlink{n-categorical background}{$n$-Categorical Background}}


\end{itemize}

}


 
\section{Motivation}

\frame{
  \hypertarget{motivation}{}


  \begin{enumerate}
            
    \item Motivation
      
      \begin{enumerate}
      
        \item
        
          \underline{
            \hyperlink{brief statement of main motivation}{Brief statement of the main motivation}
          }
      
         \item
           \underline{
             \hyperlink{Extended n-functorial QFT}{Extended $n$-functorial QFT}
           }
      
         \item
           \underline{
             \hyperlink{charged n-particle}{The ``charged $n$-particle''}
           }

         \item
           \underline{
             \hyperlink{n-transport}{Parallel $n$-Transport}
           }


      \end{enumerate}

    \item \uncover<0>{Plan}
            
    \item \uncover<0>{Parallel $n$-transport}

    \item  \uncover<0>{Lie $n$-algebra cohomology}

    \item  \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item  \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item  \uncover<0>{Conclusion}

    \item  \uncover<0>{Questions}

    \item  \uncover<0>{$n$-Categorical background}

  \end{enumerate}

}


\subsection{Brief statement of the main motivation}

\frame{
  \hypertarget{brief statement of main motivation}{}

\begin{block}{Local Motivation}

  We want to get a handle on the theory and classification
  of $n$-bundles with $n$-functorial connections, in particular
  \begin{itemize}
    \item
      String 2-bundles
    \item
      Chern-Simons 3-bundles.
  \end{itemize}

\end{block}

 \pause

\begin{block}{Global Motivation}

  We want to understand how the FRS description of 2-dimensional
  rational CFT generalizes to non-rational CFT and to SCFT.
  We have a bunch of hints that FRS is the 
  \begin{itemize}
    \item local trivialization data
    \item of a certain push-forward (``quantization'')
    \item of a transformation of parallel transport 3-functors
    \item describing a connection on a 3-bundle.
  \end{itemize}

\end{block}

}


\frame{

  There is much more to say about motivation.
  
  A couple of more details are given in 
  \underline{\hyperlink{Extended n-functorial QFT}{the following}}.

  To skip further motivation

  \begin{itemize}

    \item continue with 
      \underline{\hyperlink{plan}{the plan}} of the further discussion
      
      
    \item
    
       or go directly to the detailed discussion at
       \underline{
         \hyperlink{Lie n-algebra cohomology}{Lie $n$-algebra cohomology}
       }

    \item
    
       or jump to the 
       \underline{
         \hyperlink{conclusion}{Conclusion}
       }.    


  \end{itemize}

}


\subsection{Extended $n$-functorial quantum field theory}

\frame
{
  \frametitle{A Quantum Field Theory is a Functor}
  \hypertarget{Extended n-functorial QFT}{}

  \begin{itemize}
  \item<2-> 
      Atiyah and Segal have famously axiomatized $d$-dimensional QFTs
  \item<3->  
     as functors
     $$
       \mathrm{Z} : n\mathrm{Cob}_S \to \mathrm{Vect}
     $$
  \item<4-> 
     $$
       Z
       \hspace{5pt} 
       : 
       \hspace{5pt} 
       \left(
        \xymatrix{
          \partial_{\mathrm{in}} \Sigma 
          \ar[rr]^{(\Sigma,g)}
          &&
          \partial_{\mathrm{out}} \Sigma
        }
       \right)
       \hspace{5pt} 
         \mapsto
       \hspace{5pt} 
       \left(
       \xymatrix{
          H_{\mathrm{in}} 
          \ar[rr]^{U(\Sigma,g)}
          &&
          H_{\mathrm{out}}
       }
       \right)
       \,.
     $$
  \end{itemize}
}


\frame{
  \frametitle{Cartoon of a 1-functorial QFT}

$
  \uncover<8->{\langle} 
  \uncover<7->{\phi} 
  \uncover<6->{|} \uncover<5->{U(\Sigma)} 
  \uncover<4->{|} \uncover<3->{\psi} \uncover<2->{\rangle}
$


\only<1>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){}}="Ct";
    (30,-12)*{\makebox(14,14){}}="Vx";
    (20,-48)*{\makebox(14,14){}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
  \endxy
$$
}


\only<2>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){}}="Vx";
    (20,-48)*{\makebox(14,14){}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
  \endxy
$$
}

\only<3>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){}}="Vx";
    (20,-48)*{\makebox(14,14){}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar^{|\psi\rangle} "Ct"; "Vx"
  \endxy
$$
}

\only<4>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){$H$}}="Vx";
    (20,-48)*{\makebox(14,14){}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar^{|\psi\rangle} "Ct"; "Vx"
  \endxy
$$
}

\only<5>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){$H$}}="Vx";
    (20,-48)*{\makebox(14,14){}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\Sigma)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar^{|\psi\rangle} "Ct"; "Vx"
  \endxy
$$
}

\only<6>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){$H$}}="Vx";
    (20,-48)*{\makebox(14,14){$H$}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\Sigma)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar^{|\psi\rangle} "Ct"; "Vx"
  \endxy
$$
}

\only<7>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){$H$}}="Vx";
    (20,-48)*{\makebox(14,14){$H$}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\Sigma)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar^{|\psi\rangle} "Ct"; "Vx"
    \ar^>>>{\langle\phi|} "Vy"; "Cd"
  \endxy
$$
}


\only<8>{
$$
 \xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (30,0)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (30,-12)*{\makebox(14,14){$H$}}="Vx";
    (20,-48)*{\makebox(14,14){$H$}}="Vy";
    (25,-27)="c";
    (20,-62)*{\makebox(14,14){$\mathbb{C}$}}="Cd";
    %
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\Sigma)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar^{|\psi\rangle} "Ct"; "Vx"
    \ar^>>>{\langle\phi|} "Vy"; "Cd"
  \endxy
$$
}


}


\frame
{
  \frametitle{A Quantum Field Theory is an $n$-Functor}

    \uncover<1->{But later it was noticed that this is too imprecise}
    \uncover<2->{if we want to be able to talk about}

 \pause
 \pause

  \begin{block}{crucial requirements on QFT description}

   \begin{itemize}
     \item<3-> \emph{locality}
     \item<4-> \emph{boundary conditions}.
   \end{itemize}
   
  \end{block}

  \pause
  
  \uncover<5->{Instead:}

  \begin{block}{refined picture}
        \uncover<6->{An $n$-dimensional QFT should be an \alert<6>{$n$-functor}.}
        \\
        \uncover<7->{[Freed, Hopkins, Stolz, Teichner]}
   \end{block}

   (\hyperlink{n-categories}{\underline{remark on $n$-categories}})
   \hypertarget{towards n-functorial QFT}{}

}



\frame{
  \frametitle{Cartoon of a 2-functorial QFT}


\only<1>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){}}="Ct";
    (40,-12)*{\makebox(14,14){}}="Vx";
    (30,-48)*{\makebox(14,14){}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){}}="Ctl";
    (20,-12)*{\makebox(14,14){}}="Vxl";
    (10,-48)*{\makebox(14,14){}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
  \endxy
$$    
}

\only<2>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){}}="Vx";
    (30,-48)*{\makebox(14,14){}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){}}="Vxl";
    (10,-48)*{\makebox(14,14){}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar "Ctl"; "Ct"
  \endxy
$$    
}

\only<3>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){}}="Vx";
    (30,-48)*{\makebox(14,14){}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){}}="Vxl";
    (10,-48)*{\makebox(14,14){}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    %
    \ar "Ct"; "Vx"
    \ar "Ctl"; "Vxl"
    \ar "Ctl"; "Ct"
    %
    \ar@{=>}^{|\psi\rangle} (31,-4)+(.4,4); (31,-4)+(-.4,-4) 
  \endxy
$$    
}

\only<4>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){$H_r$}}="Vx";
    (30,-48)*{\makebox(14,14){}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){$H_l$}}="Vxl";
    (10,-48)*{\makebox(14,14){}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    %\ar@/^.4pc/@{-}^>>{U(\gamma_2)} "Vx"; "c"
    %\ar@/_.6pc/ "c"; "Vy"
    %
    \ar "Ct"; "Vx"
    \ar "Vxl"; "Vx"
    %\ar "Vy"; "Cd"
    %
    %\ar@/^.4pc/@{-}_>>{U(\gamma_1)} "Vxl"; "cl"
    %\ar@/_.6pc/ "cl"; "Vyl"
    %
    \ar "Ctl"; "Vxl"
    %\ar "Vyl"; "Cdl"
    %
    \ar "Ctl"; "Ct"
    %\ar "Cdl"; "Cd"
    %\ar@{-}@/^.3pc/ "Vxl"; "ct"
    %\ar@/_.3pc/ "ct"; "Vx"
    %\ar@{-}@/^.3pc/ "Vyl"; "cb"
    %\ar@/_.3pc/ "cb"; "Vy"
    %
    \ar@{=>}^{|\psi\rangle} (31,-4)+(.4,4); (31,-4)+(-.4,-4) 
    %\ar@{=>}^{\langle \phi |} (20,-55)+(.4,4); (20,-55)+(-.4,-4) 
    %\ar@{=>}|{U(\Sigma)} (24,-29)+(4,12); (24,-29)+(-2,-12) 
  \endxy
$$    
}


\only<5>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){$H_r$}}="Vx";
    (30,-48)*{\makebox(14,14){}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){$H_l$}}="Vxl";
    (10,-48)*{\makebox(14,14){}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\gamma_2)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar "Ct"; "Vx"
    %\ar "Vy"; "Cd"
    %
    \ar@/^.4pc/@{-}_>>{U(\gamma_1)} "Vxl"; "cl"
    \ar@/_.6pc/ "cl"; "Vyl"
    %
    \ar "Ctl"; "Vxl"
    %\ar "Vyl"; "Cdl"
    %
    \ar "Ctl"; "Ct"
    %\ar "Cdl"; "Cd"
    \ar@{-}@/^.3pc/ "Vxl"; "ct"
    \ar@/_.3pc/ "ct"; "Vx"
    %\ar@{-}@/^.3pc/ "Vyl"; "cb"
    %\ar@/_.3pc/ "cb"; "Vy"
    %
    \ar@{=>}^{|\psi\rangle} (31,-4)+(.4,4); (31,-4)+(-.4,-4) 
    %\ar@{=>}^{\langle \phi |} (20,-55)+(.4,4); (20,-55)+(-.4,-4) 
    \ar@{=>}|{U(\Sigma)} (24,-29)+(4,12); (24,-29)+(-2,-12) 
  \endxy
$$    
}

\only<6>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){$H_r$}}="Vx";
    (30,-48)*{\makebox(14,14){$H_r$}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){$H_l$}}="Vxl";
    (10,-48)*{\makebox(14,14){$H_l$}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\gamma_2)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar "Ct"; "Vx"
    %\ar "Vy"; "Cd"
    %
    \ar@/^.4pc/@{-}_>>{U(\gamma_1)} "Vxl"; "cl"
    \ar@/_.6pc/ "cl"; "Vyl"
    %
    \ar "Ctl"; "Vxl"
    %\ar "Vyl"; "Cdl"
    %
    \ar "Ctl"; "Ct"
    %\ar "Cdl"; "Cd"
    \ar@{-}@/^.3pc/ "Vxl"; "ct"
    \ar@/_.3pc/ "ct"; "Vx"
    \ar@{-}@/^.3pc/ "Vyl"; "cb"
    \ar@/_.3pc/ "cb"; "Vy"
    %
    \ar@{=>}^{|\psi\rangle} (31,-4)+(.4,4); (31,-4)+(-.4,-4) 
    %\ar@{=>}^{\langle \phi |} (20,-55)+(.4,4); (20,-55)+(-.4,-4) 
    \ar@{=>}|{U(\Sigma)} (24,-29)+(4,12); (24,-29)+(-2,-12) 
  \endxy
$$    
}



\only<7>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){$H_r$}}="Vx";
    (30,-48)*{\makebox(14,14){$H_r$}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){$H_l$}}="Vxl";
    (10,-48)*{\makebox(14,14){$H_l$}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\gamma_2)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar "Ct"; "Vx"
    \ar "Vy"; "Cd"
    %
    \ar@/^.4pc/@{-}_>>{U(\gamma_1)} "Vxl"; "cl"
    \ar@/_.6pc/ "cl"; "Vyl"
    %
    \ar "Ctl"; "Vxl"
    \ar "Vyl"; "Cdl"
    %
    \ar "Ctl"; "Ct"
    %\ar "Cdl"; "Cd"
    \ar@{-}@/^.3pc/ "Vxl"; "ct"
    \ar@/_.3pc/ "ct"; "Vx"
    \ar@{-}@/^.3pc/ "Vyl"; "cb"
    \ar@/_.3pc/ "cb"; "Vy"
    %
    \ar@{=>}^{|\psi\rangle} (31,-4)+(.4,4); (31,-4)+(-.4,-4) 
    \ar@{=>}^{\langle \phi |} (20,-55)+(.4,4); (20,-55)+(-.4,-4) 
    \ar@{=>}|{U(\Sigma)} (24,-29)+(4,12); (24,-29)+(-2,-12) 
  \endxy
$$    
}



\only<8>{
$$
\xy0;/r.21pc/:
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    %
    (40,3)*{\makebox(14,14){$\mathbb{C}$}}="Ct";
    (40,-12)*{\makebox(14,14){$H_r$}}="Vx";
    (30,-48)*{\makebox(14,14){$H_r$}}="Vy";
    (35,-27)="c";
    (30,-62)*{\makebox(14,14){$\mathbb{C}$}}="Cd";
    %
    (20,3)*{\makebox(14,14){$\mathbb{C}$}}="Ctl";
    (20,-12)*{\makebox(14,14){$H_l$}}="Vxl";
    (10,-48)*{\makebox(14,14){$H_l$}}="Vyl";
    (15,-27)="cl";
    (10,-62)*{\makebox(14,14){$\mathbb{C}$}}="Cdl";
    %
    %
    (30,-12)="ct";    
    (20,-48)="cb";    
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
    \ar@/^.4pc/@{-}^>>{U(\gamma_2)} "Vx"; "c"
    \ar@/_.6pc/ "c"; "Vy"
    %
    \ar "Ct"; "Vx"
    \ar "Vy"; "Cd"
    %
    \ar@/^.4pc/@{-}_>>{U(\gamma_1)} "Vxl"; "cl"
    \ar@/_.6pc/ "cl"; "Vyl"
    %
    \ar "Ctl"; "Vxl"
    \ar "Vyl"; "Cdl"
    %
    \ar "Ctl"; "Ct"
    \ar "Cdl"; "Cd"
    \ar@{-}@/^.3pc/ "Vxl"; "ct"
    \ar@/_.3pc/ "ct"; "Vx"
    \ar@{-}@/^.3pc/ "Vyl"; "cb"
    \ar@/_.3pc/ "cb"; "Vy"
    %
    \ar@{=>}^{|\psi\rangle} (31,-4)+(.4,4); (31,-4)+(-.4,-4) 
    \ar@{=>}^{\langle \phi |} (20,-55)+(.4,4); (20,-55)+(-.4,-4) 
    \ar@{=>}|{U(\Sigma)} (24,-29)+(4,12); (24,-29)+(-2,-12) 
  \endxy
$$    
}



}



\frame{
  \frametitle{Cartoon of a 3-functorial QFT}


\only<1>{
$$
\xy
  %% top objects $$
  %%%%%%%%%%%%%%%%%
  (-12,0)+(0,17)*{\makebox(11,11){\small $$}}="topleft";
  (12,0)+(0,17)*{\makebox(10,11){\small $$}}="topright";
  %
  (2,11.5)+(0,17)*{\makebox(8,8){\tiny $H_l$}}="topback";
  (-4,-12.8)+(0,17)*{\makebox(12,12){\large $H_r$}}="topfront";
  %
  %% bottom objects $$
  %%%%%%%%%%%%%%%%%%%%
  (-12,0)+(0,-17)*{\makebox(11,11){\small $$}}="bottomleft";
  (12,0)+(0,-17)*{\makebox(10,11){\small $$}}="bottomright";
  %
  (2,11.5)+(0,-17)*{\makebox(8,8){}}="bottomback";
  (-4,-12.8)+(0,-17)*{\makebox(12,12){}}="bottomfront";
  %
  %
  %% top morphisms %%
  %%%%%%%%%%%%%%%%%%%
  \ar|{\partial_{\mathrm{in}} \Sigma} "topleft"; "topright"
  %\ar@/_1.6pc/@{-->}|{\mbox{\tiny $\surface_2$}} "topleft"; "topright"
  %
  \ar@/_.65pc/@{-} "topback"; "topleft"="test"
  \ar@/^.7pc/@{-} "topback"; "topright"
  \ar@/_.7pc/ "topleft"; "topfront"
  \ar@/^.7pc/ "topright"; "topfront"
  %
  %\ar@{=>}|V (0,4)+(0,17); (0,-3)+(0,17)
  %
  %% bottom morphisms %%
  %%%%%%%%%%%%%%%%%%%%%%
  %\ar@{-->}|{\partial_{\mathrm{out}}\Sigma} "bottomleft"; "bottomright"
  %\ar@{-->}|{\mbox{\tiny $\surface_2$}} "bottomleft"; "bottomright"
  %
  %\ar@/_.65pc/@{--} "bottomback"; "bottomleft"
  %\ar@/^.7pc/@{--} "bottomback"; "bottomright"
  %\ar@/_.7pc/ "bottomleft"; "bottomfront"
  %\ar@/^.7pc/ "bottomright"; "bottomfront"
  %
  %\ar@{=>}|V (0,4)+(0,-17); (0,-3)+(0,-17)
  %
  %
\endxy
$$
}



\only<2>{
$$
\xy
  %% top objects $$
  %%%%%%%%%%%%%%%%%
  (-12,0)+(0,17)*{\makebox(11,11){\small $$}}="topleft";
  (12,0)+(0,17)*{\makebox(10,11){\small $$}}="topright";
  %
  (2,11.5)+(0,17)*{\makebox(8,8){\tiny $H_l$}}="topback";
  (-4,-12.8)+(0,17)*{\makebox(12,12){\large $H_r$}}="topfront";
  %
  %% bottom objects $$
  %%%%%%%%%%%%%%%%%%%%
  (-12,0)+(0,-17)*{\makebox(11,11){\small $$}}="bottomleft";
  (12,0)+(0,-17)*{\makebox(10,11){\small $$}}="bottomright";
  %
  (2,11.5)+(0,-17)*{\makebox(8,8){}}="bottomback";
  (-4,-12.8)+(0,-17)*{\makebox(12,12){}}="bottomfront";
  %
  %
  %% top morphisms %%
  %%%%%%%%%%%%%%%%%%%
  \ar|{\partial_{\mathrm{in}} \Sigma} "topleft"; "topright"
  %\ar@/_1.6pc/@{-->}|{\mbox{\tiny $\surface_2$}} "topleft"; "topright"
  %
  \ar@/_.65pc/@{-} "topback"; "topleft"="test"
  \ar@/^.7pc/@{-} "topback"; "topright"
  \ar@/_.7pc/ "topleft"; "topfront"
  \ar@/^.7pc/ "topright"; "topfront"
  %
  %\ar@{=>}|V (0,4)+(0,17); (0,-3)+(0,17)
  %
  %% bottom morphisms %%
  %%%%%%%%%%%%%%%%%%%%%%
  %\ar@{-->}|{\partial_{\mathrm{out}}\Sigma} "bottomleft"; "bottomright"
  %\ar@{-->}|{\mbox{\tiny $\surface_2$}} "bottomleft"; "bottomright"
  %
  %\ar@/_.65pc/@{--} "bottomback"; "bottomleft"
  %\ar@/^.7pc/@{--} "bottomback"; "bottomright"
  %\ar@/_.7pc/ "bottomleft"; "bottomfront"
  %\ar@/^.7pc/ "bottomright"; "bottomfront"
  %
  %\ar@{=>}|V (0,4)+(0,-17); (0,-3)+(0,-17)
  %
  %
  %% topdown morphisms %%
  %%%%%%%%%%%%%%%%%%%%%%%
  \ar "topfront"; "bottomfront"
  \ar@{-->} "topback"; "bottomback"
  %
  \ar@/^.29pc/ (12.6,3)+(0,1); (10,-5)+(0,1)
  \ar@/_.12pc/@{-->} (-11.1,3)+(0,1); (-12.6,-2.4)+(0,1)
  %
  \ar@{==>}|{U(\Sigma)} (0,2); (0,-9)
\endxy
$$
}


