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\section*{higher parallel transport}

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

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

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

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{higher_parallel_transport}{Higher parallel transport}\dotfill \pageref*{higher_parallel_transport} \linebreak
\noindent\hyperlink{higher_holonomy}{Higher holonomy}\dotfill \pageref*{higher_holonomy} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{for_trivial_circle_bundles__for_forms}{For trivial circle $n$-bundles / for $n$-forms}\dotfill \pageref*{for_trivial_circle_bundles__for_forms} \linebreak
\noindent\hyperlink{for_circle_bundles_with_connection}{For circle $n$-bundles with connection}\dotfill \pageref*{for_circle_bundles_with_connection} \linebreak
\noindent\hyperlink{nonabelian_parallel_transport_in_low_dimension}{Nonabelian parallel transport in low dimension}\dotfill \pageref*{nonabelian_parallel_transport_in_low_dimension} \linebreak
\noindent\hyperlink{1transport}{1-Transport}\dotfill \pageref*{1transport} \linebreak
\noindent\hyperlink{2transport}{2-Transport}\dotfill \pageref*{2transport} \linebreak
\noindent\hyperlink{3transport}{3-Transport}\dotfill \pageref*{3transport} \linebreak
\noindent\hyperlink{FlatInTop}{Flat $\infty$-parallel transport in $Top$}\dotfill \pageref*{FlatInTop} \linebreak
\noindent\hyperlink{InLieGrpd}{Flat $\infty$-parallel transport in $\infty LieGrpd$}\dotfill \pageref*{InLieGrpd} \linebreak
\noindent\hyperlink{parallel_transport_from_flat_differential_forms_with_values_in_chain_complexes}{$\infty$-Parallel transport from flat differential forms with values in chain complexes}\dotfill \pageref*{parallel_transport_from_flat_differential_forms_with_values_in_chain_complexes} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{in_physics}{In physics}\dotfill \pageref*{in_physics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

A [[connection on a bundle]] induces a notion of [[parallel transport]] over \emph{paths} . A [[connection on a 2-bundle]] induces a generalization of this to a notion of parallel transport over \emph{surfaces} . Similarly a [[connection on a 3-bundle]] induces a notion of parallel transport over 3-dimensional volumes.

Generally, a [[connection on an ∞-bundle]] induces a notion of parallel transport in arbitrary dimension.

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

The higher notions of [[differential cohomology]] and [[Chern-Weil theory]] make sense in any [[cohesive (∞,1)-topos]]

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma) : 
   \mathbf{H} 
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd \simeq Top
  \,.
\end{displaymath}
In every such there is a notion of [[connection on an ∞-bundle]] and of its higher parallel transport.

A typical context considered (more or less explicitly) in the literature is $\mathbf{H} =$ [[?LieGrpd]], the cohesive $(\infty,1)$-topos of [[∞-Lie groupoid|smooth ∞-groupoids]]. But other choices are possible. (See also the \hyperlink{Examples}{Examples}.)

\hypertarget{higher_parallel_transport}{}\subsubsection*{{Higher parallel transport}}\label{higher_parallel_transport}

Let $A$ be an [[∞-Lie groupoid]] such that morphisms $X \to A$ in [[?LieGrpd]] classify the $A$-[[principal ∞-bundle]]s under consideration. Write $A_{conn}$ for the differential refinement described at [[∞-Lie algebra valued form]], such that lifts

\begin{displaymath}
\itexarray{
     && A_{conn}
     \\
     & {}^{\mathllap{\nabla}}\nearrow & \downarrow
     \\
     X &\stackrel{g}{\to}& A
  }
\end{displaymath}
describe [[connections on ∞-bundles]].

\begin{udefn}
For $n \in \mathbb{N}$ say that $\nabla$ \textbf{admits parallel $n$-transport} if for all [[smooth manifold]]s $\Sigma$ of [[dimension]] $n$ and all morphisms

\begin{displaymath}
\phi : \Sigma \to X
\end{displaymath}
we have that the pullback of $\nabla$ to $\Sigma$

\begin{displaymath}
\phi^* \nabla : \Sigma \stackrel{\phi}{\to} X \stackrel{\nabla}{\to} A_{conn}
\end{displaymath}
\textbf{flat} in that it factors through the canonical inclusion $\mathbf{\flat}A \to A_{conn}$.

