# nLab higher parallel transport

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

A connection on a bundle induces a notion of parallel transport over paths . A connection on a 2-bundle induces a generalization of this to a notion of parallel transport over 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.

## Definition

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

$\left(\Pi ⊣\mathrm{Disc}⊣\Gamma \right):H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{←}}{\underset{\Gamma }{\to }}}\infty \mathrm{Grpd}\simeq \mathrm{Top}\phantom{\rule{thinmathspace}{0ex}}.$(\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \simeq Top \,.

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 $H=$ ∞LieGrpd, the cohesive $\left(\infty ,1\right)$-topos of smooth ∞-groupoids. But other choices are possible. (See also the Examples.)

### Higher parallel transport

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

$\begin{array}{ccc}& & {A}_{\mathrm{conn}}\\ & {}^{\nabla }↗& ↓\\ X& \stackrel{g}{\to }& A\end{array}$\array{ && A_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& A }

describe connections on ∞-bundles.

###### Definition

For $n\in ℕ$ say that $\nabla$ admits parallel $n$-transport if for all smooth manifolds $\Sigma$ of dimension $n$ and all morphisms

$\varphi :\Sigma \to X$\phi : \Sigma \to X

we have that the pullback of $\nabla$ to $\Sigma$

${\varphi }^{*}\nabla :\Sigma \stackrel{\varphi }{\to }X\stackrel{\nabla }{\to }{A}_{\mathrm{conn}}$\phi^* \nabla : \Sigma \stackrel{\phi}{\to} X \stackrel{\nabla}{\to} A_{conn}

flat in that it factors through the canonical inclusion $♭A\to {A}_{\mathrm{conn}}$.

In other words: if all the lower curvature $k$-forms, $1\le k\le n$ of ${\varphi }^{*}\nabla$ vanish (the higher ones vanish automatically for dimensional reasons).

Here $♭A=\left[\Pi \left(-\right),A\right]$ is the coefficient for flat differential A-cohomology.

###### Remark

This condition is automatically satisfied for ordinary connections on bundles, hence for $A=BG$ 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 $\left(n-1\right)$-connected – for instance if it is an Eilenberg-MacLane object in degree $n$, then there is no extra condition and every connection admits parallel transport. This is notably the case for circle n-bundles with connection.

###### Definition

For $\nabla :X\to {A}_{\mathrm{conn}}$ an $\infty$-connection that admits parallel $n$-transport, this is for each $\varphi :\Sigma \to X$ the morphism

$\Pi \left(\Sigma \right)\to A$\mathbf{\Pi}(\Sigma) \to A

that corresponds to ${\varphi }^{*}\nabla$ under the equivalence

$H\left(\Sigma ,♭A\right)\simeq H\left(\Pi \left(\Sigma \right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}(\Sigma, \mathbf{\flat}A ) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), A) \,.
###### Remark

The objects of the path ∞-groupoid $\Pi \left(\Sigma \right)$ are points in $\Sigma$, the morphisms are paths in there, the 2-morphisms surfaces between these paths, and so on. Hence a morphism $\Pi \left(\Sigma \right)\to A$ assigns fibers in $A$ to points in $X$, and equivalences between these fibers to paths in $\Sigma$, and so on.

### Higher holonomy

We now define thee (DR - three? where are the other two?) higher analogs of holonomy for the case that $\Sigma$ is closed.

###### Definition

Let $\nabla :X\to {A}_{\mathrm{conn}}$ be a connection with parallel $n$-transport and $\varphi :\Sigma \to X$ a morphism from a closed $n$-manifold.