\only<3>{
$$
\xy
  %% top objects $$
  %%%%%%%%%%%%%%%%%
  (-12,0)+(0,17)*{\makebox(11,11){\small $$}}="topleft";
  (12,0)+(0,17)*{\makebox(10,11){\small $$}}="topright";
  %
  (2,11.5)+(0,17)*{\makebox(8,8){\tiny $H_l$}}="topback";
  (-4,-12.8)+(0,17)*{\makebox(12,12){\large $H_r$}}="topfront";
  %
  %% bottom objects $$
  %%%%%%%%%%%%%%%%%%%%
  (-12,0)+(0,-17)*{\makebox(11,11){\small $$}}="bottomleft";
  (12,0)+(0,-17)*{\makebox(10,11){\small $$}}="bottomright";
  %
  (2,11.5)+(0,-17)*{\makebox(8,8){\tiny $H_l$}}="bottomback";
  (-4,-12.8)+(0,-17)*{\makebox(12,12){\large $H_r$}}="bottomfront";
  %
  %
  %% top morphisms %%
  %%%%%%%%%%%%%%%%%%%
  \ar|{\partial_{\mathrm{in}} \Sigma} "topleft"; "topright"
  %\ar@/_1.6pc/@{-->}|{\mbox{\tiny $\surface_2$}} "topleft"; "topright"
  %
  \ar@/_.65pc/@{-} "topback"; "topleft"="test"
  \ar@/^.7pc/@{-} "topback"; "topright"
  \ar@/_.7pc/ "topleft"; "topfront"
  \ar@/^.7pc/ "topright"; "topfront"
  %
  %\ar@{=>}|V (0,4)+(0,17); (0,-3)+(0,17)
  %
  %% bottom morphisms %%
  %%%%%%%%%%%%%%%%%%%%%%
  \ar@{-->}|{\partial_{\mathrm{out}}\Sigma} "bottomleft"; "bottomright"
  %\ar@{-->}|{\mbox{\tiny $\surface_2$}} "bottomleft"; "bottomright"
  %
  \ar@/_.65pc/@{--} "bottomback"; "bottomleft"
  \ar@/^.7pc/@{--} "bottomback"; "bottomright"
  \ar@/_.7pc/ "bottomleft"; "bottomfront"
  \ar@/^.7pc/ "bottomright"; "bottomfront"
  %
  %\ar@{=>}|V (0,4)+(0,-17); (0,-3)+(0,-17)
  %
  %
  %% topdown morphisms %%
  %%%%%%%%%%%%%%%%%%%%%%%
  \ar "topfront"; "bottomfront"
  \ar@{-->} "topback"; "bottomback"
  %
  \ar@/^.29pc/ (12.6,3)+(0,1); (10,-5)+(0,1)
  \ar@/_.12pc/@{-->} (-11.1,3)+(0,1); (-12.6,-2.4)+(0,1)
  %
  \ar@{==>}|{U(\Sigma)} (0,2); (0,-9)
\endxy
$$
}

}



\subsection{The ``charged $n$-Particle''}


\frame
{
  \hypertarget{charged n-particle}{}

  \frametitle{$n$-Particles and $(n-1)$-Branes}

  \uncover<1->{It follows that the action of the \emph{$n$-particle}\dots}

  \pause

  \begin{block}{$n$-Particle}

   \begin{itemize}
    \item<2-> \hspace{7pt}$n=1$: the point particle
    \item<3-> \hspace{7pt}$n=2$: the string
    \item<4-> \hspace{7pt}$n=3$: the membrane
    \item<5-> \hspace{7pt}$n$-particle $\simeq$ $(n-1)$-brane
   \end{itemize}

  \end{block}

}


\frame
{
  \frametitle{$n$-Bundles and $(n-1)$-Gerbes}

  
  It follows that the action of the \emph{$n$-particle}\\
  \uncover<1->{charged under an \emph{$n$-bundle with connection}\dots}


  \pause

  \begin{block}{$n$-background fields}

   \begin{itemize}
    \item<2-> \hspace{7pt}$n=1$: the electromagnetic field
    \item<3-> \hspace{7pt}$n=2$: the Kalb-Ramond field
    \item<4-> \hspace{7pt}$n=3$: the supergravity 3-form field
    \item<5-> \hspace{7pt}$n$-bundle $\simeq$ $(n-1)$-gerbe 
  \end{itemize}

  \end{block}

}

\frame
{
  \frametitle{Parallel $n$-Transport}

  \begin{itemize}
  \item<1->[] It follows that the action of the \emph{$n$-particle}
    \\
    charged under an \emph{$n$-bundle with connection}
  \item<2->[] is itself an $n$-functor
  \item<3-> \hspace{7pt}$\mathrm{tra}_1$ : 
    $
      \left(
      \xymatrix{
         x
         \ar[r]^\gamma
         &
         y
      }
      \right)
      \hspace{5pt}
        \mapsto
      \hspace{5pt}
      \left(
      \xymatrix{
         V_x
         \ar[rr]^{P \exp\left(\int_\gamma A\right)}
         &&
         V_y
      }
      \right)
    $
  \item<4-> \hspace{7pt}$\mathrm{tra}_2$ : 
    $
      \left(
      \xymatrix{
         x
         \ar@/^1.3pc/[r]^{\gamma_1}_{\ }="s"
         \ar@/_1.3pc/[r]_{\gamma_2}^{\ }="t"
         &
         y
         %
         \ar@{=>}^\Sigma "s"; "t"
      }
      \right)
      \hspace{5pt}
        \mapsto
      \hspace{5pt}
      \left(
      \xymatrix{
         V_x
         \ar@/^1.6pc/[rr]^{P \exp\left(\int_{\gamma_1} A\right)}_{\ }="s"
         \ar@/_1.6pc/[rr]_{P \exp\left(\int_{\gamma_2} A\right)}^{\ }="t"
         &&
         V_y
         %
         \ar@{=>}|{P_A \exp\left(\int_\Sigma B\right)} "s"; "t"
      }
      \right)
    $
  \end{itemize}
}


\frame
{
  \frametitle{Parallel $3$-Transport}

  \begin{itemize}
  \item<1->[] It follows that the action of the \emph{$3$-particle}
    \\
    charged under a \emph{$3$-bundle with connection}
    \\
    is itself a $3$-functor
  \item<2-> 
  \hspace{7pt}$\mathrm{tra}_3$: 
$
\left(
\xy
  (-12,0)*{\makebox(11,11){$\gamma_1$}}="left";
  (12,0)*{\makebox(10,11){$\gamma_2$}}="right";
  %
  (2,11.5)*{\makebox(8,8){\tiny $x$}}="back";
  (-4,-12.8)*{\makebox(12,12){\large $y$}}="front";
  %
  \ar@/^2pc/|{\Sigma_1} "left"; "right"
  \ar@/_1.6pc/@{-->}|{\mbox{\tiny $\Sigma_2$}} "left"; "right"
  %
  \ar@/_.65pc/@{-} "back"; "left"
  \ar@/^.7pc/@{-} "back"; "right"
  \ar@/_.7pc/ "left"; "front"
  \ar@/^.7pc/ "right"; "front"
  %
  \ar@{==>}^V (0,4); (0,-3)
\endxy
\right)
\hspace{5pt}
  \mapsto
\hspace{5pt}
\left(
\xy
  (-12,0)*{\makebox(11,11){$$}}="left";
  (12,0)*{\makebox(10,11){$$}}="right";
  %
  (2,11.5)*{\makebox(8,8){\tiny $V_x$}}="back";
  (-4,-12.8)*{\makebox(12,12){\large $V_y$}}="front";
  %
  \ar@/^2pc/|{} "left"; "right"
  \ar@/_1.6pc/@{-->}|{\mbox{\tiny $$}} "left"; "right"
  %
  \ar@/_.65pc/@{-} "back"; "left"
  \ar@/^.7pc/@{-} "back"; "right"
  \ar@/_.7pc/ "left"; "front"
  \ar@/^.7pc/ "right"; "front"
  %
  \ar@{=>}|{\mbox{\tiny $P_{A,B}\exp\left(\int_V C\right)$}} (0,6); (0,-5)
\endxy
\right)
$
\end{itemize}
}



\subsection{Parallel $n$-transport}


\frame
{
 \frametitle
 {  Parallel $n$-Transport }
 \hypertarget{n-transport}{}

  \begin{block}

   \uncover<1->{
      \alert<1>{
      A parallel $n$-transport}
      }
   \uncover<2->{\alert<2>{is} (locally)} 
   \uncover<3->
     {\alert<3>{an $n$-functor}} 
    \uncover<4->{
       from the \alert<3>{path $n$-groupoid} }
    \uncover<5->{
     to the \alert<5>{structure 
      \underline{\hyperlink{n-groups}{$n$-group}}}
    }.
    \hypertarget{back to n-groups}{}
 $$
   \uncover<1->{
     \alert<1>{
     \mathrm{tra}_n 
     }
   }
   \uncover<2->{
     \alert<2>{:}
   } 
   \uncover<4->{
     \alert<4>{\mathcal{P}_n(X)}
   }
   \uncover<3->{
     \alert<3>{\to}
   }
   \uncover<5->{
     \alert<5>{\Sigma G_{(n)}}
   }
 $$

\end{block}


\pause
\pause
\pause
\pause
\pause

\begin{block}{($n+1$)-Curvature}

   \uncover<6->{
      \alert<6>{
        Its $(n+1)$-curvature
      }
      }
   \uncover<7->{\alert<7>{is} (locally)} 
   \uncover<8->
     {\alert<8>{an $(n+1)$-functor}} 
    \uncover<9->{
       from the \alert<9>{fundamental $(n+1)$-groupoid} }
    \uncover<10->{
     to the \alert<10>{inner automorphism $(n+1)$-group} of $G_{(n)}$
    }.

 $$
   \uncover<6->{
     \alert<6>{
     d\mathrm{tra}_n := \mathrm{curv}_{(n+1)} 
     }
   }
   \uncover<7->{
     \alert<7>{:}
   } 
   \uncover<9->{
     \alert<9>{\Pi_{n+1}(X)}
   }
   \uncover<8->{
     \alert<8>{\to}
   }
   \uncover<10->{
     \alert<10>{\Sigma(\mathrm{INN}{G_{(n)}})}
   }
 $$
  

\end{block}

}




\frame
{
 \frametitle
 {  Tangent Categories }
 

\begin{block}{Inner automorphism $(n+1)$-Groups}

  \begin{itemize}
       \item<2-> Every $n$-group $G_{(n)}$ has an \alert<2>{$(n+1)$}-group 
          \alert<2>{$\mathrm{AUT}(G_{(n)})$} of automorphisms.
          
       \item<3-> This sits inside an exact sequence
         \uncover<4->{
         $
           \alert<4>{
           1 \to Z(G_{(n)}) \to \alert<5->{\mathrm{INN}(G_{(n)})} \to \mathrm{AUT}(G_{(n)})
           \to \mathrm{OUT}(G_{(n)}) \to 1
           }
         $
         }
        \item<5-> and $\mathrm{INN}_0$ plays the role of the universal $G_{(n)}$-bundle
        \uncover<6->{
          $
            \alert<6>{
              G_{(n)} \to \mathrm{INN}_0(G_{(n)}) \to \Sigma G_{(n)}
             }
          $
        }
  \end{itemize}
 
 \uncover<7->{We will re-encounter these crucial facts in their Lie $n$-algebra incarnation
     shortly.}

  \uncover<2->{[U.S., David Roberts]}

  \hypertarget{back to INN(G)}{}
  (\underline{\hyperlink{tangent categories}{on tangent categories}})
  (\underline{\hyperlink{inner automorphisms}{on inner automorphisms}})

\end{block}

}






\frame
{
 \frametitle{Some structure $n$-Groups}

\begin{block}{Important structure (1-)Groups}

\pause

   \begin{tabular}{crcl}
     \uncover<1->{electrically charged 1-particle:} 
     & \uncover<2->{$G_{(1)}$} & \uncover<2->{$=$}  & \uncover<3->{$ U(1)$}
     \\
     \uncover<4->{spinning 1-particle:} 
      & \uncover<4->{$G_{(1)}$} & \uncover<4->{$ = $} 
     & \uncover<5->{$ \mathrm{Spin}(n)$}
   \end{tabular}
  
\end{block}

\pause
\pause
\pause
\pause

  \uncover<6->{
\begin{block}{Important structure (2-)Groups}

   \begin{tabular}{crcl}
     \uncover<6->{Kalb-Ramond charged 2-particle:} 
     & \uncover<6->{$G_{(2)}$} & \uncover<6->{$ = $} & 
      \uncover<7->{$ \Sigma U(1)$}
     \\
     \uncover<8->{spinning 2-particle:} 
     & \uncover<8->{$G_{(2)}$} 
     & \uncover<8->{$ = $} & 
     \uncover<9->{$ \mathrm{String}_k(\mathrm{Spin}(n))$}
   \end{tabular}
  
\end{block}
}

  \only<6-7>{[Bartels],[Baez,S],[S,Waldorf]}
  \only<8-9>{[Baez,Crans,S,Stevenson]}


\pause
\pause
\pause
\pause

  \uncover<10->{
  \begin{block}{Important Structure 3-Groups}

   \begin{tabular}{crcl}

      \uncover<10->{Chern-Simons charged 3-particle:}
     & \uncover<10-> {$G_{(3)}$}
     & \uncover<10->{$=$} 
     & \uncover<11>{\alert<11>{ {\bf ?}}} 
   \end{tabular}
   
  \end{block}
  }



\pause 


  \uncover<12->{
    \alert<12>{
      Tough question.
    }
   }
  \uncover<13->{
    \alert<13>{
      Let's pass to the differential picture.
    }    
  }

}

\subsection{Connections with values in Lie $n$-algebras}

\hypertarget{main}{}

\frame{
  \frametitle{Finding the Chern-Simons Lie 3-algebra}

  \begin{block}{Problem}
  
   \uncover<1->{
    \alert<1>{
    Identify that class of 3-transport}} \uncover<2->{\alert<2>{-- given by its structure 3-group --}} 
    \uncover<3->{\alert<3>{which evaluates to the
    Chern-Simons functional on 3-dimensional morphisms}}.
  
  \end{block}
  

  \pause
  \pause
  \pause
  
   \begin{block}{Strategy}

    \begin{itemize}
    
      \item<4-> Differentiate. \uncover<5->{Pass from Lie $n$-groups to Lie $n$-algebras.}
      
      \item<5-> Find that Lie 3-algebra \uncover<6->{\alert<6>{$\mathrm{cs}_k(\gg)$}}
      \uncover<7->{with the property that connections taking values in it,}
      \uncover<8->{
        $
          \mathrm{Vect} \to \mathrm{cs}_k(\gg)
        $,
      }
      \uncover<9->{
        correspond to triples $(A,B,C)$ of forms 
        }\uncover<10->{such that \alert<10>{$C  = \mathrm{CS}_k(A) + d B$}.}
    \end{itemize}
  
  \end{block}

}



\frame
{
  \frametitle{From paralllel $n$-transport to Lie $n$-algebra valued connections}

  \begin{tabular}{cccccc}
  \hspace{-1.4cm}
   \uncover<2->{
    \begin{tabular}{c}
       {\bf Lie} 
       \\
       {\bf $n$-groupoids}
    \end{tabular}
    }
    &
    \uncover<4->{
      \xymatrix{\ar[r]^{\mbox{\small diff.}}&}
    }
    &
    \uncover<6->{
    \begin{tabular}{c}
      {\bf Lie $n$-algebras}
      \\
      ($\simeq$ $n$-term \\
       $L_\infty$-algebras)
    \end{tabular}
    }
    &
    \uncover<8->{
     $\simeq$
     }
    &
    \uncover<8->{
    \begin{tabular}{c}
       \hspace{-30pt}
     {\bf differential}
      \\
      \hspace{-30pt}
      {\bf algebras}
      \\
      \hspace{-30pt}
      ({\qDGCA}s)
    \end{tabular}
    }
    \\
    \hline
    \\
    \hspace{-1.4cm}
    \uncover<2->{
    $$
      \raisebox{40pt}{
      \xymatrix{
        \Sigma (\mathrm{INN}(G_{(n)}))
        \\
        \\
        \Pi_{n+1}(X)
        \ar[uu]_{F}
      }     
      }
    $$
    }
    &&
    \uncover<6->{
    $$
      \raisebox{40pt}{
      \xymatrix{
        \mathrm{inn}(\gg_{(n)})
        \\
        \\
        \mathrm{Vect}(X)
        \ar[uu]_{f}
      }     
      }
    $$
    }
    &&
    \uncover<8->{
    $$
      \raisebox{40pt}{
      \hspace{-30pt}
      \xymatrix{
        (\wedgebullet(s\gg^*_{n}\oplus ss\gg^*_{(n)}), d)
        \ar[dd]^{f^*}
        \\
        \\
        (\Omega^\bullet(X),d)
      }     
      }
    $$
    }
\end{tabular}

\vspace{6pt}

\only<1-2>{
  \alert<1-2>{
  Parallel $n$-transport is a morphism of Lie $(n+1)$-groupoids.
  }
}

\only<3-4>{
  \alert<3-4>{
  This morphism may be differentiated\dots
  }
}

\only<5-6>{
  \alert<5-6>{
  \dots to produce a morphism of Lie $(n+1)$-algebroids.  
  }
}

\only<7>{
  \alert<7>{
  These are best handled in terms of their dual maps,
  }
}

\only<8>{
  \alert<8>{
  which are morphisms of quasi-free differential-graded algebras.  
  }
}




}









\section{Plan}

\frame{
  \hypertarget{plan}{}

  \begin{enumerate}
    
        
    \item \uncover<0>{Motivation}

    \item Plan

      \begin{enumerate}
      
        \item
          \underline{\hyperlink{goal and strategy}{Goal and strategy}}
        
        \item
          \underline{\hyperlink{the bridge}{The bridge between Lie $n$-groupoids
             and differential graded algebra}}