In other words: if all the lower [[curvature]] $k$-forms, $1 \leq k \leq n$ of $\phi^* \nabla$ vanish (the higher ones vanish automatically for dimensional reasons).

\end{udefn}
Here $\mathbf{\flat}A = [\mathbf{\Pi}(-),A]$ is the coefficient for [[schreiber:path ∞-groupoid|flat differential A-cohomology]].

\begin{udefn}
This condition is automatically satisfied for ordinary [[connections on bundles]], hence for $A = \mathbf{B}G$ with $G$ an ordinary [[Lie group]]: because in that case there is only a single curvature form, namely the ordinary [[curvature]] 2-form.

But for a [[principal 2-bundle]] with connection there is in general a 2-form curvature and a 3-form curvature. A 2-connection therefore admits parallel transport only if its 2-form curvature is constrained to vanish.

Notice however that if the coefficient object $A$ happens to be $(n-1)$-[[connected]] -- for instance if it is an [[Eilenberg-MacLane object]] in degree $n$, then there is no extra condition and \emph{every} connection admits parallel transport. This is notably the case for [[circle n-bundles with connection]].

\end{udefn}
\begin{udefn}
For $\nabla : X \to A_{conn}$ an $\infty$-connection that admits parallel $n$-transport, this is for each $\phi : \Sigma \to X$ the morphism

\begin{displaymath}
\mathbf{\Pi}(\Sigma) \to A
\end{displaymath}
that corresponds to $\phi^* \nabla$ under the equivalence

\begin{displaymath}
\mathbf{H}(\Sigma, \mathbf{\flat}A ) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), A)
  \,.
\end{displaymath}
\end{udefn}
\begin{udefn}
The objects of the  $\mathbf{\Pi}(\Sigma)$ are points in $\Sigma$, the morphisms are paths in there, the 2-morphisms surfaces between these paths, and so on. Hence a morphism $\mathbf{\Pi}(\Sigma) \to A$ assigns fibers in $A$ to points in $X$, and equivalences between these fibers to paths in $\Sigma$, and so on.

\end{udefn}
\hypertarget{higher_holonomy}{}\subsubsection*{{Higher holonomy}}\label{higher_holonomy}

We now define the higher [[analogs]] of [[holonomy]] for the case that $\Sigma$ is closed.

\begin{udefn}
Let $\nabla : X \to A_{conn}$ be a connection with parallel $n$-transport and $\phi : \Sigma \to X$ a morphism from a \emph{closed} $n$-[[manifold]].

Then the \textbf{$n$-holonomy} of $\nabla$ over $\Sigma$ is the image $[\phi^* \nabla]$ of

\begin{displaymath}
\phi^* \nabla : \Pi(\Sigma) \to \Gamma(A)
\end{displaymath}
in the [[homotopy category]]

\begin{displaymath}
[\phi^* \nabla] \in [\Pi(\Sigma), \Gamma(A)]
\end{displaymath}
\end{udefn}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{for_trivial_circle_bundles__for_forms}{}\subsubsection*{{For trivial circle $n$-bundles / for $n$-forms}}\label{for_trivial_circle_bundles__for_forms}

The simplest example is the parallel transport in a [[circle n-bundle with connection]] over a [[smooth manifold]] $X$ whose underlying $\mathbf{B}^{n-1}U(1)$-bundle is trivial. This is equivalently given by a degree $n$-[[differential form]] $A \in \Omega^n(X)$. For $\phi : \Sigma_n \to X$ any [[smooth function]] from an $n$-dimensional manifold $\Sigma$, the corresponding parallel transport is simply the [[integral]] of $A$ over $\Sigma$:

\begin{displaymath}
\tra_A(\Sigma) = \exp(i \int_\Sigma \phi^* A)
  \;\;\; \in \;\; U(1)
  \,.
\end{displaymath}
One can understand higher parallel transport therefore as a generalization of integration of diifferential $n$-forms to the case where

\begin{itemize}%
\item the $n$-form is not globally defined;


\item the $n$-form takes values not in $\mathbb{R}$ but more generally is an [[∞-Lie algebroid valued differential form]].



\end{itemize}
\hypertarget{for_circle_bundles_with_connection}{}\subsubsection*{{For circle $n$-bundles with connection}}\label{for_circle_bundles_with_connection}

We show how the $n$-holonomy of [[circle n-bundles with connection]] is reproduced from the above.