Then the $n$-holonomy of $\nabla$ over $\Sigma$ is the image $\left[{\varphi }^{*}\nabla \right]$ of

${\varphi }^{*}\nabla :\Pi \left(\Sigma \right)\to \Gamma \left(A\right)$\phi^* \nabla : \Pi(\Sigma) \to \Gamma(A)

in the homotopy category

$\left[{\varphi }^{*}\nabla \right]\in \left[\Pi \left(\Sigma \right),\Gamma \left(A\right)\right]$[\phi^* \nabla] \in [\Pi(\Sigma), \Gamma(A)]

## Examples

### For trivial circle $n$-bundles / for $n$-forms

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

${tra}_{A}\left(\Sigma \right)=\mathrm{exp}\left(i{\int }_{\Sigma }{\varphi }^{*}A\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\tra_A(\Sigma) = \exp(i \int_\Sigma \phi^* A) \;\;\; \in \;\; U(1) \,.

One can understand higher parallel transport therefore as a generalization of integration of diifferential $n$-forms to the case where

• the $n$-form is not globally defined;

• the $n$-form takes values not in $ℝ$ but more generally is an ∞-Lie algebroid valued differential form.

### For circle $n$-bundles with connection

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

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

This is equivalent to a morphism

$\Pi \left(\Sigma \right)\to {ℬ}^{n}U\left(1\right),.$\Pi(\Sigma) \to \mathcal{B}^n U(1) ,.

We may map this further to its $\left(n-\mathrm{dim}\Sigma \right)$-truncation

$:\infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),{ℬ}^{n}U\left(1\right)\right)\to {\tau }_{n-\mathrm{dim}\Sigma }\infty \mathrm{Grpd}\left(\Pi \left(X\right),{ℬ}^{n}U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$:\infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \to \tau_{n-dim \Sigma} \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \,.
###### Theorem

We have

${\tau }_{n-\mathrm{dim}\Sigma }\infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),{ℬ}^{n}U\left(1\right)\right)\simeq {B}^{n-\mathrm{dim}\Sigma }U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\tau_{n-dim\Sigma} \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \simeq \mathbf{B}^{n-dim \Sigma} U(1) \,.

(This is due to an observation by Domenico Fiorenza.)

###### Proof

By general abstract reasoning (recalled at cohomology and fiber sequence) we have for the homotopy groups that

(1)${\pi }_{i}\infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),{ℬ}^{n}U\left(1\right)\right)\simeq {H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_i \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1)) \simeq H^{n-i}(\Sigma, U(1)) \,.

Now use the universal coefficient theorem, which asserts that we have an exact sequence

(2)$0\to {\mathrm{Ext}}^{1}\left({H}_{n-i-1}\left(\Sigma ,ℤ\right),U\left(1\right)\right)\to {H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\to \mathrm{Hom}\left({H}_{n-i}\left(\Sigma ,ℤ\right),U\left(1\right)\right)\to 0\phantom{\rule{thinmathspace}{0ex}}.$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 \,.

Since $U\left(1\right)$ is an injective $ℤ$-module we have

${\mathrm{Ext}}^{1}\left(-,U\left(1\right)\right)=0\phantom{\rule{thinmathspace}{0ex}}.$Ext^1(-,U(1))=0 \,.

This means that we have an isomorphism

(3)${H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\simeq {\mathrm{Hom}}_{\mathrm{Ab}}\left({H}_{n-i}\left(\Sigma ,ℤ\right),U\left(1\right)\right)$H^{n-i}(\Sigma,U(1)) \simeq Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1))

that identifies the cohomology group in question with the internal hom in Ab from the integral homology group of $\Sigma$ to $U\left(1\right)$.

For $i<\left(n-\mathrm{dim}\Sigma \right)$, the right hand is zero, so that

${\pi }_{i}\infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),{B}^{n}U\left(1\right)\right)=0\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{for}i<\left(n-\mathrm{dim}\Sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_i \infty Grpd(\Pi(\Sigma),\mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,.