          
      
      \end{enumerate}

    \item \uncover<0>{Parallel $n$-transport}
      
    \item  \uncover<0>{Lie $n$-algebra cohomology}

    \item  \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item  \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item  \uncover<0>{Conclusion}

    \item  \uncover<0>{Questions}

    \item  \uncover<0>{$n$-Categorical background}

  \end{enumerate}

}


\subsection{Goal and strategy}

\frame{

  \hypertarget{goal and strategy}{}

  Our 
  
  \begin{block}{Main goal}
    is to understand $n$-bundles with connection for given
    structure Lie $n$-algebra $\gg_{(n)} = \mathrm{Lie}(G_{(n)})$
    in terms of their differential \emph{parallel transport}.
  \end{block}

   \uncover<2->{
   using the
   
   \begin{block}{Formulation}
     in terms of (co)differential (co)algebra to facilitate
     explicit computations
   \end{block}
   }

  \uncover<3->{  
   while following the
   
   \begin{block}{Structural Guidance}
     obtained by a theory of $n$-bundles with connection in terms
     of morphisms of $n$-groupoids and parallel transport
     $n$-functors.
   \end{block}
  }
}



\subsection{The bridge between Lie $n$-groupoids and differential graded algebra}

\frame{
  \hypertarget{the bridge}{}
  
  \begin{block}{A bridge of concepts}
      \xymatrix{
        \uncover<1->{\alert<1>{\fbox{Lie $n$-groupoids}}}
        \ar@/_2pc/[dr]|>>>>>>>>>{\mbox{\uncover<3->{\alert<3>{\hspace{30pt}differentiation}}}}
        \ar@{<-}@/_1pc/[d]
        &&
        \uncover<4->{\alert<4>{\fbox{
          \begin{tabular}{c}
            codifferential\\
            coalgebra\\
            ($L_\infty$-algebra)
          \end{tabular}
        }}}
        \\
        \ar@/_2.4pc/[drr]|{\mbox{
          \only<8->{\alert<8>{
          \begin{tabular}{l}
            we shall pass 
            \\
            back and forth
            \\
            along this bridge
          \end{tabular}}
        }}}
        &
        \uncover<2->{\alert<2>{\fbox{Lie $n$-algebroids}}}
        \ar@/_2pc/[ul]|>>>>>>>>>>{\uncover<3->{\alert<3>{\mbox{integration}}}}
        \ar@{<->}[ur]|<<<<<<<<<{\uncover<5->{\alert<5>{\mbox{repackaging}}}}
        \\
        \makebox(70,10){
          \only<1>{\alert<1>{
          \begin{tabular}{c}
            \emph{the realm of}
            \\
            \emph{$n$-categories}
          \end{tabular}
          }}
          \only<2>{\alert<2>{
          \begin{tabular}{c}
            \emph{linear}
            \\
            \emph{$n$-categories}
          \end{tabular}
          }}
          \only<3>{\alert<3>{
          \begin{tabular}{c}
            \emph{categorified}
            \\
            \emph{Lie theorem}
            \\
            \emph{(unfinished)}
          \end{tabular}
          }}
          \only<4>{\alert<4>{
          \begin{tabular}{c}
            \emph{realm of}
            \\
            \emph{homotopical algebra}
          \end{tabular}
          }}
          \only<5>{\alert<5>{
          \begin{tabular}{c}
            \emph{general abstract}
            \\
            \emph{operad nonsense}
          \end{tabular}
          }}
          \only<6>{\alert<6>{
          \begin{tabular}{c}
            \emph{most physicists}
            \\
            \emph{live here}
          \end{tabular}
          }}
          \only<7>{\alert<7>{
          \begin{tabular}{c}
            \emph{simple passage}
            \\
            \emph{to dual vector space}
          \end{tabular}
          }}
        }
        &
        &
        \uncover<6->{\alert<6>{\fbox{
          \begin{tabular}{c}
            differential\\
            algebra\\
            (qDGCA)
          \end{tabular}
        }}}        
        \ar@{<->}[uu]|{\uncover<7->{\alert<7>{\mbox{dualization}}}}
      }
    
    
  \end{block}

}


\frame{
  \frametitle{Bridging schools of thought}
  


  \begin{block}{How to use the bridge}
    \begin{tabular}{ccc}
        Lie $n$-groupoids
        &
        $\stackrel{\mbox{the bridge}}{\leftrightarrow}$
        &
        differential algebra
        \\
        \hline
        \\
        \begin{tabular}{c}
          \uncover<3->{conceptual} 
          \\ \uncover<3->{understanding}
          \\
          \\
          \uncover<5->{\emph{What is going on?}}
       \end{tabular}
       &&
       \begin{tabular}{c}
         \uncover<4->{computational}
         \\ 
         \uncover<4->{accessibility}
         \\
         \\
         \uncover<6->{\emph{How does it work?}}
       \end{tabular}
        \\
        \\
        \begin{tabular}{c}
          \uncover<7->{diagrammatics} 
          \\ \uncover<8->{arrow theory}
       \end{tabular}
       &&
       \begin{tabular}{c}
         \uncover<9->{implementation}
       \end{tabular}
     \end{tabular}
  \end{block}


}



\section{Parallel $n$-transport}

\frame{
  \hypertarget{parallel transport}{}

  \begin{itemize}
  
  
    \item \uncover<0>{Motivation}

    \item \uncover<0>{Plan}
    
    \item {Parallel $n$-transport}

     \begin{itemize}

        \item \underline{\hyperlink{locally trivializable n-transport}{Locally trivializable $n$-transport}}

        \item \underline{\hyperlink{1-transport}{Parallel 1-transport}}

        \item \underline{\hyperlink{2-transport}{Parallel 2-transport}}

        \item \underline{\hyperlink{curvature}{$(n+1)$-Curvature}}
        
        \item \underline{\hyperlink{3-transport}{Parallel 3-transport}}


     \end{itemize}

      
    \item  \uncover<0>{Lie $n$-algebra cohomology}

    \item  \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item  \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item \uncover<0>{Conclusion}

    \item  \uncover<0>{Questions}

    \item \uncover<0>{$n$-Categorical Background}


 \end{itemize}

}

\subsection{Locally trivializable $n$-transport}


\frame{
  \hypertarget{locally trivializable n-transport}{}

  \begin{block}{Terminology}
  
    \begin{itemize}
       
       \item Write $P_n(X)$ for a Lie $n$-groupoid that plays the role
        of $n$-paths in $X$.
        
       \item Write $T$ for a given Lie $n$-groupoid that a parallel $n$-
         transport might take values in.
       
       \item Write $G_{(n)}$ for a given Lie $n$-group which plays the
         role of the structure Lie $n$-group of the $n$-transport.

    \end{itemize}
  
  \end{block}

}



\frame{

  \begin{block}{Definition}
    
    Let $\pi : \xymatrix{P_n(Y) \ar@{->>}[r] & P_n(X)}$ be an epimorphism
    and $i : \xymatrix{\Sigma G_{(n)} \ar@{^{(}->}[r] & T}$ a monomorphism.
    Then 
    an $n$-functor $\mathrm{tra} : P_n(X) \to T$ is called a $\pi$-locally
    $i$-trivial $n$-transport functor if there exists a square\vspace{-10pt}
    $$
      \xymatrix{
        P_n(Y) \ar@{->>}[rr]^\pi_>{\ }="s"
        \ar[dd]_{\mathrm{triv}}^>{\ }="t" 
        && 
         P_n(X)
         \ar[dd]^{\mathrm{tra}}
        \\
        \\
        \Sigma G_{(n)} \ar[rr]_{i} && T
        %
        \ar@{=>}^{\simeq}_t "s"; "t"
      }
    $$
  such that the induced transition
  $
    g := \pi_2^* t^{-1} \circ \pi_1^* t
  $
  is in components itself a locally trivializabel $(n-1)$-transport.
    
  \end{block}

}

\frame{

  \begin{block}{Definition}

    Let $S^\infty$ be a closed category of smooth spaces. 
    (We consider Chen-smooth spaces.) Then an $n$-functor
    $\mathrm{tra}$ is smoothly $\pi$-locally $i$-trivializable
    if the local trivialization $\mathrm{triv}$ is smooth and 
    if the transition $g$ is smoothly locally
    $i$-trivializable.
  \end{block}

  \pause

  \begin{block}{Definition}

    There is a more or less obvious $n$-category 
    $\mathrm{Desc}_n^i(\pi)$ of $\pi$-local $i$-descent data.
    
  \end{block}


}






\subsection{Parallel $1$-transport}



\frame{
  \hypertarget{1-transport}{}

\begin{block}{Parallel transport is a functor}
  The parallel transport induced by a connection on a principal bundle
  $P \to X$
$$
  \xy
    (0,0)="nw";
    (60,0)="ne";
    (0,-30)="sw";
    (60,-30)="se";
    (0,-40)="sws";
    (60,-40)="ses";
    %
    (-10,-15)*{P};
    (-10,-40)*{X};
    %
    (10,-24)*{\;\bullet^b}="b";
    (50,-18)*{\;\;\bullet^{b'}}="b'";
    (10,-12)*{\;\;\;\bullet^{gb}}="gb";
    (50,-6)*{\;\;\;\;\bullet^{gb'}}="gb'";
    (10,-39.8)*{\;\bullet^x}="x";
    (50,-39.1)*{\;\bullet^{x'}}="x'";
    %
    \ar@{-} "nw"; "ne"
    \ar@{-} "ne"; "se"
    \ar@{-} "nw"; "sw"
    \ar@{-} "sw"; "se"
    %
    \ar@{-} "sws"; "ses" 
    %
    \ar@{--} (10,0); (10,-30)
    \ar@{--} (50,0); (50,-30)
    %
    \ar@{-}@/_1.6pc/ "b"; "b'"
    \ar@{-}@/_1.6pc/ "gb"; "gb'"
  \endxy
$$
is a functorial map from paths to fiber morphisms.

\end{block}
}





\frame{


  \begin{block}{Definition}
    The smooth path 1-groupoid $\mathcal{P}_1(X)$ is that whose morphisms
     $\gamma : x \to y$ are thin homotopy
    classes of paths in $X$.\vspace{-10pt} 
$$
  \xy
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    (6,-14)*{
      \xymatrix{
        x
         \ar@/_2pc/[rr]_{\gamma}
         &&
       x'
     }
     };
    %
    (3,-44)*{X};
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
  \endxy
$$
  \end{block}



}

\frame{


  \begin{block}{Proposition}
    For $G$ a Lie group, smooth 1-functors 
    $\mathcal{P}_1(X) \to \Sigma G$ are in bijection with 
    1-forms $A \in \Omega^1(X,\mathrm{Lie}(G))$.
  \end{block}


  \begin{block}{Proposition}
    Let $G$ be a Lie group and $i : \Sigma G \hookrightarrow T$
    a monomorphic equivalence. Then
    locally $i$-trivializabel transport functors 
    $\mathrm{tra} : \mathcal{P}_1(X) \to \Sigma G$ are equivalent
    to $G$-bundles with connection.
  \end{block}

}


\frame{


  \begin{tabular}{cc}   
     \begin{tabular}{c}
       local connection 1-form
       \\
       $A \in \Omega^1(U,\mathfrak{g})$
     \end{tabular}
  &
     \begin{tabular}{c}
       smooth transport functor
       \\
       $\mathrm{tra}_U : \mathcal{P}_1(U) \to \Sigma(G)$
    \end{tabular}
   \\
   \hline
     \begin{tabular}{c}
       transition function 
       \\
       $g \in \Omega^0(U^{[2]},G)$
     \end{tabular}
  &
     \begin{tabular}{c}
       natural isomorphism
       \\
       $g : p_1^*\mathrm{tra}_U \to p_2^*\mathrm{tra}_U$
    \end{tabular}
   \\
   \hline
   $A_i = g_{ij}A_j g_{ij}^{-1} + g_{ij}d g_{ij}^{-1}$
   &
   \raisebox{20pt}{
   \xymatrix{
     \bullet
     \ar[r]^{g_{ij}(x)}
     \ar[d]_{\mathrm{tra}_i(\gamma)}
     &
     \bullet
     \ar[d]^{\mathrm{tra}_j(\gamma)}
     \\
     \bullet
     \ar[r]_{g_{ij}(y)}
     &
     \bullet
   }
   }
   \\
   \hline
   $g_{ij}g_{jk} = g_{ik}$
   &
   \raisebox{20pt}{
   \xymatrix{
     & 
     p_2^*\mathrm{tra}_U
     \ar[dr]^{p_{23}^* g}
     \\
     p_1^*\mathrm{tra}_U  
     \ar[rr]_{p_{13}^* g}
     \ar[ur]^{p_{12}^* g}
     &&
     p_3^*\mathrm{tra}_U  
   }
   }
  \end{tabular}


}



\subsection{Parallel $2$-transport}

\frame{
  \hypertarget{2-transport}{}


  \begin{block}{Defintion}
    $\mathcal{P}_2(X)$ is the (strict) 2-groupoid whose 2-morphisms 
    $S : \gamma \to \gamma'$are
    thin homotopy classes of cobounding surfaces.
$$
  \xy
    (0,0)="nw";
    (50,0)="ne";
    (0,-50)="sw";
    (50,-50)="se";
    (6,-14)*{
      \xymatrix{
        x
         \ar@/^2pc/[rr]^\gamma
         \ar@/_2pc/[rr]_{\gamma'}
         &&
       x'
       %
       \ar@{=>}^S (18,-14)+(0,3); (18,-14)+(0,-3)
     }
     };
    %
    (3,-44)*{X};
    %
    \ar@/_1pc/@{--} "nw"; "ne"
    \ar@/_1pc/@{--} "nw"; "sw"
    \ar@/_1pc/@{--} "sw"; "se"
    \ar@/_1pc/@{--} "se"; "ne"
    %
  \endxy
$$
  \end{block}


}


\frame{

  \begin{block}{Proposition}
    For $G_{(2)} = (\xymatrix{H \ar[r]^t & G})$ a strict Lie group, smooth 2-functors 
    $\mathcal{P}_2(X) \to \Sigma G_{(2)}$ are in bijection with 
    forms $(A,B) \in \Omega^1(X,\mathrm{Lie}(G)) \times 
     \Omega^2(X,\mathrm{Lie}(H))$ such that $F_A + t_* \circ B = 0$.

$$
\raisebox{36pt}{
\xymatrix{
  0 \ar[rr]^{\path_1} \ar[dd]_{\path_3} && x \ar[dd]^{\path_2}
  \\
  \\
  y \ar[rr]_{\path_4} && x+y
  %
  \ar@{=>} (14,-10); (12,-12)
}
}
\hspace{2pt}
\mapsto
\hspace{2pt}
\raisebox{36pt}{
\xymatrix{
  \bullet 
   \ar[rr]^{1 + A\of{\path_1} + \cdots} 
   \ar[dd]_{1+ A\of{\path_3} + \cdots} 
   && 
    \bullet \ar[dd]^{1 + A\of{\path_2} + \cdots}
  \\
  \\
  \bullet \ar[rr]_{1 + A\of{\path_4} + \cdots} && \bullet
  %
  \ar@{=>}^{\hspace{-34pt}1+B\of{\path_1,\path_3} + \cdots} (14,-10); (12,-12)
}
}
$$

\end{block}



}


\frame{

  \begin{block}{}
   
     Descent data is now a 3-simplex\vspace{-10pt}
$$
\xy(0,0)*{
\xy
  (0,0)*{\makebox(35,15){$p_1^*\transport_U$}}="dl";
  (0,30)*{\makebox(35,15){$p_2^*\transport_U$}}="ul";  
  (30,30)*{\makebox(35,15){$p_3^*\transport_U$}}="ur";  
  (30,0)*{\makebox(35,15){$p_4^*\transport_U$}}="dr";
  \ar^{p_{12}^*g} "dl"; "ul"
  \ar^{p_{23}^*g} "ul"; "ur"
  \ar^{p_{34}^*g} "ur"; "dr"
  \ar_{p_{14}^*g} "dl"; "dr"
  \ar|{p_{13}^*g} "dl"; "ur"
  \ar@{<=}^{p_{123}^*\trivializationTransitionModification} (16,18)+(-4,0); (12, 22)+(-4,0)
  \ar@{<=}_{p_{134}^*\trivializationTransitionModification} (21,7); (21, 13)
\endxy
}\endxy
\hspace{10pt}
=
\hspace{10pt}
\xy(0,0)*{
\xy
  (0,0)*{\makebox(35,15){$p_1^*\transport_U$}}="dl";
  (0,30)*{\makebox(35,15){$p_2^*\transport_U$}}="ul";  
  (30,30)*{\makebox(35,15){$p_3^*\transport_U$}}="ur";  
  (30,0)*{\makebox(35,15){$p_4^*\transport_U$}}="dr";
  \ar^{p_{12}^*g} "dl"; "ul"
  \ar^{p_{23}^*g} "ul"; "ur"
  \ar^{p_{34}^*g} "ur"; "dr"
  \ar_{p_{14}^*g} "dl"; "dr"
  \ar|{p_{24}^*g} "ul"; "dr"
  \ar@{<=}^{p_{234}^*\trivializationTransitionModification} (30,29)+(-10,-10); (35, 34)+(-10,-10)
  \ar@{<=}^{p_{124}^*\trivializationTransitionModification} (10,7); (10, 13)
\endxy
}\endxy
$$
    
  \end{block}

}



\frame{

  \begin{block}{Proposition}
     The descent catgeory for $\Sigma U(1)$ 2-transport is canonically
     equivalent to $U(1)$-bundle gerbes with connection.
  \end{block}

  \begin{block}{Proposition}
     The descent catgeory for $\mathrm{AUT}(H)$ 2-transport is canonically
     equivalent to fake flat nonabelian bundle gerbes with connection.
  \end{block}

}



\subsection{$(n+1)$-Curvature}

\frame{
  \hypertarget{curvature}{}

   \begin{block}{Observation}
     
      The right notion for $G_{(n)}$ $n$-bundles with connection is
      as locally trivializable $\mathrm{INN}_0(G_{(n)})$ $(n+1)$-transport
      whose transitions factor through the inclusion 
      $G_{(n)} \hookrightarrow \mathrm{INN}_0(G_{(n)})$.
     
   \end{block}


}


\subsection{Parallel $3$-transport}

\frame{
  \hypertarget{3-transport}{}


  \begin{block}{Proposition}
    Descent data for smoothly locally trivializable 
   $\mathrm{INN}_0(\mathrm{AUT}(H))$ 3-transport whose transitions
   factor through $\mathrm{AUT}(H) \hookrightarrow 
   \mathrm{INN}_0(\mathrm{AUT}(H))$ is equivalent to 
   the Breen-Messing data.