Let $\phi^* \nabla : \mathbf{\Pi}(\Sigma) \to \mathbf{B}^n U(1)$ be the parallel transport for a [[circle n-bundle with connection]] over a $\phi : \Sigma \to X$.

This is equivalent to a morphism

\begin{displaymath}
\Pi(\Sigma) \to \mathcal{B}^n U(1)
  ,.
\end{displaymath}
We may map this further to its $(n-dim \Sigma)$-[[nLab:truncated|truncation]]

\begin{displaymath}
:\infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) 
  \to
  \tau_{n-dim \Sigma} \infty Grpd(\Pi(X), \mathcal{B}^n U(1))
  \,.
\end{displaymath}
\begin{utheorem}
We have

\begin{displaymath}
\tau_{n-dim\Sigma} \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1))
  \simeq
  \mathbf{B}^{n-dim \Sigma} U(1)
  \,.
\end{displaymath}
\end{utheorem}
(This is due to an observation by [[nLab:Domenico Fiorenza]].)

\begin{proof}
By general abstract reasoning (recalled at [[nLab:cohomology]] and [[nLab:fiber sequence]]) we have for the [[nLab:homotopy group]]s that

\begin{displaymath}
\pi_i \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))
  \simeq 
  H^{n-i}(\Sigma, U(1))
  \,.
\end{displaymath}
Now use the [[nLab:universal coefficient theorem]], which asserts that we have an [[nLab:exact sequence]]

\begin{displaymath}
0
  \to 
  Ext^1(H_{n-i-1}(\Sigma,\mathbb{Z}),U(1))
  \to 
  H^{n-i}(\Sigma,U(1))
  \to 
  Hom(H_{n-i}(\Sigma,\mathbb{Z}),U(1))
  \to 0
  \,.
\end{displaymath}
Since $U(1)$ is an [[nLab:injective object|injective]] $\mathbb{Z}$-[[nLab:module]] we have

\begin{displaymath}
Ext^1(-,U(1))=0
  \,.
\end{displaymath}
This means that we have an [[nLab:isomorphism]]

\begin{displaymath}
H^{n-i}(\Sigma,U(1))
  \simeq 
  Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1))
\end{displaymath}
that identifies the [[nLab:cohomology group]] in question with the [[nLab:internal hom]] in [[nLab:Ab]] from the integral [[nLab:homology]] group of $\Sigma$ to $U(1)$.

For $i\lt (n-dim \Sigma)$, the right hand is zero, so that

\begin{displaymath}
\pi_i \infty Grpd(\Pi(\Sigma),\mathbf{B}^n U(1)) =0 \;\;\;\;
  for i\lt (n-dim \Sigma)
  \,.
\end{displaymath}
For $i=(n-dim \Sigma)$, instead, $H_{n-i}(\Sigma,\mathbb{Z})\simeq \mathbb{Z}$, since $\Sigma$ is a closed $dim \Sigma$-manifold and so

\begin{displaymath}
\pi_{(n-dim\Sigma)} \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))\simeq U(1)
  \,.
\end{displaymath}
\end{proof}
\begin{udef}
The resulting morphism

\begin{displaymath}
\mathbf{H}(\Sigma, A_{conn})
  \stackrel{\exp(i S(-))}{\to}
  \mathbf{B}^{n-dim\Sigma} U(1)
\end{displaymath}
in [[nLab:∞Grpd]] we call the \textbf{$\infty$-Chern-Simons action} on $\Sigma$.

\end{udef}
Here in the language of [[nLab:quantum field theory]]

\begin{itemize}%
\item the [[nLab:object]]s of $\mathbf{H}(\Sigma,A_{conn})$ are the [[nLab:gauge field]] on $\Sigma$;


\item the [[nLab:morphism]]s in $\mathbf{H}(\Sigma, A_{conn})$ are the [[nLab:gauge transformation]]s;.



\end{itemize}
\hypertarget{nonabelian_parallel_transport_in_low_dimension}{}\subsubsection*{{Nonabelian parallel transport in low dimension}}\label{nonabelian_parallel_transport_in_low_dimension}

At least in low categorical dimension one has the definition of the [[path n-groupoid]] $\mathbf{P}_n(X)$ of a smooth manifold, whose $n$-morphisms are [[thin homotopy]]-classes of smooth functions $[0,1]^n \to X$. Parallel $n$-transport with only the $(n+1)$-curvature form possibly nontrivial and all the lower curvature degree 1- to $n$-forms nontrivial may be expressed in terms of smooth $n$-functors out of $\mathbf{P}_n$ (\hyperlink{SWI}{SWI}, \hyperlink{SWII}{SWII}, \hyperlink{MartinsPickenI}{MartinsPickenI}, \hyperlink{MartinsPickenII}{MartinsPickenII}).