For $i=\left(n-\mathrm{dim}\Sigma \right)$, instead, ${H}_{n-i}\left(\Sigma ,ℤ\right)\simeq ℤ$, since $\Sigma$ is a closed $\mathrm{dim}\Sigma$-manifold and so

${\pi }_{\left(n-\mathrm{dim}\Sigma \right)}\infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),{ℬ}^{n}U\left(1\right)\right)\simeq U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_{(n-dim\Sigma)} \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))\simeq U(1) \,.
###### Definition

The resulting morphism

$H\left(\Sigma ,{A}_{\mathrm{conn}}\right)\stackrel{\mathrm{exp}\left(iS\left(-\right)\right)}{\to }{B}^{n-\mathrm{dim}\Sigma }U\left(1\right)$\mathbf{H}(\Sigma, A_{conn}) \stackrel{\exp(i S(-))}{\to} \mathbf{B}^{n-dim\Sigma} U(1)

in ∞Grpd we call the $\infty$-Chern-Simons action on $\Sigma$.

Here in the language of quantum field theory

• the objects of $H\left(\Sigma ,{A}_{\mathrm{conn}}\right)$ are the gauge field on $\Sigma$;

• the morphisms in $H\left(\Sigma ,{A}_{\mathrm{conn}}\right)$ are the gauge transformations;.

### Nonabelian parallel transport in low dimension

At least in low categorical dimension one has the definition of the path n-groupoid ${P}_{n}\left(X\right)$ of a smooth manifold, whose $n$-morphisms are thin homotopy-classes of smooth functions $\left[0,1{\right]}^{n}\to X$. Parallel $n$-transport with only the $\left(n+1\right)$-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 ${P}_{n}$ (SWI, SWII, MartinsPickenI, MartinsPickenII).

#### 2-Transport

We work now concretely in the category $2\mathrm{DiffeoGrpd}$ of 2-groupoids internal to the category of diffeological spaces.

Let $X$ be a smooth manifold and write ${P}_{2}\left(X\right)\in 2\mathrm{DiffeoGrpd}$ for its path 2-groupoid. Let $G$ be a Lie 2-group and $BG\in 2\mathrm{DiffeoGrpd}$ its delooping 1-object 2-groupoid. Write $𝔤$ for the corresponding Lie 2-algebra.

Assume now first that $G$ is a strict 2-group given by a crossed module $\left({G}_{1}\to {G}_{0}\right)$. Corresponding to this is a differential crossed module $\left({𝔤}_{1}\to {𝔤}_{0}\right)$.

We describe now how smooth 2-functors

$\mathrm{tra}:{P}_{2}\left(X\right)\to BG$tra : \mathbf{P}_2(X) \to \mathbf{B}G

i.e. morphisms in $2\mathrm{DiffeoGrpd}$ are characterized by Lie 2-algebra valued differential forms on $X$.

###### Definition

Given a morphism $F:{P}_{2}\left(X\right)\to BG$ we construct a ${𝔤}_{1}$-valued 2-form ${B}_{F}\in {\Omega }^{2}\left(X,{𝔤}_{1}\right)$ as follows.

To find the value of ${B}_{F}$ on two vectors ${v}_{1},{v}_{2}\in {T}_{p}X$ at some point, choose any smooth function

$\Gamma :{ℝ}^{2}\to X$\Gamma : \mathbb{R}^2 \to X

with

• $\Gamma \left(0,0\right)=p$

• $\frac{d}{ds}{\mid }_{s=0}\Gamma \left(s,0\right)={v}_{1}$

• $\frac{d}{dt}{\mid }_{t=0}\Gamma \left(0,t\right)={v}_{2}$.

Notice that there is a canonical 2-parameter family

${\Sigma }_{ℝ}:{ℝ}^{2}\to 2\mathrm{Mor}{P}_{2}\left({ℝ}^{2}\right)$\Sigma_{\mathbb{R}} : \mathbb{R}^2 \to 2Mor \mathbf{P}_2(\mathbb{R}^2)

of classes of bigons on the plane, given by sending $\left(s,t\right)\in {ℝ}^{2}$ to the class represented by any bigon (with sitting instants) with straight edges filling the square

${\Sigma }_{ℝ}\left(s,t\right)=\left(\begin{array}{ccc}\left(0,0\right)& \to & \left(0,t\right)\\ ↓& & ↓\\ \left(s,0\right)& \to & \left(s,t\right)\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\Sigma_{\mathbb{R}}(s,t) = \left( \array{ (0,0) &\to& (0,t) \\ \downarrow && \downarrow \\ (s,0) &\to& (s,t) } \right) \,.