The 2- and 3-curvature is now unrestricted\vspace{-10pt}
$$
  \raisebox{25pt}{
  \xy
    (0,0)*{\makebox(11,11){$\bullet$}}="nw";
    (25,0)*{\makebox(11,11){$\bullet$}}="ne";
    (0,-25)*{\makebox(11,11){$\bullet$}}="sw";
    (25,-25)*{\makebox(11,11){$\bullet$}}="se";
    %
    (17,17)*{\makebox(10,8){\tiny$\bullet$}}="l";
    (17,17)+(25,0)*{\makebox(10,8){\tiny$\bullet$}}="c";
    (17,17)+(25,-25)*{\makebox(10,8){\tiny $\bullet$}}="r";
    %
    \ar "nw"; "ne"    
    \ar "sw"; "se"    
    \ar "nw"; "sw"    
    \ar "ne"; "se"
    %
    \ar|{\mathrm{tra}_A(\path_3)} "l"; "nw"    
    \ar "c"; "ne"    
    \ar "r"; "se"    
    \ar|{\tiny \mbox{$\transport_A\of{\path_1}$}} "l"; "c"    
    \ar "c"; "r" 
    %
    \ar@{=>}^{\hspace{-14pt}\transport_{A,B}\of{\surface_t}} (14,-12); (11,-15)   
    \ar@{=>}^{\hspace{-14pt}\transport_{A,B}\of{\surface_1}} (22,10); (19,8)
    \ar@{=>}^{\hspace{-14pt}\transport_{A,B}\of{\surface_2}} (36,-2); (33,-3)
  \endxy
  }
  \hspace{10pt}
  \to
  \hspace{10pt}
  \raisebox{25pt}{
  \xy
    (0,0)*{\makebox(11,11){$\bullet$}}="nw";
    (17,17)+(0,-25)*{\makebox(11,11){\tiny$\bullet$}}="ne";
    (0,-25)*{\makebox(11,11){$\bullet$}}="sw";
    (25,-25)*{\makebox(11,11){$\bullet$}}="se";
    %
    (17,17)*{\makebox(10,8){\tiny$\bullet$}}="l";
    (17,17)+(25,0)*{\makebox(10,8){\tiny$\bullet$}}="c";
    (17,17)+(25,-25)*{\makebox(10,8){\tiny $\bullet$}}="r";
    %
    \ar|{\mbox{\tiny $\transport_A\of{\path_2}$}} "l"; "ne"    
    \ar "sw"; "se"    
    \ar "nw"; "sw"    
    \ar "ne"; "r"
    %
    \ar|{\transport_A\of{\path_3}} "l"; "nw"    
    \ar "ne"; "sw"    
    \ar "r"; "se"    
    \ar|{\mbox{\tiny $\transport\of{\path_1}$}} "l"; "c"    
    \ar "c"; "r" 
    %
    \ar@{=>}^{\hspace{-16pt}\mbox{\tiny $\transport_{A,B}\of{\surface_s}$}} (14,-12)+(18,18); (11,-15)+(18,18)   
    \ar@{=>}^{\hspace{-18pt}\transport_{A,B}\of{\surface_4}} (21,11)+(2,-25); (19,8)+(2,-25)
    \ar@{=>}^{\hspace{-16pt}\transport_{A,B}\of{\surface_3}} (36,-2)+(-24,0); (33,-3)+(-24,0)
  \endxy
  }
$$


  \end{block}
  

}




\section{Lie $n$-Algebra Cohomology}


\frame{
  \hypertarget{Lie n-algebra cohomology}{}
  
  \begin{enumerate}
            
    \item \uncover<0>{Motivation}

    \item \uncover<0>{Plan}

    \item \uncover<0>{Parallel $n$-transport}
      
    \item  Lie $n$-algebra cohomology

     \begin{enumerate}

       \item \underline{\hyperlink{Lie n-algebras}{Lie $n$-algebras}}
       
       \item\underline{
         \hyperlink{the inn construction}{The $\mathrm{inn}(\cdot)$-construction}
       }

       \item \underline{
           \hyperlink{Lie alg cohomol with inng}{
           Lie algebra cohomology in terms of the Weil algebra $\mathrm{inn}(\gg)^*$}}
       
       \item 
             \underline{
               \hyperlink{String, Chern-Simons and Chern}{
                String, Chern-Simons and Chern Lie $n$-algebras}
             }
       
       \item 
             \underline{
                \hyperlink{Lie n-algebra cohomology sub}{Lie $n$-algebra cohomology}
             }
       
       \item
          \underline{
            \hyperlink{the algebra bgn}{The algebra $b\gg_{(n)}^*$ of 
            invariant polynomials}
           }
     
       \item
           \underline{ 
             \hyperlink{inv polynomials of string and chern}{
               Invariant polynomials of String and Chern Lie $n$-algebras}
           }

     \end{enumerate}

    \item  \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item  \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item \uncover<0>{Conclusion}

    \item  \uncover<0>{Questions}

    \item \uncover<0>{$n$-Categorical Background}

  \end{enumerate}

}



\subsection{Lie $n$-Algebras}



\frame{
  \frametitle{Infinitesimal higher dimensional algebra}
  \hypertarget{Lie n-algebras}{}


\begin{block}{The concept of Lie $n$-algebras}
  
  \begin{itemize}

    \item<1-> A Lie algebra $\gg$ is the infinitesimal version of 
      a Lie group $G$:\vspace{-10pt}
    $$
      \gg = \mathrm{Lie}(G)\vspace{-10pt}
    $$
    
    \item<2-> A group $G$ is a one-object groupoid $\Sigma G$.
    
    \item<3-> An $n$-group $G_{(n)}$ is a one-object $n$-groupoid $\Sigma G_{(n)}$.

    \item<4-> A Lie $n$-algebra $\gg_{(n)}$ is the infinitesimal version of a 
      one-object Lie $n$-groupoid:\vspace{-10pt}
      $$
        \gg_{(n)} = \mathrm{Lie}(G_{(n)})
        \,.\vspace{-10pt}
      $$

  \end{itemize}
  
\end{block}

\pause \pause \pause \pause 

\begin{block}{Definition}
  A semistric Lie $n$-algebra is an $n$-category $\gg_{(n)}$ internal to 
  $\mathrm{Vect}$ equipped with a skew symmetric functor
  $[\cdot,\cdot] : \gg_{(n)} \times \gg_{(n)} \to \gg_{(n)}$  which satisfies the Jacobi
  identity up to coherent isomorphism.
\end{block}

}




\frame{
  \frametitle{The relation between Lie $n$-groupoids and Lie $n$-algebras}

    Caveat: To what extent, and under which conditions, it is
    true that
  \begin{block}{Expected Statement: $n$-Lie's third theorem}
    \begin{itemize}
      \item 
         Every Lie $n$-algebra integrates to a Lie $n$-groupoid.
      \item
         Every Lie $n$-groupoid gives rise to a semistrict Lie $n$-algebra.
    \end{itemize}
  \end{block}
    is still being investigated. Special cases are understood.
    
    The statement hinges on

    \begin{block}{Issues still being discussed}
      \begin{itemize}
        \item The precise definition of Lie $n$-groupoids.
        \item The question whether and when one may assume strict
         skew symmetry.
      \end{itemize}
    \end{block}

}




\subsection{The Bridge: categorical Lie algebra to differential algebra}

\frame{
  \frametitle{The Bridge: categorical Lie algebra to differential algebra}

  \begin{block}{Principle}
     $
      \left.
      \mbox{  \begin{tabular}{c}
          \emph{Lie $n$-algebra} $\gg_{(n)}$ --
          \\
          higher categorical
          \\
          Lie algebra
        \end{tabular}}
        \right\rbrace
        \leftrightarrow
        \left\lbrace
        \mbox{
        \begin{tabular}{c}
           graded-commutative \\
          (co)differential
          \\
          (co)algebra $(\wedgebullet sV^*, d_{\gg_{(n)}})$
        \end{tabular}
        }
        \right.
    $
  \end{block}

  \pause
    
  \begin{block}{Dictionary}
     \begin{tabular}{ccc}
       $\mathrm{Mor}_k(L)$ & $\simeq$ & $(sV)_1 \oplus \cdots \oplus (sV)_k$
       \\
       structure morphisms &$\leftrightarrow$& $d_{\gg_{(n)}}$
       \\
       coherence &$\leftrightarrow$& $(d_{\gg_{(n)}})^2 = 0$
     \end{tabular}
  \end{block}

  \pause  



}


\frame{
  \frametitle{$L_\infty$ and qDGCA}

  More precisely

  \begin{block}{Definition and Proposition}
     An $n$-term $L_\infty$-algebra is a free graded commutative co-algebra
     $
       S^c(sV)
     $
     on graded vector space $V = V_0 \oplus \cdots V_{n-1}$, which is 
     equipped with a degree -1 codifferential
     $
       D : S^c(sV) \to S^c(sV)
     $
     that squares to 0:
     $
       D^2 = 0
       \,.
     $
  \end{block}

  \pause

  \begin{block}{Dual statement}
     Dually, this is the exterior algebra $\wedge^\bullet (s V^*) $
     equipped with the differential
     $d \omega := - \omega(D(\cdot))$.    
     This we call an $n$-term quasi-free graded-commutative differential algebra,
     or $n$-term qDGCAs, for short.
  \end{block}

}


\frame{
  \frametitle{The standard example}

  \begin{block}{Example: ordinary Lie algebra as $L_\infty$-algebra}
     For $\gg$ an ordinary Lie (1-)algebra, the codifferential
     on the free graded-commutative coalgebra $S^c(s \gg)$ acts as\vspace{-10pt}
     $$
       D(s x_1 \vee s x_2) = s[x_1,x_2]\vspace{-10pt}
     $$
     on all products of two generators $x_1,x_2 \in \gg$
     and is freely extended as a codifferential to higher products
     of generators. The statement
     $$
       D^2(s x_1 \vee s x_2 \vee s x_3) = 0
     $$
     for a triples of generators is the Jacobi identity.
  \end{block}



}


\frame{
  \frametitle{The standard example}

  \begin{block}{Example: ordinary Lie algebra as qDGCA}
     For $\gg$ an ordinary Lie (1-)algebra, the differential
     on the exterior algebra $\wedge^\bullet(s \gg^*)$ acts as\vspace{-10pt}
     $$
       d_{\gg} t^a = - \frac{1}{2}C^a{}_{bc}t^b \wedge t^c\vspace{-10pt}
     $$
     for $\{t^a\}$ any basis of $s \gg^*$ and $C^a{}_{bc}$ the structure
     constants of $\gg$ in the corresponding dual basis.

     Of course this is nothing but the qDGCA of left-invariant forms
     on the group $G$\vspace{-10pt}
     $$
       \gg^* := (\wedge^\bullet (s \gg^*), d_\gg) \simeq 
       \Omega_{\mathrm{li}}^\bullet(G)\,.
     $$
  \end{block}



}



\frame{
  \frametitle{The Bridge in more detail}

  \begin{block}{The Bridge again, more precisely}
    Semistrict Lie $n$-algebras are ``the same'' as $n$-term $L_{\infty}$-algebras,
    which in turn are dual (for finite dimensions) to
    $n$-term qDGCAs.
  \end{block}

  \pause

  \begin{block}{Caveat: ``Semistrictness''}
    Here ``semistrict'' refers to the fact that the Jacobi identity
    is coherently weakened, while the skew symmetry is taken to hold
    strictly.
  \end{block}

  \pause
 
  \begin{block}{Caveat: higher morphisms}
    The general statement follows from general abstract operad nonsense.
    But explicit details on how \emph{higher morphisms} pass over
    the bridge are hard to come by.
  \end{block}
  
  
}

\frame{
  \frametitle{The oidified Bridge: many objects}


  \begin{block}{Lie algebroid version}
    On the qDGCA side the rather obvious generalization yields
    what should be addressed as Lie $n$-algebroids: in the
    literature the many-object qDGCAs are also known as 
    \emph{NQ-manifolds}.
  \end{block}

   \pause
 
  \begin{block}{The tangent Lie algebroid}
    The only Lie algebroid which we need here is the tangent
    algebroid $\mathrm{Vect}(X)$ of a manifold $X$. This is the differential
    of the \emph{fundamental groupoid}\vspace{-10pt}
    $$
       \mathrm{Vect}(X) := \mathrm{Lie}(\Pi_1(X))\,.\vspace{-10pt}
    $$
    This is very conveniently handled in its dual incarnation -- there
    it is simply the deRham complex \vspace{-10pt}
    $$
      \mathrm{Vect}(X)^* = (\Omega^\bullet(X), d)\,.
    $$
  \end{block}


}


\subsection{The $\mathrm{inn}(\cdot)$-construction}


\frame{
  \frametitle{The $\mathrm{inn}(\cdot)$-construction}
  \hypertarget{the inn construction}{}


  \begin{block}{Definition. (Inner derivation Lie $(n+1)$-algebra)}
  
    \only<1>{
         $
           \mathrm{inn}(\gg_{(n)})^*  \simeq  
           (\bigwedge (s \gg_{(n)}^* \oplus ss \gg_{(n)}^*), 
           \left(
              \begin{array}{cc}
                d & 0
                \\
                \mathrm{Id} & d
              \end{array}
           \right))
         $}
    \only<2->{
         $
           \mathrm{inn}(\gg_{(n)})^*  \simeq  
           (\bigwedge (s \gg_{(n)}^*) \oplus ss \gg_{(n)}^*), 
             d_{\mathrm{inn}(\gg_{(n)})})
         $}
         corresponds to the mapping cone of the identity on $\gg_{(n)}$
  \end{block}

  \pause

  \begin{block}{Proposition}
  
    \begin{itemize}
      \item There is a canonical injection 
      $\gg_{(n)} \hookrightarrow \mathrm{inn}(\gg)$.
      
      \item $\mathrm{inn}(\gg_{(n)})$ is \emph{contractible}
      
      \item $(\bigwedge (s \gg^* \oplus ss \gg^*), d_{\mathrm{inn}(\gg)})$
        is the \emph{Weil algebra} of $\gg_{(n)} = \gg$
    \end{itemize}
  \end{block}

 \pause
   \begin{block}{Remark.}
     Hence $\mathrm{inn}(\gg_{(1)})^*$ plays the role of differential forms on the 
     universal $G$-bundle.
   \end{block}
}


\frame{
  \frametitle{The $\mathrm{inn}(\cdot)$-construction}

  \begin{block}{The qDGCA of $\mathrm{inn}(\gg)$: the Weil algebra}
        
    $\mathrm{inn}(\gg)^* \simeq (\wedgebullet (s\gg^* \oplus ss\gg^*),d)$ 
    is spanned by generators
    $\{t^a\}$ in degree 1 and $\{r^a\}$ in degree 2,
    with differential
    \begin{eqnarray*}
      d t^a = - \frac{1}{2}C^a{}_{bc} t^b \wedge t^c - r^a
      \\
      d r^a = - C^a{}_{bc} t^b \wedge r^c
      \,.
    \end{eqnarray*}
  
  \end{block}




}



\subsection{Lie algebra cohomology in terms of the Weil algebra $\mathrm{inn}(\gg)^*$}


\frame{
  \hypertarget{Lie alg cohomol with inng}{}
  
  \begin{block}
  
    We will now
    
    \begin{itemize}
    
      \item express the Lie algebra cohomology of $\gg$ in terms of the cohomology
      of the qDGCA underlying $\mathrm{inn}(\gg)$.
      
      \item use the insight gained thereby to describe three families of
        Lie $n$-algebras: one for each cocycle, one for each invariant polynomial
        and one for each transgression element.
        
      \item then show that for the canonical 3-cocycle on a semisimple
        Lie algebra, connections with values in the Lie 3-algebra obtained this
        way describe the Chern-Simons parallel transport which we are after.
    
    \end{itemize}
    
  \end{block}

}


\frame{
  \begin{block}{Lie algebra cohomology in terms of $\mathrm{inn}(\gg)$}
  
     
     \begin{itemize}
       
       \item
            A Lie algebra $n$-cocycle $\mu$ 
            is
            $$
              d|_{\wedgebullet (s\gg^*)} \mu = 0
              \,.
            $$ \vspace{-10pt}
        \item
          An invariant degree $n$-polynomial $k$ is
           $$
              d|_{\wedgebullet (ss\gg^*)} k = 0
              \,.
            $$ \vspace{-10pt}
          \item
           A transgression element $\mathrm{cs}$ is
             \begin{eqnarray*}
               \mathrm{cs}|_{\wedgebullet s\gg^*} &&= \mu
               \\
               d \mathrm{cs} &&= k
               \,.
             \end{eqnarray*}
             \vspace{-10pt}
    \end{itemize}



  
  \end{block}

}


\frame{
  
  \begin{block}{The homotopy operator}

     \begin{itemize}
     
       \item Recall that we said that $\mathrm{inn}(\gg_{(n)})$ is
       trivializable.
       
       \item This means there is a homotopy
       $$
         \xymatrix{
           \mathrm{inn}(\gg_{(n)})
           \ar@/^1.8pc/[rr]^{0}_{\ }="s"
           \ar@/_1.8pc/[rr]_{\mathrm{Id} = [d,\tau]}^{\ }="t"
           &&
           \mathrm{inn}(\gg_{(n)})
           %
           \ar@{=>}^\tau "s"; "t"
         }
       $$
       
       \item We have $\tau$ \emph{explicitly}
         (see \underline{\hyperlink{higher morphisms}{Higher morphisms of Lie $n$-algebras}})
         and hence
         an effective algorithm to always solve
         $k = d\mathrm{cs}$ as
         $$
            \mathrm{cs} := \tau(k) + dq
            \,.
         $$ 
         \vspace{-10pt}
        
        \item The only nontrivial condition is hence 
         $\mathrm{cs}|_{\wedgebullet s\gg^*} = \mu$.
     \end{itemize}

  \end{block}

}

\frame{
 \begin{block}{A map of the cocycle situation}
$$
  \xymatrix{
    \mbox{
      cocycle
    }
    &
    \mbox{Chern-Simons}
    &
    \mbox{inv. polynomial}
    \\
    (\wedgebullet (s\gg^*), d_\gg)
    & 
    (\wedgebullet (s\gg^* \oplus ss\gg^*),d_{\mathrm{inn}(\gg)})
    \ar@{->>}[l]_<<<<{i^*} 
    & 
    (\wedgebullet (ss\gg)^*)
    \ar@{_{(}->}[l]_<<<<<{p^*}
    \\
    & 0
    \\
    0
    &
    p^* k
    \ar@{|->}[u]_{d_{\mathrm{inn}(\gg)}}
    \ar@/_1pc/[d]_\tau
    &
    k
    \ar@{|->}[l]^{p^*}
    \\
    \mu
    \ar@{|->}[u]_{d_\gg}
    &
    cs
    \ar@{|->}[l]_{i^*}
    \ar@{|->}[u]_{d_{\mathrm{inn}(\gg)}}
    &
  }
$$

 \end{block}


}


\subsection{String, Chern-Simons and Chern Lie $n$-algebras}

\frame{
  \frametitle{Lie $n$-algebras from cocycles}
   \hypertarget{String, Chern-Simons and Chern}{}  
  
   In the following we discuss
  
   \begin{block}{Definition and Proposition}
        From elements of $\mathrm{inn}(\gg)^*$-cohomology we obtain
     Lie $n$-algebras:
   
     \begin{tabular}{cc|cc}
       \hline
       Lie algebra cocycle & $\mu$ & Baez-Crans Lie $n$-algebra & $\gg_{\mu}$
       \\
       invariant polynomial & $k$ & Chern Lie $n$-algebra 
          & $\mathrm{ch}_k(\gg)$
       \\
       transgression element & $\mathrm{cs}$ & Chern-Simons Lie $n$-algebra
         & $\mathrm{cs}_k(\gg)$
         \\
         \hline
     \end{tabular}