\hypertarget{1transport}{}\paragraph*{{1-Transport}}\label{1transport}

See [[parallel transport]].

\hypertarget{2transport}{}\paragraph*{{2-Transport}}\label{2transport}

We work now concretely in the category $2DiffeoGrpd$ of [[2-groupoid]]s [[internalization|internal to]] the category of [[diffeological space]]s.

Let $X$ be a [[smooth manifold]] and write $\mathbf{P}_2(X) \in 2DiffeoGrpd$ for its [[path 2-groupoid]]. Let $G$ be a [[Lie 2-group]] and $\mathbf{B}G \in 2DiffeoGrpd$ its [[delooping]] 1-object 2-groupoid. Write $\mathfrak{g}$ for the corresponding [[Lie 2-algebra]].

Assume now first that $G$ is a [[strict 2-group]] given by a [[crossed module]] $(G_1 \to G_0)$. Corresponding to this is a [[differential crossed module]] $(\mathfrak{g}_1 \to \mathfrak{g}_0)$.

We describe now how smooth 2-functors

\begin{displaymath}
tra : \mathbf{P}_2(X) \to \mathbf{B}G
\end{displaymath}
i.e. morphisms in $2DiffeoGrpd$ are characterized by [[Lie 2-algebra valued differential forms]] on $X$.

\begin{udefn}
Given a morphism $F : \mathbf{P}_2(X) \to \mathbf{B}G$ we construct a $\mathfrak{g}_1$-valued 2-form $B_F \in \Omega^2(X, \mathfrak{g}_1)$ as follows.

To find the value of $B_F$ on two vectors $v_1, v_2 \in T_p X$ at some point, \emph{choose} any smooth function

\begin{displaymath}
\Gamma : \mathbb{R}^2 \to X
\end{displaymath}
with

\begin{itemize}%
\item $\Gamma(0,0) = p$


\item $\frac{d}{d s}|_{s = 0} \Gamma(s,0) = v_1$


\item $\frac{d}{d t}|_{t = 0} \Gamma(0,t) = v_2$.



\end{itemize}
Notice that there is a canonical 2-parameter family

\begin{displaymath}
\Sigma_{\mathbb{R}} : \mathbb{R}^2 \to 2Mor \mathbf{P}_2(\mathbb{R}^2)
\end{displaymath}
of classes of bigons on the plane, given by sending $(s,t) \in \mathbb{R}^2$ to the class represented by any bigon (with sitting instants) with straight edges filling the square

\begin{displaymath}
\Sigma_{\mathbb{R}}(s,t) = 
  \left(
    \itexarray{
      (0,0) &\to& (0,t)
      \\
      \downarrow && \downarrow
      \\
      (s,0) &\to& (s,t)
    }
  \right)
  \,.
\end{displaymath}
Using this we obtain a smooth function

\begin{displaymath}
F_\Gamma : 
  \mathbb{R}^2 \stackrel{\Sigma_{\mathbb{R}}}{\to}
  2Mor \mathbf{P}_2(\mathbb{R}^2)
  \stackrel{\Gamma_*}{\to}
  2Mor \mathbf{P}_2(X)
  \stackrel{F}{\to}
  G_0 \times G_1
  \stackrel{p_2}{\to}
  G_1
  \,.
\end{displaymath}
Then set

\begin{displaymath}
B_F(v_1, w_1) := 
   \frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(0,0)}
  \,.
\end{displaymath}
\end{udefn}
\begin{uprop}
This is well defined, in that $B_F(v_1,v_2)$ does not depend on the choices made. Moreover, the 2-form defines this way is smooth.

\end{uprop}
\begin{proof}
To see that the definition does not depend on the choice of $\Gamma$, proceed as follows.

For given vectors $v_1,v_2 \in \T_X X$ let $\Gamma, \Gamma' : \mathbb{R}^2 \to X$ be two choices of smooth maps as in the defnition. By restricting, if necessary, to a neighbourhood of the origin of $\mathbb{R}^2$, we may assume without restriction that these maps land in a single coordinate patch in $X$. Using the vector space structure of $\mathbb{R}^n$ defined by such a patch, define a smooth homotopy

\begin{displaymath}
\tau : [0,1]^3 \to X : (x,y,z) \mapsto (1-z)\Gamma(x,y) + z \Gamma'(x,y)
\end{displaymath}
Let

\begin{displaymath}
Z = \{(x,y,w) \in [0,1]^3 | 0 \leq w \leq \frac{1}{2}(x^2 + y^2) \}
\end{displaymath}
and consider the map $f : [0,1]^3 \to Z$ given by

\begin{displaymath}
f : (x,y,z) \mapsto (x,y, \frac{1}{2}(x^2 + y^2) z)
\end{displaymath}
and the map $g : Z \to X$ given away from $(x^2 + y^2) = 0$ by

\begin{displaymath}
g : (x,y,w) \mapsto \tau(x,y, 2 \frac{w}{x^2 + y^2})
  \,.
\end{displaymath}
Using [[Hadamard's lemma]] and the fact that by constructon $\tau$ has vanishing 0th and 1st order differentials at the origin it follows that this is indeed a [[smooth function]].

We want to similarly factor the smooth family of bigons $[0,1]^3 \to 2Mor(\mathbf{P}_2(X))$ given by

\begin{displaymath}
[0,1]^3 \times [0,1]^2 \to X
\end{displaymath}
\begin{displaymath}
((x,y,z),(s,t)) \mapsto \tau(s x, t y, z)
\end{displaymath}
as $[0,1]^3 \times [0,1]^2 \to Z \times [0,1]^2 \to Z \to X$

\begin{displaymath}
((x,y,z),(s,t)) 
    \mapsto
  ((x, y, \frac{1}{2}(x^2 + y^2)), (s,t))
    \mapsto
  (s x , t y, \frac{1}{2}((s x)^2 + (t y)^2)z)
    \mapsto

  \tau(s x, s y, z)
  \,.
\end{displaymath}
As before using Hadamard's lemma this is a sequence of smooth functions. To make this qualify as a family of bigons (which are maps from the square that are constant in a [[neighbourhood]] of the left and right boundary of the square) furthermore precompose this with a suitable smooth function $[0,1]^2 \to [0,1]^2$ that realizes a square-shaped bigon.

Under the hom-adjunction it corresponds to a factorization of $G_\Gamma : [0,1]^3 \to 2 Mor(\mathbf{P}_2(X))$ into

\begin{displaymath}
G_\Gamma :  [0,1]^3 \stackrel{f}{\to} Z \to 2 Mor(\mathbf{P}_2(X))
  \,.
\end{displaymath}
By the above construction we have the the push-forwards

\begin{displaymath}
f_* : \frac{\partial}{\partial x}(x=0,y=0,z)  
    \mapsto
    \frac{\partial}{\partial x}(x= 0, y = 0, w = 0)
\end{displaymath}
and similarly for $\frac{\partial}{\partial y}$ are indendent of $z$. It follows by the [[chain rule]] that also

\begin{displaymath}
\frac{\partial^2 G_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}
\end{displaymath}
is independent of $z$. But at $z = 0$ this equals $\frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}$, while at $z = 1$ it equals $\frac{\partial^2 F_{\Gamma'}}{\partial x \partial y}|_{(x=0,y=0)}$. Therefore these two are equal.

\end{proof}
\hypertarget{3transport}{}\paragraph*{{3-Transport}}\label{3transport}

see [[3-groupoid of Lie 3-algebra valued forms]]

\hypertarget{FlatInTop}{}\subsubsection*{{Flat $\infty$-parallel transport in $Top$}}\label{FlatInTop}

Even though it is a degenerate case, it can be useful to regard the [[(∞,1)-topos]] [[Top]] explicitly a [[cohesive (∞,1)-topos]]. For a discussion of this see [[discrete ∞-groupoid]].