     For every transgression element $\mathrm{cs}$ these fit into a weakly exact
     sequence\vspace{-10pt}
     $$
       \gg_{\mu_k} \to \mathrm{cs}_k(\gg) \to \mathrm{ch}_k(\gg)
       \,.
     $$   
   \end{block}
  
  
}


\frame{
  \frametitle{Baez-Crans Lie $n$-algebras from cocycles}

  \begin{block}{Definition and proposition [Baez,Crans]}

    \uncover{
     For every Lie algebra $(n+1)$-cocycle $\mu$ of the Lie algebra $\gg$}
    \uncover{ 
     there is a skeletal Lie $n$-algebra
     $$
       \gg_\mu
       \,.
     $$  }
  \end{block}

  \pause
  
  \begin{block}{Construction.}
  
    Set $\gg_\mu \simeq (\wedgebullet (s\gg^* \oplus s^{n}\mathbb{R}^*), d)$
    such that 
    the differential is given by
    \begin{eqnarray*}
      d t^a &&= - \frac{1}{2}C^a{}_{bc} t^b \wedge t^c
      \\
      d b && = -\mu
    \end{eqnarray*}
  \end{block}


}


\frame{
  \frametitle{Chern Lie $n$-algebras from invariant polynomials}


  \begin{block}{Definition and proposition }

     For every degree $(n+1)$ Lie algebra invariant polynomial $k$ 
     of the Lie algebra $\gg$ 
     there is a Lie $(2n+1)$-algebra
     $$
       \mathrm{ch}_k(\gg)
       \,.
     $$  
     \vspace{-17pt}
  \end{block}

  \pause
  
  \begin{block}{Construction.}
  
    Set $\mathrm{ch}_k(\gg) 
      \simeq (\wedgebullet (s\gg^* \oplus ss\gg^* \oplus s^{(2n+1)}\mathbb{R}^*), d)$
     such that we have
    \begin{eqnarray*}
      d t^a &&= - \frac{1}{2}C^a{}_{bc} t^b \wedge t^c - r^a
      \\
      d r^a && = - C^a{}_{bc}t^b \wedge t^c 
      \\
      d c && = k
    \end{eqnarray*}
  \end{block}




}


\frame{
  \frametitle{Chern-Simons Lie $n$-algebras from transgression elements}


  \begin{block}{Definition and proposition }

     For every transgression element $q$ of degree $(2n+1)$
     there is a Lie $(2n+1)$-algebra
     $$
       \mathrm{cs}_k(\gg)
       \,.
     $$  
     \vspace{-17pt}
  \end{block}

  \pause
  \pause
  
  \begin{block}{Construction.}
  
    Set $\mathrm{cs}_k(\gg) 
      \simeq (\wedgebullet (s\gg^* \oplus ss\gg^* \oplus 
      \oplus s^{2n}\mathbb{R}^* \oplus s^{(2n+1)}\mathbb{R}^*), d)$
     such that
    \begin{eqnarray*}
      d t^a &&= - \frac{1}{2}C^a{}_{bc} t^b \wedge t^c - r^a
      \\
      d r^a && = - C^a{}_{bc}t^b \wedge t^c 
      \\
      db &&= -\mathrm{cs} + c
      \\
      d c && = k
    \end{eqnarray*}
  \end{block}





}


\frame{

  \begin{block}{Theorem}
    Whenever they exist, these Lie $(2n+1)$-algebras form a (weakly) short
    exact sequence:
    $$
      0 \to \gg_{\mu_k} \to \mathrm{cs}_k(\gg) \to \mathrm{ch}_k(\gg) \to 0
      \,.
    $$
  \end{block}

   \pause

  \begin{block}{Theorem}
    Moreover, we have an isomorphism
    $$
      \mathrm{cs}_k(\gg) \simeq \mathrm{inn}(\gg_{\mu_k})
      \,.
    $$
  \end{block}

}



\subsection{Lie $n$-algebra cohomology}


\frame{
  \hypertarget{Lie n-algebra cohomology sub}{}


  The way we obtained Lie algebra cohomology from 
  $\mathrm{inn}(\gg)^*$ has a straightforward generalization 
  with $\mathrm{inn}(\gg)^*$ replaced by $\mathrm{inn}(\gg_{(n)})^*$,
  for $\gg_{(n)}$ any Lie $n$-algebra.

  

}


\frame{
  \frametitle{Lie $n$-algebra cohomology from $\mathrm{inn}(\gg_{(n)})^*$}


     \begin{itemize}
       
       \item
            A Lie $\gg_{(n)}$-cocycle $\mu$ 
            is
            $$
              d_{\gg_{\mu}} \mu = 0
              \,.
            $$ \vspace{-10pt}
        \item
          A $\gg_{(n)}$ invariant polynomial $k$ is
            $$
              d_{\mathrm{inn}(\gg_{(n)})}|_{\wedgebullet (ss\gg_{(n)}^*)} k = 0
              \,.
            $$ \vspace{-10pt}
          \item
           A {transgression element} $\mathrm{cs}$ is
             \begin{eqnarray*}
               \mathrm{cs}|_{\wedgebullet s\gg_{(n)}^*} &&= \mu
               \\
               d_{\mathrm{inn}(\gg_{(n)})} \mathrm{cs} &&= k
               \,.
             \end{eqnarray*}
             \vspace{-10pt}
    \end{itemize}



}


\frame{
  \frametitle{Generalized String, Chern-Simons and Chern Lie $n$-algebras}

  \begin{block}{Remark}
    The entire construction of String, Chern-Simons and Chern Lie
    $n$-algebras from ordinary Lie algebra cohomology
    accordinly has a straightforward analog for Lie $n$-algebra
    cohomology.
    
    \begin{itemize}
      \item
        $(\gg_{(n)})_\mu$, $\mathrm{cs}_{k}(\gg_{(n)})$, 
        $\mathrm{ch}_k(\gg_{(n)})$
    \end{itemize}
    
  \end{block}

  For the present discussion, however, we only need $\gg_{(n)}$ invariant
  polynomials. And we need to make manifest the qDGCA which they span.

}


\subsection{The algebra $b \gg_{(n)}^*$ of invariant polynomials}

\frame{
  \frametitle{Coboundaries for invariant polynomials}
  \hypertarget{the algebra bgn}{}

  The qDGCA of $\gg_{(n)}$ invariant polynomials will turn out to 
  play the role of differential forms on the classifying space
  of $\gg_{(n)}$-bundles. 

  Therefore we will denote it $b \gg_{(n)}^*$.
  
  Before defining this, we need to define coboundaries of
  $\gg_{(n)}$ invariant polynomials.

}

\frame{
  \frametitle{Coboundaries for invariant polynomials}


  \begin{block}{Definition}
    An $\gg_{(n)}$ invariant polynomial 
    $k \in \wedgebullet(s s \gg_{(n)}^*)$ is a coboundary
    of invariant polynomials if it has a potential $L$ such that\vspace{-10pt}
    $$
      k = d_{\mathrm{inn}(\gg_{(n)})} L
      \,,
    $$
    which vanishes ``on the fibers'' in that\vspace{-10pt}
    $$
      L|_{\wedgebullet s \gg_{(n)}^*} = 0
      \,.
    $$
  \end{block}

}

\frame{
  \frametitle{Coboundaries for invariant polynomials}

  \begin{block}{Remark}
    Recall that, due to the existence of the trivializing homotopy 
    $
      \xymatrix{
        \tau : 0 \to \mathrm{Id}_{\mathrm{inn}(\gg)_{(n)}}
      },
    $
    every $d_{\mathrm{inn}(\gg_{(n)})}$ closed element $k$ is 
    $d_{\mathrm{inn}(\gg_{(n)})}$-exact\vspace{-10pt}
    $$
      k = d (\tau k)
      \,.
      \vspace{-15pt}
    $$
    \begin{itemize}
      \item
        When $\mu \simeq (\tau k)|_{\wedgebullet (s \gg_{(n)}^*) }$
        is closed, then $\mathrm{cs} \simeq \tau k$ is a transgression element.
      \item
         When $L \simeq (\tau k)$ vanishes on $\wedgebullet(s \gg_{(n)}^*)$
         it is a coboundary of invariant polynomials.
    \end{itemize}
    Hence coboundaries of invariant polynomials are transgression elements
    for the trivial cocycle.
  \end{block}

}


\frame{
  \frametitle{
    The algebra of $\gg_{(n)}$ invariant polynomials
  }

  \begin{block}{Proposition}
    \begin{itemize}
    \item
    The strict kernel
    \only<1>{
    $$
      \xymatrix{
        \gg_{(n)}^*
        &&
        \mathrm{inn}(\gg_{(n)})^*
        \ar[ll]_{i^*}
        &&
        \mathrm{ker}(i^*)
        \ar[ll]
      }
    $$}
    \only<2->{
    $$
      \xymatrix{
        \gg_{(n)}^*
        &&
        \mathrm{inn}(\gg_{(n)})^*
        \ar[ll]_{i^*}
        &&
        \alert<2>{b \gg_{(n)}^*}
        \ar[ll]
      }
    $$}
    is\vspace{-10pt}
    $$
      b \gg_{(n)}^* := [\mathrm{inv}(\gg_{(n)})]
      \,,
    $$
    which is the qDGCA freely generated from the nontrivial generators of the
    invariant polynomials of $\gg_{(n)}$, equipped with the trivial differential.
    \item<2->
      The degree of $b \gg_{(n)}$ is that of the highest degree invariant polynomial.
    \end{itemize}
  \end{block}
}


\frame{
  \frametitle{
    The algebra of $\gg_{(n)}$ invariant polynomials
  }

  
  \begin{block}{Example and Remark}
    The notation is derived from the important special abelian case
    where $\gg_{(n)} := \mathrm{Lie}(\Sigma^{n-1} U(1))$. In that
    case\vspace{-10pt}
    $$
      b \mathrm{Lie}(\Sigma^{n-1} U(1)) = \mathrm{Lie}(\Sigma^{n} U(1))
      \,,
    $$
    mimicking the fact that the classifying ``space'' of the
    $n$-group $\Sigma^{(n-1) U(1)}$ is the $(n+1)$-group
    $\Sigma^n U(1)$.
  \end{block}

} 


\frame{
  \frametitle{
    The algebra of $\gg_{(n)}$ invariant polynomials
  }


  \begin{block}{Remark}
    A morphism\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        b \gg_{(n)}^*
        \ar[ll]_{\{K_i\}}
      }
    $$
    is precisely the choice of closed $r$-forms $K_i$ on $X$,
    one for each
    degree $r$ generator $k_i$ of $b \gg_{(n)}^*$.
    
    There is a canonical morphism\vspace{-10pt}
    $$
      \xymatrix{
         \mathrm{ch}_{k_i}(\gg_{(n)})^*
         &&
         b \gg_{(n)}^*
         \ar[ll]
      }
    $$
    for each $k_i$, and composing this with a 
    \underline{\hyperlink{connection}{connection}}\vspace{-10pt}
    $$
      \xymatrix{
         \Omega^\bullet(X)
         &
         \mathrm{inn}(\gg_{(n)})^*
         \ar[l]_{(A,F_A)}
         &
         \mathrm{ch}_k(\gg_{(n)})^*
         \ar[l]
         &
         b \gg_{(n))}^*
         \ar[l]
         \ar@/^	2pc/[lll]_{k_i(F_A)}
      }
    $$
    picks out the Chern form of $A$ with respect to $k_{i}$.    
  \end{block}

}


\frame{
  \frametitle{
    The algebra of $\gg_{(n)}$ invariant polynomials
  }

  \begin{block}{Remark}
  
    For Lie 1-algebras $\gg_{(n)} = \gg$, the morphism
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        b \gg^*
        \ar[ll]_{\{K_i\}}
      }
    $$
    is essentially the Chern-Weil homomorphism, once we impose the
    conditions described in 
    \underline{
      \hyperlink{definition gn-bundle}{Definition of $\gg_{(n)}$-bundles}
    }.

 \end{block}

}



\subsection{Invariant polynomials of String and Chern-Simons Lie $n$-algebras}

\frame{
  \hypertarget{inv polynomials of string and chern}{}
  

  With these definitions in hand, we can now set out and 
  try to explicitly compute
  $b \gg_{(n)}^*$ for concrete examples.
  
  This will allow us then to make statements about the
  characteristic classes of $\gg_{(n)}$-bundles.  

}


\frame{
  \frametitle{Invariant polynomials of the String Lie 2-algebra}
  \hypertarget{invariant polynomials of string Lie 2}{}

  \begin{block}{Proposition}
    Let $\gg$ be a Lie algebra with transgressive 
    invariant polynomial $k$. Then
    the algebra of invariant polynomials of 
    the corresponding String (Baez-Crans type) 
    Lie 2-algebra $\gg_{\mu_k}$ is that of $\gg$ modulo
    $k$:\vspace{-10pt}
    $$
      b \gg_{\mu_k}^* \simeq b \gg^* / [k]
      \,.
    $$\vspace{-10pt}
  \end{block}
  
  \pause
  
  \begin{block}{Sketch of proof}
    In $\mathrm{inn}(\gg_{\mu_k}), $$k$ becomes a coboundary of invariant polynomials:\vspace{-12pt}
    \begin{eqnarray*}
      k &&= d_{\mathrm{inn}(\gg_{\mu_k})} \mathrm{cs}\\
      &&=
      d_{\mathrm{inn}(\gg_{\mu_k})} ((\mathrm{cs}- \mu) + \mu)\\
      &&=
      d_{\mathrm{inn}(\gg_{\mu_k})} ((\mathrm{cs}- \mu) + c)
    \end{eqnarray*}
  \end{block}


}


\frame{
  \frametitle{Invariant polynomials of the String Lie 2-algebra}


  \begin{block}{Interpretation}
     In 
     \underline{\hyperlink{n-bundles with connection}{Bundles with Lie $n$-algebra connection}}
     we find that morphisms\vspace{-10pt}
     $$
       \xymatrix{
          \Omega^\bullet(X)
          &&
          b \gg_{(n)}^*
          \ar[ll]_{\{K_i\}}
       }
     $$
     yield the characteristic classes of $\gg_{(n)}$-bundles. The
     above statement then amounts to saying that the characteristic classes
     of String bundles ($\gg_{\mu_k}$-bundles) are those of 
     $\gg$-bundles modulo the element $k$.
     
     Conversely, a $\gg$-bundle cannot be lifted to a 
     $\gg_{\mu_k}$-bundle unless its characteristic class corresponding to
     $k$ vanishes.
  \end{block}

}


\section{$n$-Bundles with Lie $n$-algebra connection}


\frame{

  \hypertarget{n-bundles with connection}{}
  \hypertarget{n-bundles with g-connection}{}

  \begin{enumerate}
            
    \item \uncover<0>{Motivation}

    \item \uncover<0>{Plan}
      
    \item \uncover<0>{Parallel $n$-transport}

    \item  \uncover<0>{Lie $n$-algebra cohomology}

    \item  Bundles with Lie $n$-algebra connection

      \begin{enumerate}
      
        \item
          \underline{
            \hyperlink{connection and curvature}{$\gg_{(n)}$-Connection and curvature}
          }

        \item
          \underline{
            \hyperlink{examples of connection n-forms}{Examples of connection $n$-forms}
          }


        \item
          \underline{
            \hyperlink{bundles with connection}{Bundles with $\gg_{(n)}$-connection}
          }

      
      \end{enumerate}

    \item  \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item  \uncover<0>{Conclusion}

    \item  \uncover<0>{Questions}

    \item  \uncover<0>{$n$-Categorical background}



  \end{enumerate}

}





\subsection{$\gg_{(n)}$-Connection and curvature}


\frame{
  \frametitle{Connection and Curvature}
  \hypertarget{connection}{}
  \hypertarget{connection and curvature}{}

  \begin{block}{Definition}
    For $X$ some manifold and $\gg_{(n)}$ a Lie $n$-algebra,
    a $\gg_{(n)}$-connection on the trivial $\gg_{(n)}$-bundle
    over $X$ is a morphism\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        \mathrm{inn}(\gg_{(n)})^*
        \ar[ll]_{(A,F_A)}
      }
      \vspace{-10pt}
      \,.
    $$
    Morphisms of connections are higher qDGCA morphisms 
    \vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        \mathrm{inn}(\gg_{(n)})^*
        \ar@/_1.9pc/[ll]_{(A,F_A)}^{\ }="s"
        \ar@/^1.9pc/[ll]^{(A',F_{A'})}_{\ }="t"
        %
        \ar@{=>} "s"; "t"
      }
      \vspace{-10pt}
    $$
    which vanish when pulled back along 
    $\mathrm{inn}(\gg_{(n)})^* \leftarrow b \gg_{(n)}^*$.
  \end{block}

}

\frame{
  \frametitle{Connection and Curvature}


   \begin{block}{Example}
     For $\gg_{(n)} = \gg_{(1)} = \gg$ an ordinary Lie algebra,
     connections
     $$
       \xymatrix{
         \Omega^\bullet(X)
         &&
         \mathrm{inn}(\gg_{(n)})^*
         \ar[ll]_{(A,F_A)}
       }
       \vspace{-10pt}
     $$
     are in bijection with $\gg$-valued 1-forms on $X$, and 
     morphisms of them are linearized gauge transformations of these.
   \end{block}
   
   We have the following situation
   $$
     \xymatrix{
       \gg_{(n)}^* 
       \ar[d]_{(A,F_A = 0)}
       && 
       \mathrm{inn}(\gg_{(n)})^* \ar[ll]
       \ar[d]_{(A,F_A)}
       \\
       \Omega^\bullet(X)
       &&
       \Omega^\bullet(X)
     }
   $$
   
   
}

\frame{
  \frametitle{Connection and Curvature}

  \begin{block}{Remark}
     Recall that $\mathrm{inn}(\gg_{(n)})$ is
     trivializable. This makes the full
     $\mathrm{Hom}(\mathrm{inn}(\gg_{(n)})^*, \Omega^\bullet(X))$
     also trivializable. But by restricting higher morphisms
     to those whose pullback along 
     $\mathrm{inn}(\gg_{(n)})^* \leftarrow b \gg_{(n)}^*$
     vanishes the crucial information is retained.
  \end{block}
    
  \begin{block}{Definition}
      \begin{tabular}{cc}
      \begin{tabular}{l}
    A morphism $\xymatrix{0 \ar[r]^<<<<{(e,\nabla)} & (A,F_A)}$\\
    is a section $e$ (of the trivial \\
    $\gg_{(n)}$-bundle) together \\ 
    with its covariant derivative \\
    $\nabla e$ with respect to the  \\
    connection $A$.
    \end{tabular}
    &
      \xymatrix{
        \Omega^\bullet(X)
        &&
        \mathrm{inn}(\gg_{(n)})^*
        \ar@/_1.9pc/[ll]_{0}^{\ }="s"
        \ar@/^1.9pc/[ll]^{(A,F_{A})}_{\ }="t"
        %
        \ar@{=>}|{(e, \nabla e)} "s"; "t"
      }
    \end{tabular}
  \end{block}

}

\frame{
  \frametitle{Connection and Curvature}

  \begin{block}{Definition}
    The $r$-form\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &
        \mathrm{inn}(\gg_{(n)})^*
        \ar[l]_{(A,F_A)}
        &
        \mathrm{ch}_k(\gg_{(n)})
        \ar[l]
        &
        b \gg_{(n)}^*
        \ar[l]
        \ar@/^2pc/[lll]_{k(F_A)}
      }
    $$
    for $k$ a degree $r$ invariant polynomial on $\gg_{(n)}$
    is the Chern-form of the connection $A$ with respect to $k$.
  \end{block}


  

}


\subsection{Examples of connection $n$-forms}

\frame{
  \hypertarget{examples of connection n-forms}{}
 
  \begin{block}{Observation}

   A connection\vspace{-10pt}
   $$
    \xymatrix{
      \Omega^\bullet(X)
      &&
      \mathrm{inn}(\gg_{(n)})
      \ar[ll]_{(A,F_A)}
    }
   $$
   on a trivial $\gg_{(n)}$-bundle is determined by
   an $n$-tuple of differential forms\vspace{-10pt}
   $$
     A \in \Omega^1(X,V_1)\times \Omega^2(X,V_2) \times \cdots \times
      \Omega^n(X,V_n)
      \,,
   $$
   where $V_k$ is the degree $k$ part of the graded vector space
   underlying $\gg_{(n)}$.