For $\mathbf{H} =$ [[Top]] lots of structure of cohesive $(\infty,1)$-topos theory degenerates, since by the [[homotopy hypothesis]]-theorem here the [[global section]] [[(∞,1)-geometric morphism]]

\begin{displaymath}
(\Pi \dashv \Delta \dashv \Gamma) : Top 
   \stackrel{\overset{\Pi}{\leftarrow}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \in
 \infty Grpd
\end{displaymath}
an [[equivalence in an (∞,1)-category|equivalence]]. The abstract [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\Pi$ is here the ordinary [[fundamental ∞-groupoid]]

\begin{displaymath}
\Pi : Top \stackrel{\simeq}{\to} \infty Grpd
  \,.
\end{displaymath}
If both [[(∞,1)-topos]]es here are [[presentable (∞,1)-category|presented]] by their standard [[model category]] models, the standard [[model structure on simplicial sets]] and the standard [[model structure on topological spaces]], then $\Pi$ is presented by the [[singular simplicial complex]] functor in a [[Quillen equivalence]]

\begin{displaymath}
(|-| \dashv Sing) : Top \stackrel{\leftarrow}{\overset{\simeq_{Quillen}}{\to}}
  Top
  \,.
\end{displaymath}
This means that in this case many constructions in [[topology]] and classical [[homotopy theory]] have equivalent reformulations in terms of $\infty$-parallel transport.

For instance: for $F \in Top$ and $Aut(F) \in Top$ its [[automorphism ∞-group]], $F$-fibrations over a base space $X \in Top$ are classfied by morphisms

\begin{displaymath}
g : X \to B Aut(F)
\end{displaymath}
into the [[delooping]] of $Aut(F)$. The corresponding fibration $P \to X$ itself is the [[homotopy fiber]] of this cocycles, given by the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
     P &\to& * 
     \\
     \downarrow && \downarrow
     \\
     X &\stackrel{g}{\to}&  B Aut(F)
  }
\end{displaymath}
in [[Top]], as described at [[principal ∞-bundle]].

Using the [[fundamental ∞-groupoid]] functor we may send this equivalently to a [[fiber sequence]] in [[∞Grpd]]

\begin{displaymath}
\Pi(P) \to \Pi(X) \to B Aut(\Pi(F))
  \,.
\end{displaymath}
One may think of the morphism $\Pi(X) \to B Aut(\Pi(F))$ now as the $\infty$-parallel transport coresponding to the original fibration:

\begin{itemize}%
\item to each point in $X$ it assigns the unique object of $B Aut(\Pi(F))$, which is the fiber $F$ itself;


\item to each path $(x \to y)$ in $X$ it assigns an equivalence between the fibers $F_x to F_y$ etc.



\end{itemize}
If one presents $\Pi$ by $Sing : Top \to sSet_{Quillen}$ as above, then one may look for explicit simplicial formulas that express these morphisms. Such are discussed in \hyperlink{Stasheff}{Stasheff}.

We may embed this example into the smooth context by regarding $Aut(F)$ as a [[discrete space|discrete]] [[∞-Lie groupoid]] as discussed in the section \hyperlink{InLieGrpd}{Flat ∞-Parallel transport in ?LieGrpd}.

For that purpose let

\begin{displaymath}
(\Pi_{smooth} \dashv Disc_{smooth} \dashv \Gamma_{smooth}) : 
   \infty LieGrpd
    \stackrel{\overset{\Pi_{smooth}}{\to}}{\stackrel{\overset{Disc_{smooth}}{\leftarrow}}{\underset{\Gamma_{smooth}}{\to}}}
  \infty Grpd \simeq Top
\end{displaymath}
be the [[global section]] [[(∞,1)-geometric morphism]] of the [[cohesive (∞,1)-topos]] [[?LieGrpd]].

We may reflect the [[∞-group]] $Aut(F)$ into this using the [[constant ∞-stack]]-functor $Disc$ to get the discrete [[∞-Lie group]] $Disc Aut(F)$. Let then $X$ be a [[paracompact topological space|paracompact]] [[smooth manifold]], regarded naturally as an object of [[?LieGrpd]]. Then we can consider [[cocycle]]s/classifying morphisms

\begin{displaymath}
X \to \mathbf{B} Disc Aut(F)
  \,,
\end{displaymath}
now in the smooth context of $\infty LieGrpd$.

\begin{uprop}
The [[∞-groupoid]] of $F$-fibrations in [[Top]] is equivalent to the $\infty$-groupoid of $Disc Aut(F)$-[[principal ∞-bundle]]s in [[?LieGrpd]]:

\begin{displaymath}
\infty LieGrpd(X, \mathbf{B} Disc Aut(F))
  \simeq
  Top(X, B Aut(F))
  \,.
\end{displaymath}
Moreover, all the [[principal ∞-bundle]]s classified by the morphisms on the left have canonical extensions to  in $\infty LieGrpd$, in that the flat parallel $\infty$-transport $\nabla_{flat}$ in

\begin{displaymath}
\itexarray{
    X &\stackrel{g}{\to}& \mathbf{B} Disc Aut(F)
    \\
    \downarrow & \nearrow_{\nabla_{flat}}
    \\
    \mathbf{\Pi}(X)
  }
\end{displaymath}
always exists.