   The corresponding curvatures forms\vspace{-10pt}
   $$
     F_A \in \Omega^2(X,V_1)\times \Omega^3(X,V_2) \times \cdots \times
      \Omega^{n+1}(X,V_n)
   $$
   are uniquely fixed.

  \end{block}

}


\frame{

  The following lists some examples of 
  $\gg_{(n)}$-connections and the nature of the differential
  form data corresponding to it.

}


\frame{
  \frametitle{Ordinary connection 1-forms}

  \only<1>{

  \begin{block}{Ordinary connection 1-forms}
  
$$  
  \xymatrix{
    \mbox{n=1} 
    \\
    \gg
    \\
    \mathrm{Vect}(X)
    \ar[u]^>>>>{ (A) }|<<< {F_A = 0}
  }
  \hspace{70pt}
$$
for $A \in \Omega^1(X,\gg)$.

  
\end{block}

Morphisms into $\gg_{(1)}$ come from \emph{flat} connection 1-forms.

}


  \only<2>{

  \begin{block}{Ordinary connection 1-forms}
  
$$  
  \xymatrix{
    \mbox{n=1} 
    &
    \mbox{n=2}
    \\
    \gg
    \ar@{^{(}->}[r]
    &
    \mathrm{inn}(\gg)
    \\
    \mathrm{Vect}(X)
    \ar[u]^>>>>{ (A) }|<<< {F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[u]^{\hspace{1.4cm} (A)  }
  }
$$
for $A \in \Omega^1(X,\gg)$.

  
\end{block}

  Morphisms into $\mathrm{inn}(\gg_{(1)})$ come from \emph{arbitrary} connection 1-forms.


}

}



\frame{
  \frametitle{General Chern-Simons-like connections}

  \only<1-2>{
  \begin{block}{Theorem}
    For every degree $(2n+1)$ Lie algebra transgressive
    element, $(2n+1)$-connections with values in $\mathrm{cs}_k(\gg)$
    are in bijection with $\gg$-Chern-Simons forms.
  \end{block}
  
  \uncover<2>{This means...}
  
  }


  \only<3>{
  \begin{block}{}  
    \vspace{-20pt}
    $$
  \xymatrix{
    \hspace{2pt}
    \\
    \mbox{$1$}
    \\
    \gg 
    \\
    \\
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A) }|<<<<<<<<{F_A = 0}
  }
  \hspace{248pt}
$$


  \end{block}
  }
  

  \only<4>{
  \begin{block}{}  
    \vspace{-20pt}
    $$
  \xymatrix{
    &
    \mbox{Baez-Crans}
    \\
    \mbox{$1$}
    &
    \mbox{$2n$}
    \\
    \gg 
    & 
    \gg_{\mu_k}
    \ar@{->>}[l] 
    \\
    \\
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A) }|<<<<<<<<{F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A,B) }
      |<<<<<<<{F_A = 0 \atop dB + \mathrm{CS}_k(A) = 0}  
  }
  \hspace{168pt}
$$


  \end{block}
  }



  \only<5>{
  \begin{block}{}  
    \vspace{-20pt}
    $$
  \xymatrix{
    &
    \mbox{Baez-Crans}
    &
    \mbox{Chern-Simons}
    \\
    \mbox{$1$}
    &
    \mbox{$2n$}
    &
    \mbox{$2n+1$}
    \\
    \gg 
    & 
    \gg_{\mu_k}
    \ar@{->>}[l] 
    \ar@{^{(}->}[r] 
    &
    \mathrm{cs}_k(\gg)    
    \\
    \\
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A) }|<<<<<<<<{F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A,B) }
      |<<<<<<<{F_A = 0 \atop dB + \mathrm{CS}_k(A) = 0}  
    &
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A,B,C)}
      |<<<<<<<<{ C = dB + \mathrm{CS}_k(A) }
  }
  \hspace{75pt}
$$


  \end{block}
  }


  \only<6>{
  \begin{block}{}  
    \vspace{-20pt}
    $$
  \xymatrix{
    &
    \mbox{Baez-Crans}
    &
    \mbox{Chern-Simons}
    &
    \mbox{Chern}
    \\
    \mbox{$1$}
    &
    \mbox{$2n$}
    &
    \mbox{$2n+1$}
    &
    \mbox{$2n+1$}
    \\
    \gg 
    & 
    \gg_{\mu_k}
    \ar@{->>}[l] 
    \ar@{^{(}->}[r] 
    &
    \mathrm{cs}_k(\gg)    
    \ar@{->>}[r]
    &
    \mathrm{ch}_k(\gg)    
    \\
    \\
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A) }|<<<<<<<<{F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A,B) }
      |<<<<<<<{F_A = 0 \atop dB + \mathrm{CS}_k(A) = 0}  
    &
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A,B,C)}
      |<<<<<<<<{ C = dB + \mathrm{CS}_k(A) }
    &
    \mathrm{Vect}(X)
    \ar[uu]^>>>>>>>>{(A,C)}
      |<<<<<<<{ dC = k( (F_A)^{n+1})}
  }
$$


  \end{block}
  }


}


\frame{
   \frametitle{The standard Chern-Simons 3-connection}


  \begin{block}{Finally: the case we wanted to understand}

      Let now $\gg$ be semisimple and let 
       $$
         \mu = \langle \cdot , [\cdot, \cdot]\rangle
       $$
       be the canonical 3-cocycle.
         
  \end{block}


  \pause
  
  \begin{block}{Theorem (Baez, Crans, S, Stevenson)}
    
    The corresponding Baez-Crans Lie 2-algebra $\gg_{\mu}$
    is equivalent to that of the corresponding String 2-group
    $$
      \gg_\mu \simeq \mathrm{Lie}(\mathrm{String}_k(G))
      \,.
    $$
    
  \end{block}


}

\frame{
   \frametitle{The standard Chern-Simons 3-connection}


 \only<1>{
 \begin{block}{}

{\small
$$
 \raisebox{120pt}{
  \xymatrix{
    \gg 
    \ar@{=}[d]
    \\
    \gg 
    \\
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A) }|<<<{F_A = 0}
  }
  }
  \hspace{233pt}
$$}


\end{block}

}


 \only<2>{
 \begin{block}{}

{\small
$$
 \raisebox{120pt}{
  \xymatrix{
    \gg 
    \ar@{=}[d]
    & 
    \mathrm{string}_k(\gg) 
    \ar@{->>}[l] 
    \ar@{=}[d]^\sim
    \\
    \gg 
    & 
    \gg_k 
    \ar@{->>}[l] 
    \\
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A) }|<<<{F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A,B) }
      |<<<{F_A = 0 \atop dB + k\mathrm{CS}(A) = 0}  
  }
  }
  \hspace{66pt}\hspace{95pt}
$$}


 \end{block}
 }



 \only<3>{
 \begin{block}{}

{\small
$$
 \raisebox{120pt}{
  \xymatrix{
    \gg 
    \ar@{=}[d]
    & 
    \mathrm{string}_k(\gg) 
    \ar@{->>}[l] 
    \ar@{^{(}->}[r] 
    \ar@{=}[d]^\sim
    &
    \mathrm{inn}(\mathrm{string}_k(\gg))
    \\
    \gg 
    & 
    \gg_k 
    \ar@{->>}[l] 
    \ar@{^{(}->}[r] 
    &
    \mathrm{cs}_k(\gg)    
    \ar@{=}[u]_\sim
    \\
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A) }|<<<{F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A,B) }
      |<<<{F_A = 0 \atop dB + k\mathrm{CS}(A) = 0}  
    &
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A,B,C)}
      |<<<{ C = dB + k \mathrm{CS}(A) }
  }
  }
  \hspace{66pt}
$$}
 \end{block}
 }



 \only<4>{
 \begin{block}{}

{\small
$$
 \raisebox{120pt}{
  \xymatrix{
    \gg 
    \ar@{=}[d]
    & 
    \mathrm{string}_k(\gg) 
    \ar@{->>}[l] 
    \ar@{^{(}->}[r] 
    \ar@{=}[d]^\sim
    &
    \mathrm{inn}(\mathrm{string}_k(\gg))
    \\
    \gg 
    & 
    \gg_k 
    \ar@{->>}[l] 
    \ar@{^{(}->}[r] 
    &
    \mathrm{cs}_k(\gg)    
    \ar@{=}[u]_\sim
    \ar@{->>}[r]
    &
    \mathrm{ch}_k(\gg)    
    \\
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A) }|<<<{F_A = 0}
    &
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A,B) }
      |<<<{F_A = 0 \atop dB + k\mathrm{CS}(A) = 0}  
    &
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A,B,C)}
      |<<<{ C = dB + k \mathrm{CS}(A) }
    &
    \mathrm{Vect}(X)
    \ar[u]^>>>{(A,C)}
      |<<<{ dC = \langle F_A \wedge F_A \rangle }
  }
  }
$$}


 \end{block}
 }



}


\frame{
  \frametitle{General Chern-Simons-like connections}

  
   \begin{block}{Remark.}
   
     The relevance of this statement is that this means that 
     under the integration morphism\vspace{-10pt}
     $$
       \xymatrix{
         \fbox{Lie $n$-algebroids}
         \ar[rr]^{\mbox{integration}}
         &&
         \fbox{Lie $n$-groupoids}
       }
     $$
     a morphism\vspace{-10pt}
     $$
       \xymatrix{
         \Omega^\bullet(X)
         &&&
         \mathrm{ch}_k(\gg)
         \ar[lll]_{(C = \mathrm{CS}_k(A) + d B)}
       }
     $$
     should turn into a 4-functor\vspace{-10pt}
     $$
       \xymatrix{
         \Pi_4(X)
         \ar[rrr]^{\mathrm{tra}_{C}}
         &&&
         G_{(4)}
       }
     $$
     which on 3-dimensional volumes $V$ 
     acts as the Chern-Simons \emph{functional}\vspace{-10pt}
     $$
       V \mapsto \exp(i \int_V \mathrm{CS}(A))
       \,.
     $$
   \end{block}


}



\subsection{$\gg_{(n)}$-Bundles with connection}



\frame{
  \frametitle{$\gg_{(n)}$-Connections on nontrivial bundles}
  \hypertarget{bundles with connection}{}

  \begin{block}{Remark}
    For $\gg_{(n)}$ any Lie $n$-algebra, the sequence
    $$
      \xymatrix{
        \gg_{(n)}^*
        &
        \mathrm{inn}(\gg_{(n))})^*
        \ar[l]_{i_u^*}
        &
        b \gg_{(n)}^*
        \ar[l]_{p_u^*}
      }
    $$
    plays the role of differential forms on the universal $\gg_{(n)}$-$n$-bundle.
  \end{block}

  For more background on this, see
  \begin{itemize}
        \item \underline{\hyperlink{universal bundles in terms of groupoids}{Universal $G_{(n)}$-bundles in terms of $n$-groupoids}}
        
        \item \underline{\hyperlink{Gn bundles with connections}{$G_{(n)}$-bundles with connection}}
  \end{itemize}
  in \underline{\hyperlink{n-categorical background}{$n$-C	ategorical background}}. 

}




\frame{
  \frametitle{Bundles with $\gg_{(n)}$-connection}
  \hypertarget{definition gn-bundle}{}  

  \begin{block}{Definition}
    \begin{tabular}{cc}
     \begin{tabular}{l}
     \uncover<1->{A bundle $p : P \to X$} \\
     \uncover<1->{with $\gg_{(n)}$-connection is }\\
     \uncover<2->{a morphism $(A,F_A)$}\\
     \uncover<3->{and a morphism $i^*$}\\
     \uncover<4->{such that $i^*A \simeq i_u^*$};\\
     \uncover<5->{and a morphism $p^*$}\\
     \uncover<6->{and a choice} \\
     \uncover<6->{of $r$-forms $\{K_i\}$}\\
     \uncover<7->{such that $p^*K_i \simeq k_i(F_A)$.}
     \end{tabular}
     &
     $
      \raisebox{50pt}{
      \xymatrix{
       \uncover<3->{\alert<3>{\Omega^\bullet_{\mathrm{li}}(|G_{(n)}|)}}
       &&
       \alert<3>{\gg_{(n)}^*}
       \ar[ll]_{\uncover<3->{\alert<3>{\simeq}}}^<{\ }="s1"
       \\
       \\
       \uncover<2->{\alert<2>{\Omega^\bullet(P)}}
       \ar[uu]^{\uncover<3->{\alert<3>{i^*}}}_<{\ }="t1"
       &&
       \alert<2>{\mathrm{inn}(\gg_{(n)})^*}
       \ar[uu]^{i_u^*}
       \ar[ll]_{\uncover<2->{\alert<2>{(A,F_A)}}}^<{\ }="s2"
       \\
       \\
       \uncover<5->{\alert<5>{\Omega^\bullet(X)}}
       \ar[uu]^{\uncover<5->{\alert<5>{p^*}}}_<{\ }="t2"
       &&
       b \gg_{(n)}^*
       \ar[ll]_{\uncover<6->{\alert<6>{\{K_i\}}}}
       \ar[uu]^{p_u^*}
       %
       \ar@{=>}_{\uncover<4->{\alert<4>{=}}} "s1"; "t1"
       \ar@{=>}_{\uncover<7->{\alert<7>{\simeq}}} "s2"; "t2"
     }}$
     \end{tabular}
  \end{block}
  

}


\section{Examples of $\gg_{(n)}$-bundles}



\frame{
  \hypertarget{examples of gn bundles}{}


  \begin{enumerate}
 
    \item \uncover<0>{Motivation}

    \item \uncover<0>{Plan}

    \item \uncover<0>{Parallel $n$-transport}

    \item \uncover<0>{Lie $n$-algebra cohomology}

    \item \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item {Examples of $\gg_{(n)}$-bundles}

      \begin{enumerate}
      
        \item \underline{\hyperlink{ordinary bundles}{Ordinary bundles}}

        \item \underline{\hyperlink{line 2-bundles}{Line 2-bundles (abelian gerbes)}}

        \item \underline{\hyperlink{string 2-bundles}{String 2-bundles}}
      
        \item \underline{\hyperlink{Chern-Simons 3-bundles}{Chern-Simons 3-bundles}}

      \end{enumerate}


    \item \uncover<0>{Conclusion}

    \item  \uncover<0>{Questions}

    \item \uncover<0>{$n$-Categorical background}



  \end{enumerate}


}



\subsection{Ordinary bundles}

\frame{
  \frametitle{Ordinary bundles}
  \hypertarget{ordinary bundles}{}

  \begin{block}{Example}
    For an ordinary Lie algebra $\gg_{(n)} = \gg$ this reproduces the
    definition of a Cartan-Ehresmann connection:
  \end{block}
    
    \only<2>{
      The morphism\vspace{-10pt}
      $$
        \xymatrix{
         \Omega^\bullet(P)
         &&
         \mathrm{inn}(\gg)^*
         \ar[ll]_{(A,F_A)}
        }
      $$
      is a $\gg$-valued 1-form $A$ on the total space $P$ of the bundle.
    }
    
    \only<3>{
      The square\vspace{-10pt}
      $$\xymatrix{
        \Omega_{\mathrm{li}}^\bullet(G)
        &&
        \gg^*
        \ar[ll]_{\simeq}
        \\
        \\
        \Omega^\bullet(P)
        \ar[uu]_{i^*}
        &&
        \mathrm{inn}(\gg)^*
        \ar[ll]_{(A,F_A)}
        \ar[uu]
      }
      $$
      says that $A$ restricted to the fiber is the canonical 1-form on $G$.
    }
    \only<4>{
      The square\vspace{-10pt}
      $$\xymatrix{
        \Omega^\bullet(P)
        &&
        \mathrm{inn}(\gg)^*
        \ar[ll]_{(A,F_A)}^<{\ }="s"
        \\
        \\
        \Omega^\bullet(X)
        \ar[uu]^{p^*}_<{\ }="t"
        &&
        b \gg^*
        \ar[ll]_{\{K_i = k_i(F_A)\}}
        \ar[uu]
        %
        \ar@{=>}_{\simeq} "s"; "t"
      }
      $$
      says that the Chern forms $k_i(F_A)$ on the total space have to
      descend to the characteristic classes on the base space.
      A sufficient condition for this is the $\gg$-equivariance of $A$.
    }
    

}


\subsection{Line 2-bundles (abelian gerbes)}

\frame{
  \frametitle{Line 2-bundles (abelian gerbes)}
  \hypertarget{line 2-bundles}{}

  \begin{block}{Example}
    For $\gg_{(2)} = \mathrm{Lie}(\Sigma U(1))$
    the morphism\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        b \gg_{(2)}^*
        \ar[ll]_K
      }
    $$
    defines a closed 3-form on $X$. 
    
    The condition\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet_{\mathrm{li}}(|G_{(2)}|)
        &&
        \gg_{(2)}^*
        \ar[ll]_\simeq
      }
    $$
    says that the fibers have\vspace{-10pt}
    $$
      H^\bullet(|G_{(2)}|) = H^2(|G_{(2)}|) \simeq \mathbb{R}
      \,.
    $$
    They look like $PU(H)$.
  \end{block}

}


\subsection{String 2-bundles}


\frame{
  \frametitle{String 2-bundles}
  \hypertarget{string 2-bundles}{}

  \begin{block}{Example}
    For $\gg$ simple and
    $\gg_{(2)} = \gg_{\langle\cdot, [\cdot,\cdot]\rangle}$
    the String Lie 2-algebra,
    the morphism\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        b \gg_{(2)}^*
        \ar[ll]_{\{K_i = k_i(F_A)\}}
      }
    $$
    assigns, due to the nature of the  
    \underline{\hyperlink{invariant polynomials of string Lie 2}{
     invariant polynomials of the String Lie 2-algebra}}, 
    the characteristic classes of a $\gg$-bundle with 
    $[\langle F_A \wedge F_A\rangle]$ vanishing.
    
    \uncover<2->{
    The condition\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet_{\mathrm{li}}(|G_{(2)}|)
        &&
        \gg_{(2)}^*
        \ar[ll]_\simeq
      }
    $$
    says that the fibers are like $G$ but with \vspace{-10pt}
    $$
      H^3(|G_{(2)}|) \simeq 0
      \,.
    $$
    This says they look like the 
    \underline{\hyperlink{string group}{String group}}.
    }
  \end{block}

}


\subsection{Chern-Simons 3-bundles}


\frame{
  \frametitle{Chern-Simons 3-bundles}
  \hypertarget{Chern-Simons 3-bundles}{}

  \begin{block}{Example}
    For $\gg$ simple and
    $\gg_{(2)} = \mathrm{ch}_{\langle\cdot,\cdot\rangle}(\gg)$
    the Chern Lie 3-algebra corresponding to the Killing form,
    the morphism\vspace{-10pt}
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        b \gg_{(2)}^*
        \ar[ll]_{\{K_i = k_i(F_A)\}}
      }
    $$
    assigns the Pontryagin class of a $\gg$-bundle.
    
  \end{block}

}


\section{Conclusion}

\frame{
  \hypertarget{conclusion}{}


 \begin{enumerate}
 
   \item \uncover<0>{Motivation}

   \item \uncover<0>{Plan}

   \item \uncover<0>{Parallel $n$-transport}

   \item \uncover<0>{Lie $n$-algebra cohomology}
 
   \item \uncover<0>{Bundles with Lie $n$-algebras connection}

   \item \uncover<0>{Examples of $\gg_{(n)}$-bundles}

   \item Conclusion
   
     \begin{enumerate}
     
       \item \underline{\hyperlink{n-transport conclusion}{Integral picture: parallel $n$-transport}}
      
       \item \underline{\hyperlink{n-Lie theory}{$n$-Lie theory}}
      
       \item \underline{\hyperlink{g-n connection conclusion}{Differential picture: Lie $n$-algebra connections}}
      
     \end{enumerate}

    \item  \uncover<0>{Questions}

   \item \uncover<0>{$n$-Categorical background}


 \end{enumerate}

}

\subsection{Integral picture: parallel $n$-transport}


\frame{
  \hypertarget{n-transport conclusion}{}
  
  \begin{block}{$n$-Bundles with connection}

      $G_{(n)}$ $n$-bundles ($(n-1)$-gerbes) with connection are 
      \begin{itemize}
        \item
          locally trivializable
          parallel transport $n$-functors
        \item
          or rather their curvature $(n+1)$-functors
        \item
          from the fundamental $(n+1)$-groupoid of the base space
        \item
          to (a representation of) the structure Lie $n$-group $G_{(n)}$
        \item
          or rather (locally) to its inner automorphism $(n+1)$-group
          $\mathrm{INN}_0(G_{(n)})$.
      \end{itemize}      
  
 \end{block}

}


\frame{

  \begin{block}{Examples of $n$-Bundles with connection}


     \begin{itemize}
        
        \item Ordinary bundles with connection are parallel transport
          1-functors
          
        \item $U(1)$ bundle gerbes with connection are descent data
          of $\Sigma U(1)$ 2-transport.
          
        \item Line bundle gerbes with connection are descent data of 
           $1d\mathrm{Vect}$ 2-transport.
           
        \item Aschieri-Jurco nonabelian bundle gerbes with connection are descent data
          of $\mathrm{Bitor}(H)$ 2-transport.
           
        \item Breen-Messing nonabelian gerbe connection data is descent data
          for $\mathrm{INN}_0(\mathrm{AUT}(G))$ 3-curvature of 2-transport.
         
        \item Stolz-Teichner String connection is like associated
           $\mathrm{String}_k(G)$ 2-transport.
        
     \end{itemize}
  \end{block}

}

\frame{

  \begin{block}{Universal $n$-bundles in terms of $n$-groupoids}

   \begin{itemize}
     \item
        For every $n$-group $G_{(n)}$ there is an $(n+1)$-group 
        $\mathrm{INN}_0(G_{()})$ of inner automorphisms.
        
     \item
        It sits in a sequence\vspace{-10pt}
        $$
          Z(G_{(n)}) \to \mathrm{INN}_0(G_{(n)})
          \to
          \mathrm{AUT}(G_{(n)})
          \to
          \mathrm{OUT}(G_{(n)})
        $$
     \item
       Its underlying $n$-groupoid plays the role of the universal
       $G_{(n)}$-bundle\vspace{-10pt}
       $$
         G_{(n)} \to \mathrm{INN}_0(G_{(n)}) \to \Sigma G_{(n)}
       $$
     \item 
       For $n=1$ shown by Segal in the 60s:\vspace{-10pt}
       $$
         \stackrel{|\cdot|}{\mapsto}
         (G \to E G \to B G)
       $$
     \item
       For $n=2$ discussed in [RobertsSchreiber].
    
    \end{itemize}
  \end{block}

}


\subsection{$n$-Lie theory}

\frame{
  \hypertarget{n-Lie theory}{}


  \begin{block}{Passage between Lie $n$-groupoids and Lie $n$-algebroids}
  
   \begin{itemize}

    
    \item
      Lie $n$-algebras and Lie $n$-algebroids are to Lie $n$-groups and
      Lie $n$-groupoids like Lie algebras are to Lie groups.
           

    \item
      A full $n$-Lie theorem -- concerning differentiation of Lie $n$-groupoids
      and integration of Lie $n$-algebroids --  is expected, even though
      only partially understood so far.


    \item
      Still, we can transfer structural understanding between 
      the two realsm.
      
    \item
      Parallel $n$-transport is a morphism of Lie $n$-groupoids. Hence
      it corresponds differentially to a morphisms of Lie $n$-algebroids.

   \end{itemize}
  
  \end{block}

}

\frame{

  \begin{block}{Passage between Lie $n$-algebras and differential algebra}

     \begin{itemize}

        \item General abstract operad nonsense implies equivalence between
         Lie $n$-algebras and $n$-term $L_{\infty}$-algebras, or their
         duals: free graded commutative algebras with a nilpotent degree +1
         differential (qDGCAs).

       \item 
          qDGCAs are useful for concrete computations. 
          
       \item
         qDGCAs prevail in physics literature (compare in particular AKSZ-BV).
         Making the explicit $n$-categorical structure explicit is
         often useful.

       \item
          For instance pairing the qDGCA description with its understanding
          in terms of Lie $n$-algebra yields understanding of Lie $n$-algebra
          cohomology and $n$-characteristic classes.
     
     \end{itemize}


  \end{block}

}


\frame{
  
  \begin{block}{Lie $n$-algebra cohomology}


     
       The notion of Lie-cocycle, invariant polynomial and transgression
         elements can be generalized to Lie $n$-algebras.
         
      \begin{tabular}{cc|cc}
       \hline
       Lie algebra cocycle & $\mu$ & Baez-Crans Lie $n$-algebra & $\gg_{\mu}$
       \\
       invariant polynomial & $k$ & Chern Lie $n$-algebra 
          & $\mathrm{ch}_k(\gg)$
       \\
       transgression element & $\mathrm{cs}$ & Chern-Simons Lie $n$-algebra
         & $\mathrm{cs}_k(\gg)$
         \\
         \hline
     \end{tabular}

   
     For every transgression element $\mathrm{cs}$ these fit into a weakly exact
     sequence\vspace{-10pt}
     $$
       \gg_{\mu_k} \to \mathrm{cs}_k(\gg) \to \mathrm{ch}_k(\gg)
       \,.
     $$   
    


  \end{block}


}


\frame{

  \begin{block}{Cokernels, mapping cones and inner derivations}

    \begin{itemize}
    
      \item
        Crucial for considerations of $\gg_{(n)}$-connections is the strict cokernel\vspace{-10pt}
        $$
          \xymatrix{
            \ff_{(n)} \ar@{^{(}->}[r]^{t} & \gg_{(n)} \ar@{->>}[r] & \mathrm{coker}(t) 
          }
        $$
        of Lie $n$-algebra injections


      \item
        and its weak analog, the mapping cone Lie $(n+1)$-algebra
        $
          (\xymatrix{
            \ff_{(n)} \ar@{^{(}->}[r]^{t} & \gg_{(n)} 
          })
        $.  
       \item
          \begin{itemize}
            \item
              $
              (\xymatrix{
                \gg_{(n)} \ar[r]^{\mathrm{Id}} & \gg_{(n)} 
              }) = \mathrm{inn}(\gg_{(n)})
              $
            is the inner derivation Lie $(n+1)$-algebra of $\gg_{(n)}$ --
            codomain for $\gg_{(n)}$-connections
           \item
              $
              \mathrm{coker}(\xymatrix{
                \gg_{(n)} \ar@{^{(}->}[r] & \mathrm{inn}(\gg_{(n)}) 
              }) = b \gg_{(n)}
              $
              is the Lie $n'$-algebra generated from the classes of invariant 
              $\gg_{(n)}$ polynomials -- it plays the role of the classifying space
              for $\gg_{(n)}$
            \item
              $
                \mathrm{coker}(
                  \gg_{(n)} \hookrightarrow
                   (\xymatrix{
                     \ff_{(n)} \ar[r]^{t} & \gg_{(n)} 
                   })              )
              $
              is the structure Lie $n'$-algebra for obstructions of extensions through
              $\gg_{(n)} \to \mathrm{coker}(t)$.
         \end{itemize}         



    \end{itemize}


  \end{block}

}



\subsection{Differential picture: Lie $n$-algebra connections}

\frame{
  \hypertarget{g-n connection conclusion}{}

  \begin{block}{$\gg_{(n)}$-Bundles with connection}
  
    After passing from Lie $n$-groupoids to Lie $n$-algebroids 

    \begin{itemize}
    
      \item
        The curvature $(n+1)$-functor\vspace{-10pt}
        $$
          \mathrm{curv} : \Pi_{n+1}(P) \to \Sigma \mathrm{INN}_0(G_{(n)})
        $$
        turns into a qDGCA morphism\vspace{-10pt}
        $$
          \xymatrix{
            \Omega^\bullet(P)
            &&
            \mathrm{inn}(\gg_{(n)})^*
            \ar[ll]_{(A,F)}
          }
          \,.
        $$
       \item
         The $n$-groupoid version of the universal $G_{(n)}$-bundle\vspace{-10pt}
         $$
           G_{(n)} \to \mathrm{INN}(G_{(n)}) \to \Sigma \mathrm{INN}(G_{(n)})
         $$
         turns into the sequence
         $$
           \xymatrix{
             \gg_{(n)}^*
             &&
             \mathrm{inn}(\gg_{(n)})^*
             \ar[ll]
             &&
             b \gg_{(n)}^*
             \ar[ll]
           }
         $$
    \end{itemize}

  \end{block}


}

\frame{
  \hypertarget{g-n connection conclusion}{}

  \begin{block}{The $n$-Ehresmann conditon}

    And the descent condition on $(A,F_A)$ says we have a 
    cone of the universal $G_{(n))}$-bundle in that
    $$
    \xymatrix{
      \Omega^\bullet_{\mathrm{li}}(|G_{(n)}|)
      &&
      \gg_{(n)}^*
      \ar[ll]_{\simeq}^<{\ }="s1"
      \\
      \\
      \Omega^\bullet(P)
      \ar[uu]^{i^*}_<{\ }="t1"
      &&
      \mathrm{inn}(\gg_{(n)})^*
      \ar[ll]_{(A,F_A)}^<{\ }="s2"
      \ar[uu]
      \\
      \\
      \Omega^\bullet(X)
      \ar[uu]^{p^*}_<{\ }="t2"
      &&
      b \gg_{(n)}^*
      \ar[uu]
      \ar[ll]_{\{K_i = k_i(F_A)\}}
      %
      \ar@{=>}_= "s1"; "t1"
      \ar@{=>}_\simeq "s2"; "t2"
    }
    $$

  \end{block}

}



\frame{

 \begin{block}{$n$-Chern-Weil and characteristic classes}
 
   \begin{itemize}
   
     \item Here 
     $\xymatrix{
        \Omega^\bullet(X) && b \gg_{(n)}^*
        \ar[ll]_{\{K_i = k_i(F_A)\}}
      }
     $
     is the $n$-Chern-Weil homomorphism, assigning the characteristic
     classes $K_i$ to the given $(n+1)$-curvature $F_A$.
   
     \item
       For instance: the characteristic classes of $\gg_{\mu_k}$-bundles
       (String 2-bundles) are those of the underlying $\gg$-bundles,
       but modulo $K = k(F_A) = \langle F_A \wedge F_A\rangle$\vspace{-10pt}
       $$
         b \gg_{\mu_k}^* \simeq b \gg/[k]
         \,.
       $$   
   
   \end{itemize}
 
 \end{block}


}


\section{Questions}

\frame{
  \hypertarget{questions}{}

  \begin{enumerate}
 
    \item \uncover<0>{Motivation}

    \item \uncover<0>{Plan}

    \item \uncover<0>{Parallel $n$-transport}

   \item \uncover<0>{Lie $n$-algebra cohomology}

    \item \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item \uncover<0>{Conclusion}

    \item {Questions}

       \begin{enumerate}
       
         \item \underline{\hyperlink{sugra}{11-Dimensional supergravity}}
       
       \end{enumerate}

    \item \uncover<0>{$n$-Categorical background}



  \end{enumerate}
  


}

\subsection{11-Dimensional supergravity}

\frame{
  \hypertarget{sugra}{}
 
   \begin{block}{Remark.}
     
      There is an obvious and straightforward generalization of
      all Lie $n$-algebra construction from the world of vector
      spaces to that of super vector spaces (i.e. to the category
      of $\mathbb{Z}_2$-graded vector spaces equipped with the
      unique nontrivial symmetric braiding).
     
   \end{block}

   \pause
 
   \begin{block}{The supergravity Lie 3-algebra}

     D'Auria and Fr{\'e} noticed that (rephrased in our language)
     11-dimensional supergravity is governed by 
     the Baez-Crans type Lie 3-algebra\vspace{-10pt}
     $$
       \mathrm{sugra}_{11}
       :=
       s\mathfrak{iso}(11)_{\mu}
     $$
     coming from a 4-cocylce $\mu$
     of the super-Poincar{\'e} Lie algebra $s\mathfrak{iso}(11)$.

   \end{block}

}

\frame{

  \begin{block}{Sugra configurations are $\mathrm{sugra}_{11}$-connections}
  
    A field configuration of supergravity is nothing but 
    a $\mathrm{sugra}_11$-connection
    $$
      \xymatrix{
        \Omega^\bullet(X)
        &&
        \mathrm{inn}(\mathrm{sugra}_{11})^*
        \ar[ll]_{(A,F_A)}
      }
      \,,
    $$
    where $A$ encodes
    \begin{itemize}
      \item
        the graviton, in terms of
        \begin{itemize}
          \item
             the vielbein
          \item
              the spin connection
        \end{itemize}
       \item
         the gravitino
       \item
         the 3-form field\,.
    \end{itemize}
  \end{block}

}

\frame{

  \begin{block}
  
    This suggests that 11-dimensional supergravity is a theory
    of $\mathfrak{siso}(11)_\mu$-bundles with connection.
    
    The \underline{\hyperlink{definition gn-bundle}{$n$-Ehresmann condition}}
    would give the global description.
  
  \end{block}

}



\section{$n$-Categorical background}


\frame{
  
  \hypertarget{n-categorical background}{}


 \begin{enumerate}

    \item \uncover<0>{Motivation}

    \item \uncover<0>{Plan}

    \item \uncover<0>{Parallel $n$-transport}

    \item \uncover<0>{Lie $n$-algebra cohomology}

    \item \uncover<0>{Bundles with Lie $n$-algebra connection}

    \item \uncover<0>{Examples of $\gg_{(n)}$-bundles}

    \item \uncover<0>{Conclusion}

    \item \uncover<0>{Questions}

    \item $n$-Categorical background

     \begin{enumerate}

        \item  \underline{\hyperlink{tangent categories}{Tangent categories}}

        \item \underline{\hyperlink{inner automorphism groups}{Inner autmorphism $(n+1)$-groups}}

        \item \underline{\hyperlink{universal bundles in terms of groupoids}{Universal $G_{(n)}$-bundles in terms of $n$-groupoids}}
        
        \item \underline{\hyperlink{Gn bundles with connections}{$G_{(n)}$-bundles with connection}}

      \end{enumerate}

  \end{enumerate}

}



\subsection{Tangent Categories}


\frame{
  \hypertarget{tangent categories}{}
  
  The sequences of Lie $n$-algebras
  which appeared in 
  \underline{\hyperlink{n-bundles with connection}{Bundles with Lie $n$-algebra connection}}
  and which were related to universal $\gg_{(n)}$-bundles have their origin
  in a very fundamental $n$-categorical construction which we address
  as the construction of \emph{tangent $n$-categories}.


}


\frame{

  \begin{block}{Definition}
    Let 
    $$
      2 := \{ \xymatrix{
         \bullet \ar[r] & \circ
      }\}
    $$
    be the category wtih two objects and one nontrivial morphism,
    going between them.
  \end{block}


  \begin{block}{Definition}
    For $C$ any category, the tangent category $TC$ is the strict pullback
    $$
      \xymatrix{
        T C 
        \ar[r]
        \ar[d]
        &
        C^2
        \ar@{->>}[d]
        \\
        C_0 
        \ar@{^{(}->}[r]
        &
        C
      }
    $$
    in $\mathrm{Cat}$.
  \end{block}

}


\frame{

  \begin{block}{Proposition}
  
    \begin{itemize}
      \item
        $\mathrm{Mor}(C) \to T C \to C$ is exact
      \item
        for $C$ a (Lie) groupoid, $T C \simeq C_0$
      \item
        sections $\Gamma(T C)$ of $T C \to C_0$ inherit a 2-group structure through
        the inclusion $\Gamma(T C ) \hookrightarrow T_{\mathrm{Id}} \mathrm{End}(C)$
      \item
        $\Gamma_\mathbb{R}(T C) := \mathrm{Hom}(\mathbb{R}, \Gamma (T C))$ 
        is the Lie algebroid of $C$
      \item
        for $C = \Sigma G$, $T C := \mathrm{INN}(G)$ is the inner automorphism
        2-group of $G$.
    \end{itemize}

  \end{block}


}


\frame{

  \begin{block}{Remark.}
  
    These statements have more or less obvious generalizations to 
    $n > 1$. For $n=2$ this is done in [RobertsSchreiber]
  
  \end{block}
}


\subsection{Inner automorphisms $(n+1)$-groups}


\frame
{
  \hypertarget{inner automorphism groups}{}

\begin{block}{Inner automorphism $(n+1)$-Groups}

  \begin{itemize}
       \item Every $n$-group $G_{(n)}$ has an {$(n+1)$}-group 
          {$\mathrm{AUT}(G_{(n)})$} of automorphisms.
          