\end{uprop}
\begin{proof}
The first statement is a special case of that spelled out at [[?LieGrpd]] and [[nonabelian cohomology]]. The second follows using that in a [[∞-connected (∞,1)-topos|connected]] [[locally ∞-connected (∞,1)-topos]] the functor $Disc$ is a [[full and faithful (∞,1)-functor]].

\end{proof}
\hypertarget{InLieGrpd}{}\subsubsection*{{Flat $\infty$-parallel transport in $\infty LieGrpd$}}\label{InLieGrpd}

(\ldots{})

\hypertarget{parallel_transport_from_flat_differential_forms_with_values_in_chain_complexes}{}\subsubsection*{{$\infty$-Parallel transport from flat differential forms with values in chain complexes}}\label{parallel_transport_from_flat_differential_forms_with_values_in_chain_complexes}

A typical choice for an [[(∞,1)-category]] of ``$\infty$-vector spaces'' is that [[presentable (∞,1)-category|presented]] by the a [[model structure on chain complexes]] of modules. In a geometric context this may be replaced by some stack of complexes of vector bundles over some site.

If we write $Mod$ for this stack, then the $\infty$-parallel transport for a flat $\infty$-vector bundle on some $X$ is a morphism

\begin{displaymath}
\mathbf{\Pi}(X) \to Mod
  \,.
\end{displaymath}
This is typically given by differential form data with values in $Mod$.

A discussion of how to integrate flat differential forms with values in chain complexes -- a representation of the [[tangent Lie algebroid]] as discussed at [[representations of ∞-Lie algebroids]] -- to flat $\infty$-parallel transport $\mathbf{\Pi}(X) \to Mod$ is in (\hyperlink{abadSchaetz}{AbadSchaetz}), building on a construciton in (\hyperlink{Igusa}{Igusa}).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{in_physics}{}\subsubsection*{{In physics}}\label{in_physics}

In [[physics]] various [[action functional]]s for [[quantum field theories]] are nothing but higher parallel transport.

\begin{itemize}%
\item The gauge interaction part of the action functional for the particle charged under a background [[electromagnetic field]], which is a [[circle n-bundle with connection|circle bundle with connection]] $\nabla$, is the parallel 1-transport of $\nabla$.


\item The gauge interaction part of the action functional for the string charged under a background [[Kalb-Ramond field]], which is a [[circle n-bundle with connection|circle 2-bundle with connection]] $\nabla$, is the parallel 2-transport of $\nabla$.


\item The gauge interaction part of the action functional for the membrane charged under a background [[supergravity C-field]], which is a [[circle n-bundle with connection|circle 3-bundle with connection]] $\nabla$, is the parallel 3-transport of $\nabla$.


\item The action functional of [[Chern-Simons theory]] is the parallel 3-transport of a [[Chern-Simons circle 3-bundle]].



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

\begin{itemize}%
\item [[connection on a bundle]], [[connection on a 2-bundle]], [[connection on an infinity-bundle]],


\item [[parallel transport]], \textbf{higher parallel transport}


\item [[holonomy]]

\begin{itemize}%
\item [[holonomy group]]


\item [[special holonomy]]



\end{itemize}

\item [[Wilson surface]]



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

For references on ordinary 1-dimensional parallel transport see [[parallel transport]].

For references on parallel 2-transport in [[bundle gerbes]] see [[connection on a bundle gerbe]].

The description of parallel $n$-transport in terms of $n$-functors on the [[path n-groupoid]] for low $n$ is in

\begin{itemize}%
\item [[Urs Schreiber]], [[Konrad Waldorf]], \emph{Smooth Functors and Differential Forms}, Homology, Homotopy Appl., 13(1), 143-203 (2011) (\href{http://arxiv.org/abs/0802.0663}{arXiv:0802.0663})


\item [[João Faria Martins]], [[Roger Picken]], \emph{On 2-dimensional holonomy} (\href{http://arxiv.org/abs/0710.4310}{arXiv})



\end{itemize}
\begin{itemize}%
\item [[João Faria Martins]], [[Roger Picken]], \emph{The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module} (\href{http://arxiv.org/abs/0907.2566}{arXiv})

\end{itemize}
The description of [[connections on a 2-bundle]] in terms of such parallel 2-transport

\begin{itemize}%
\item [[John Baez]], [[Urs Schreiber]], \emph{Higher gauge theory}, in A. Davydov et al. (eds.), \emph{Categories in Algebra, Geometry and Mathematical Physics}, Contemp Math 431, AMS, Providence, Rhode Island (2007) pp 7-30 (\href{http://arxiv.org/abs/math/0511710}{arXiv:0511710}, \href{http://arxiv.org/abs/hep-th/0412325}{arXi:hep-th/0412325hep-th/0412325})


\item [[Urs Schreiber]], [[Konrad Waldorf]] \emph{Connections on nonabelian gerbes and their holonomy}, Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540 (\href{http://arxiv.org/abs/0808.1923}{arXiv:0808.1923}, \href{http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html}{TAC})



\end{itemize}
Much further discussion and illustration and relation to [[tensor networks]] is in

\begin{itemize}%
\item [[Arthur Parzygnat]], \emph{Two-dimensional algebra in lattice gauge theory} (\href{https://arxiv.org/abs/1802.01139}{arXiv:1802.01139})

\end{itemize}
Applications are discussed in

\begin{itemize}%
\item [[Arthur Parzygnat]], \emph{Gauge invariant surface holonomy and monopoles}, Theory and Applications of Categories, Vol. 30, 2015, No. 42, pp 1319-1428 (\href{http://www.tac.mta.ca/tac/volumes/30/42/30-42abs.html}{TAC})

\end{itemize}
Parallel transport for [[circle n-bundles with connection]] is discussed generally in

\begin{itemize}%
\item [[Kiyonori Gomi]] and Yuji Terashima, \emph{Higher dimensional parallel transport} Mathematical Research Letters 8, 25--33 (2001) (\href{http://mrlonline.org/mrl/2001-008-001/2001-008-001-004.pdf}{pdf})


\item David Lipsky, \emph{Cocycle constructions for topological field theories} (2010) ([[LipskyThesis.pdf:file]])



\end{itemize}
see also the discussion at \emph{[[fiber integration in ordinary differential cohomology]]}.

Realization of this as an [[extended TQFT]] is discussed in

\begin{itemize}%
\item [[Nersés Aramyan]], Research statement (\href{http://math.illinois.edu/~aramyan2/research.pdf}{pdf})

\end{itemize}
Parallel transport with [[coefficients]] in [[crossed complexes]]/[[strict infinity-groupoids]] is discussed in

\begin{itemize}%
\item [[Mikhail Kapranov]], \emph{Membranes and higher groupoids} (\href{http://arxiv.org/abs/1502.06166}{arXiv:1502.06166})

\end{itemize}
The integration of flat differential forms with values in chain complexes to flat $\infty$-parallel transport on $\infty$-vector bundles is in

\begin{itemize}%
\item [[Camilo Arias Abad]], [[Florian Schaetz]], \emph{The $A_\infty$ de Rham theorem and integration of representations up to homotopy} (\href{http://arxiv.org/abs/1011.4693}{arXiv})

\end{itemize}
based on

\begin{itemize}%
\item Jonathan Block, Aaron Smith, \emph{A Riemann Hilbert correspondence for infinity local systems} (\href{http://arxiv.org/abs/0908.2843}{arXiv})

\end{itemize}
in turn based on constructions in

\begin{itemize}%
\item [[Kiyoshi Igusa]], \emph{Iterated integrals of superconnections} (\href{http://arxiv.org/abs/0912.0249}{arXiv})

\end{itemize}
Remarks on $\infty$-parallel transport in [[Top]] are in

\begin{itemize}%
\item [[Jim Stasheff]], \emph{[[StasheffParallelTransportv02.pdf:file]]}

\end{itemize}
[[!redirects surface holonomy]] [[!redirects surface holonomies]]

[[!redirects volume holonomy]] [[!redirects volume holonomies]]

[[!redirects higher holonomy]] [[!redirects higher holonomies]]

[[!redirects higher volume holonomy]] [[!redirects higher volume holonomies]]



\end{document}