       \item This sits inside an exact sequence
         {
         $
           \alert<4>{
           1 \to Z(G_{(n)}) \to \alert<5->{\mathrm{INN}(G_{(n)})} \to \mathrm{AUT}(G_{(n)})
           \to \mathrm{OUT}(G_{(n)}) \to 1
           }
         $
         }
        \item and $\mathrm{INN}_0$ plays the role of the universal $G_{(n)}$-bundle
        \uncover{
          $
            {
              G_{(n)} \to \mathrm{INN}_0(G_{(n)}) \to \Sigma G_{(n)}
             }
          $
        }
  \end{itemize}
 

  {[David Roberts, U.S.]}


\end{block}

}


\subsection{Universal $n$-bundles in terms of $n$-groupoids}


\frame{
  \hypertarget{universal bundles in terms of groupoids}{}


  \begin{block}{Observation}

  Given a cover $Y \to X$ and a $G$-coycle 
  $
    g : Y^{[2]} \to \Sigma G
  $
  its pullback\vspace{-10pt}
  $$
    \xymatrix{
      Y^{[2]}\times_g \mathrm{INN}(G)
      \ar[rr] 
      \ar[dd]
      &&
      \mathrm{INN}(G)
      \ar[dd]
      \\
      \\
      Y^{[2]}
      \ar[rr]^g
      &&
      \Sigma G
    }
  $$
  plays the role of the total space of the $G$-bundle clasified by $g$.

  \end{block}
  
  Analogous statements hold for $n > 1$.
}


\subsection{$G_{(n)}$-bundles with connection}

\frame{
  \hypertarget{Gn bundles with connections}{}

  The following presents the arrow-theory of universal $n$-bundles and their
  pullbacks and connections (explicitly only for $n=1$)
  in a way that shows how the 
  \underline{\hyperlink{definition gn-bundle}{definition}} 
  of bundles with $\gg_{(n)}$-connection arises.
     
}



\frame{

\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{ \mbox{ \tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  %(55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  %(90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  %(20,-58)+(-16,20)*{Y^{[2]}}="y2";
  %(55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  %(90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  %\ar_g "y2"+(0,-3); "sg"+(0,3)  
  %\ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  %\ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  %\ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  %\ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  %\ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  %\ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\alert<1>{The universal $G$ 1-bundle.}
\only<2>{\alert<2>{Now suppose that $G = U(1)$ }.}
\only<3>{Then $\Sigma G$ is itself a 2-group.}
\only<4>{And what used to be the classifying space for $G$ 1-bundles\dots}
}


\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  %(20,-58)+(-16,20)*{Y^{[2]}}="y2";
  %(55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  %(90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  %\ar_g "y2"+(0,-3); "sg"+(0,3)  
  %\ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  %\ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  %\ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  %\ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1>{\dots becomes the fiber of the universal $\Sigma G$ 2-bundle.}
\only<2>{\alert<1>{Given a space $X$, let $\Pi_2(X)$ be its fundamental 2-groupoid.}}
}

\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  %(20,-58)+(-16,20)*{Y^{[2]}}="y2";
  %(55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  %\ar_g "y2"+(0,-3); "sg"+(0,3)  
  %\ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  %\ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  %\ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  %\ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1-2>{Given a space $X$, let $\Pi_2(X)$ be its fundamental 2-groupoid.}
\only<3>{Then a (smooth) morphism from $\Pi_2(X)$ to $\Sigma \Sigma G$}
}


\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  %(20,-58)+(-16,20)*{Y^{[2]}}="y2";
  %(55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  %\ar_g "y2"+(0,-3); "sg"+(0,3)  
  %\ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  %\ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  %\ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1>{Then a (smooth) morphism from $\Pi_2(X)$ to $\Sigma \Sigma G$}
\only<2>{Is a choice of 2-form $K \in \Omega^2(X)$ on $X$.}
\only<3>{This we may regard as a trivial $\Sigma^2 G$ 2-bundle with connection on $X$.}
\only<4->{Hence we may ask if we can lift the structure 2-group}
\only<5->{through $\Sigma \mathrm{INN}(G) \to \Sigma \Sigma G$.}
\only<6->{We can, if we can form}
}

\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  %(20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  %\ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  %\ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1>{Hence we may ask if we can lift the structure 2-group
through $\Sigma \mathrm{INN}(G) \to \Sigma \Sigma G$.
We can, if we can form this.}
\only<2>{Here $\pi : Y \to X$ is a choice of cover of $X$.}
\only<3>{And $\mathcal{C}_2(Y)$ is generated from $\Pi_2(Y)$ and from $Y^{[2]}$,
modulo an obvious relation.}
}



\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  %\ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1>{And $\mathcal{C}_2(Y)$ is generated from $\Pi_2(Y)$ and from $Y^{[2]}$,
modulo an obvious relation.}
}


\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1>{
 Hence $g : Y^{[2]} \to \Sigma G$ is the classifying map of a $G$ 1-bundle. 
}
\only<2>{
  While the smooth parallel transport 2-functor
  $(\mathrm{tra},\mathrm{curv}) : \Pi_2(Y) \to \Sigma \mathrm{INN}(G)$
  encodes a compatible connection 1-form $A$ and its curvature 2-form $F_A$.
} 
\only<3>{
  Requiring the left square to commute is the gluing condition on 
  a $G$-bundle with connection.
}
\only<4>{
  Requiring the \alert<4>{right} square to commute 
  says that the 2-form $K = F_A$ is the curvature 2-form of this
  connection.
}
\only<5>{
  Requiring the right square to commute 
  \alert<5>{up to natural isomorphism}
  says that $K$ represents the Chern class of 
  $g$.
}
\only<6>{
  Finally, we obtain the total ``space'' of the $G$-bundle thus
  classified by pulling back $g$ along $\mathrm{INN}(G) \to \Sigma G$.
}
}

\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma (G)$}};
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  (20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  \ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  \ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar "igt"+(0,-2); "iig"+(1,3)
\endxy
\vspace{6pt}
\only<1>{
  Finally, we obtain the total space of the $G$-bundle thus
  classified by pulling back $g$ along $\mathrm{INN}(G) \to \Sigma G$.
}
\only<2>{
  There is in fact an entire lattice of universal $n$-bundles in the 
  background.
}
}


\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  (55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  (90,0)*{\mbox{\tiny $\Sigma G$}}="sgt";
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  (55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  (90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  (20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  \ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  \ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  \ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  \ar "y2ig"; "y2"+(1,3)
  %
  \ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  \ar@{^{(}->} "igt"+(0,-2); "iig"+(1,3)
  %
  \ar@{^{(}->} "g"+(4,0); "igt"+(-6,0)
  \ar@{->>} "igt"+(6,0); "sgt"+(-3,0)
  %
  \ar@{^{(}->} "sgt"+(-2,-3); "sigt"+(0,3)
  \ar@{->>} "sigt"; "ssg"+(1,3)
  %
  \ar@{=>}_>>>>>>>>>>{\simeq} "sgt"+(-6,-3); "iig"+(6,3)
\endxy
\vspace{6pt}
\only<1>{
  There is in fact an entire lattice of universal $n$-bundles in the 
  background.
}
\only<2>{Where the middle row and column give the universal $\mathrm{INN}(G)$ 2-bundle.}
\only<3>{Notice that, since $\mathrm{INN}(G)$ is trivializable, 
 that universal 2-bundle admits a canonical 2-section $e$.}
}

\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  (55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  (90,0)*{\mbox{\tiny $\Sigma G$}}="sgt";
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  (55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  (90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  (20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  \ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  \ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  \ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  \ar "y2ig"; "y2"+(1,3)
  %
  \ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  \ar@{^{(}->} "igt"+(0,-2); "iig"+(1,3)
  %
  \ar@{^{(}->} "g"+(4,0); "igt"+(-6,0)
  \ar@{->>} "igt"+(6,0); "sgt"+(-3,0)
  %
  \ar@{^{(}->} "sgt"+(-2,-3); "sigt"+(0,3)
  \ar@{->>} "sigt"; "ssg"+(1,3)
  %
  \ar@{=>}_>>>>>>>>>>{\simeq} "sgt"+(-6,-3); "iig"+(6,3)
  %
  \ar@{_{(}->}@/_1.5pc/|>>>>>>>>>>>{\makebox(14,14){}}_e "sig"+(2,4); "iig"+(2,-2)
\endxy
\vspace{6pt}
\only<1>{Notice that, since $\mathrm{INN}(G)$ is trivializable, that universal 2-bundle 
admits a canonical 2-section $e$.}
\only<2>{We can further pull back our data along this lattice, for instance
in the middle.}
}




\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  (55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  (90,0)*{\mbox{\tiny $\Sigma G$}}="sgt";
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  (55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  (90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  (20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  (55,-25)+(-12,16)*{\mbox{\small $\mathcal{C}_2(Y) \times_{g}\mathrm{INN}(\mathrm{INN}(G))$}}="c2iig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  \ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  \ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  \ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  \ar "y2ig"; "y2"+(1,3)
  %
  \ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "igt"+(0,-2); "iig"+(1,3)
  %
  \ar@{^{(}->} "g"+(4,0); "igt"+(-6,0)
  \ar@{->>} "igt"+(6,0); "sgt"+(-3,0)
  %
  \ar@{^{(}->} "sgt"+(-2,-3); "sigt"+(0,3)
  \ar@{->>} "sigt"; "ssg"+(1,3)
  %
  \ar@{=>}_>>>>>>>>>>{\simeq} "sgt"+(-6,-3); "iig"+(6,3)
  %
  \ar "c2iig"; "c2"+(0,3)
  \ar "c2iig"; "iig"+(-2,3)
  %
  \ar@{_{(}->}@/_1.5pc/|>>>>>>>>>>>{\makebox(14,14){}}_e "sig"+(2,4); "iig"+(2,-2)
\endxy
\vspace{6pt}
\only<1>{We can further pull back our data along this lattice, for instance
in the middle.}
\only<2>{
  This yields essentially the Atiyah groupoid 
  $\mathcal{C}_2(Y) \times_g \mathrm{INN}(\mathrm{INN}(G))$.
}
\only<3>{
  And we find that the choice $(g,\mathrm{tra}, \mathrm{curv})$
  lifts the canonical section $e$ to a splitting of the Atiyah groupoid
  projection.
}
}


\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  (20,0)*{\mbox{\tiny $G$}}="g";
  (55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  (90,0)*{\mbox{\tiny $\Sigma G$}}="sgt";
  %
  (20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  (55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  (90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  (20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  (55,-25)+(-12,16)*{\mbox{\small $\mathcal{C}_2(Y) \times_{g}\mathrm{INN}(\mathrm{INN}(G))$}}="c2iig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  \ar "y2ig"; "ig"+(0,3)
  %
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  \ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  \ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  \ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  \ar "y2ig"; "y2"+(1,3)
  %
  \ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  \ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "igt"+(0,-2); "iig"+(1,3)
  %
  \ar@{^{(}->} "g"+(4,0); "igt"+(-6,0)
  \ar@{->>} "igt"+(6,0); "sgt"+(-3,0)
  %
  \ar@{^{(}->} "sgt"+(-2,-3); "sigt"+(0,3)
  \ar@{->>} "sigt"; "ssg"+(1,3)
  %
  \ar@{=>}_>>>>>>>>>>{\simeq} "sgt"+(-6,-3); "iig"+(6,3)
  %
  \ar "c2iig"; "c2"+(0,3)
  \ar "c2iig"; "iig"+(-2,3)
  %
  \ar@{_{(}->}@/_1.5pc/|>>>>>>>>>>>{\makebox(14,14){}}_e "sig"+(2,4); "iig"+(2,-2)
  \ar@<+.5pt>@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
  \ar@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
  \ar@<-.5pt>@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
\endxy
\vspace{6pt}
\only<1>{
  And we find that the choice $(g,\mathrm{tra}, \mathrm{curv})$
  lifts the canonical section $e$ to a splitting of the Atiyah groupoid
  projection.
}
\only<2-5>{
  We should probably read this as follows:\\
}
\only<3>{
  $\Sigma \mathrm{INN}(G)$ plays the role of the fundamental 2-groupoid of
  $B G$.  
}
\only<4>{
  The section $e$ of the $\mathrm{INN}(G)$ 2-bundle plays the role of the
  universal connection on the universal $G$-bundle.
}
\only<5>{
  The choice $(g,\mathrm{tra},\mathrm{curv})$ pulls back the universal connection.
}
\only<6>{
  Finally, recall that we assumed $G$ to be abelian. 
}
\only<7>{
  The reason is that otherwise the 2-groupoid $\Sigma \Sigma G$ does not exist. 
}
\only<8>{
  But we shall pass to the differential picture now,\dots
}
\only<9>{
  \dots and find that for nonabelian $G$, $\Sigma \Sigma G$ may be thought of
  as being replaced
  by an $r$-groupoid\dots
}
\only<10>{
  \dots for $r$ the degree of the highest generator of the algebra
  of invariant polynomials of $\gg = \mathrm{Lie}(G)$.
}
\only<11>{
  To get there, we first suppress everything except for the front face of
  our diagram\dots
}
}


\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  %(20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma G$}}="sgt";
  %
  %(20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\Sigma G}="sg";
  (55,-58)+(-8,0)*{\Sigma \mathrm{INN}(G)}="sig";
  (90,-58)+(-8,0)*{\Sigma \Sigma G}="ssg";
  %
  (20,-58)+(-16,20)*{Y^{[2]}}="y2";
  (55,-58)+(-16,20)*{\mathcal{C}_2(Y)}="c2";
  (90,-58)+(-16,20)*{\Pi_2(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %(55,-25)+(-12,16)*{\mbox{\small $\mathcal{C}_2(Y) \times_{g}\mathrm{INN}(\mathrm{INN}(G))$}}="c2iig";
  %
  %
  \ar_g "y2"+(0,-3); "sg"+(0,3)  
  \ar|{\hspace{-18pt}(g,\mathrm{tra},\mathrm{curv})} "c2"+(0,-3); "sig"+(0,3)  
  \ar^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{^{(}->} "sg"+(6,0); "sig"+(-10,0)
  \ar@{->>} "sig"+(10,0); "ssg"+(-6,0)
  %
  \ar@{^{(}->} "y2"+(6,0); "c2"+(-6,0)
  \ar@{->>} "c2"+(6,0); "pix"+(-6,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  %\ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "igt"+(0,-2); "iig"+(1,3)
  %
  %\ar@{^{(}->} "g"+(4,0); "igt"+(-6,0)
  %\ar@{->>} "igt"+(6,0); "sgt"+(-3,0)
  %
  %\ar@{^{(}->} "sgt"+(-2,-3); "sigt"+(0,3)
  %\ar@{->>} "sigt"; "ssg"+(1,3)
  %
  %\ar@{=>}_>>>>>>>>>>{\simeq} "sgt"+(-6,-3); "iig"+(6,3)
  %
  %\ar "c2iig"; "c2"+(0,3)
  %\ar "c2iig"; "iig"+(-2,3)
  %
  %\ar@{_{(}->}@/_1.5pc/|>>>>>>>>>>>{\makebox(14,14){}}_e "sig"+(2,4); "iig"+(2,-2)
  %\ar@<+.5pt>@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
  %\ar@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
  %\ar@<-.5pt>@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
\endxy
\vspace{6pt}
\only<2>{
  \dots and then restrict attention to the special case where
  we take the cover $Y$ to be the total space $P$ of the $G$-bundle
  $P \to X$ itself, $Y = P$.
}
\only<3>{
  Then the cocycle data $g : Y^{[2]} \to \Sigma G$ is canonically given
  as $g : (p,p \cdot g_1) \mapsto g_1$.
}
\only<4>{This way we should arrive at the following differential formulation\dots}
}

\frame{
\xy
  (0,0)*{};
  (100,0)*{};
  (0,-60)*{};
  (100,-60)*{};
  %
  %(20,0)*{\mbox{\tiny $G$}}="g";
  %(55,0)*{\mbox{\tiny $\mathrm{INN}(G)$}}="igt";
  %(90,0)*{\mbox{\tiny $\Sigma G$}}="sgt";
  %
  %(20,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(G)$}}="ig";
  %(55,-25)+(-4,0)*{\mbox{\small $\mathrm{INN}(\mathrm{INN}(G))$}}="iig";
  %(90,-25)+(-4,0)*{\mbox{\small $\Sigma \mathrm{INN}(G)$}}="sigt";
  %
  (20,-58)+(-8,0)*{\gg^*}="sg";
  (55,-58)+(-8,0)*{\mathrm{inn}(\gg)^*}="sig";
  (90,-58)+(-8,0)*{b \gg^*}="ssg";
  %
  (20,-58)+(-16,20)*{\Omega^\bullet_{\mathrm{li}(G)}}="y2";
  (55,-58)+(-16,20)*{\Omega^\bullet(P)}="c2";
  (90,-58)+(-16,20)*{\Omega^\bullet(X)}="pix";
  %
  %(20,-25)+(-12,16)*{\mbox{\small $Y^{[2]} \times_{\Sigma G} \mathrm{INN}(G)$}}="y2ig";
  %(55,-25)+(-12,16)*{\mbox{\small $\mathcal{C}_2(Y) \times_{g}\mathrm{INN}(\mathrm{INN}(G))$}}="c2iig";
  %
  %
  \ar@{<-}_{\simeq} "y2"+(0,-3); "sg"+(0,3)  
  \ar@{<-}|{\hspace{0pt}(A, F_A)} "c2"+(0,-3); "sig"+(0,3)  
  \ar@{<-}^{K} "pix"+(0,-3); "ssg"+(0,3)
  %
  \ar@{<<-} "sg"+(3,0); "sig"+(-6,0)
  \ar@{<-^{)}} "sig"+(6,0); "ssg"+(-3,0)
  %
  \ar@{<<-}^{i^*} "y2"+(6,0); "c2"+(-6,0)
  \ar@{<-^{)}}^{p^*} "c2"+(6,0); "pix"+(-8,0)
  %
  %\ar "y2ig"; "ig"+(0,3)
  %
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "ig"; "sg"+(1,4)
  %
  %\ar@{^{(}->} "ig"+(7,0); "iig"+(-12,0)
  %\ar@{->>} "iig"+(12,0); "sigt"+(-8,0)
  %\ar@{->>}|<<<<<<<<<<<<{\makebox(12,12){}} "iig"; "sig"+(1,3)
  %
  %\ar "y2ig"; "y2"+(1,3)
  %
  %\ar@{=>}|<<<<<<<<<<<<<{\makebox(24,24){}}_>>>>>>>>>>\simeq "iig"+(-5,-5); "sg"+(5,5)
  %
  %\ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "g"+(0,-2); "ig"+(1,3)
  %
  %\ar@{^{(}->}|<<<<<<<<{\makebox(19,19){}} "igt"+(0,-2); "iig"+(1,3)
  %
  %\ar@{^{(}->} "g"+(4,0); "igt"+(-6,0)
  %\ar@{->>} "igt"+(6,0); "sgt"+(-3,0)
  %
  %\ar@{^{(}->} "sgt"+(-2,-3); "sigt"+(0,3)
  %\ar@{->>} "sigt"; "ssg"+(1,3)
  %
  %\ar@{=>}_>>>>>>>>>>{\simeq} "sgt"+(-6,-3); "iig"+(6,3)
  %
  %\ar "c2iig"; "c2"+(0,3)
  %\ar "c2iig"; "iig"+(-2,3)
  %
  %\ar@{_{(}->}@/_1.5pc/|>>>>>>>>>>>{\makebox(14,14){}}_e "sig"+(2,4); "iig"+(2,-2)
  %\ar@<+.5pt>@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
  %\ar@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
  %\ar@<-.5pt>@{_{(}->}@/_1.5pc/ "c2"+(2,4); "c2iig"+(2,-2)
\endxy
\vspace{6pt}
}






\end{document}
    