Using this we obtain a smooth function

${F}_{\Gamma }:{ℝ}^{2}\stackrel{{\Sigma }_{ℝ}}{\to }2\mathrm{Mor}{P}_{2}\left({ℝ}^{2}\right)\stackrel{{\Gamma }_{*}}{\to }2\mathrm{Mor}{P}_{2}\left(X\right)\stackrel{F}{\to }{G}_{0}×{G}_{1}\stackrel{{p}_{2}}{\to }{G}_{1}\phantom{\rule{thinmathspace}{0ex}}.$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 \,.

Then set

${B}_{F}\left({v}_{1},{w}_{1}\right):=\frac{{\partial }^{2}{F}_{\Gamma }}{\partial x\partial y}{\mid }_{\left(0,0\right)}\phantom{\rule{thinmathspace}{0ex}}.$B_F(v_1, w_1) := \frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(0,0)} \,.
###### Proposition

This is well defined, in that ${B}_{F}\left({v}_{1},{v}_{2}\right)$ does not depend on the choices made. Moreover, the 2-form defines this way is smooth.

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

$\tau :\left[0,1{\right]}^{3}\to X:\left(x,y,z\right)↦\left(1-z\right)\Gamma \left(x,y\right)+z\Gamma \prime \left(x,y\right)$\tau : [0,1]^3 \to X : (x,y,z) \mapsto (1-z)\Gamma(x,y) + z \Gamma'(x,y)

Let

$Z=\left\{\left(x,y,w\right)\in \left[0,1{\right]}^{3}\mid 0\le w\le \frac{1}{2}\left({x}^{2}+{y}^{2}\right)\right\}$Z = \{(x,y,w) \in [0,1]^3 | 0 \leq w \leq \frac{1}{2}(x^2 + y^2) \}

and consider the map $f:\left[0,1{\right]}^{3}\to Z$ given by

$f:\left(x,y,z\right)↦\left(x,y,\frac{1}{2}\left({x}^{2}+{y}^{2}\right)z\right)$f : (x,y,z) \mapsto (x,y, \frac{1}{2}(x^2 + y^2) z)

and the map $g:Z\to X$ given away from $\left({x}^{2}+{y}^{2}\right)=0$ by

$g:\left(x,y,w\right)↦\tau \left(x,y,2\frac{w}{{x}^{2}+{y}^{2}}\right)\phantom{\rule{thinmathspace}{0ex}}.$g : (x,y,w) \mapsto \tau(x,y, 2 \frac{w}{x^2 + y^2}) \,.

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 $\left[0,1{\right]}^{3}\to 2\mathrm{Mor}\left({P}_{2}\left(X\right)\right)$ given by

$\left[0,1{\right]}^{3}×\left[0,1{\right]}^{2}\to X$[0,1]^3 \times [0,1]^2 \to X
$\left(\left(x,y,z\right),\left(s,t\right)\right)↦\tau \left(sx,ty,z\right)$((x,y,z),(s,t)) \mapsto \tau(s x, t y, z)

as $\left[0,1{\right]}^{3}×\left[0,1{\right]}^{2}\to Z×\left[0,1{\right]}^{2}\to Z\to X$

$\left(\left(x,y,z\right),\left(s,t\right)\right)↦\left(\left(x,y,\frac{1}{2}\left({x}^{2}+{y}^{2}\right)\right),\left(s,t\right)\right)↦\left(sx,ty,\frac{1}{2}\left(\left(sx{\right)}^{2}+\left(ty{\right)}^{2}\right)z\right)↦\tau \left(sx,sy,z\right)\phantom{\rule{thinmathspace}{0ex}}.$((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) \,.

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 $\left[0,1{\right]}^{2}\to \left[0,1{\right]}^{2}$ that realizes a square-shaped bigon.

Under the hom-adjunction it corresponds to a factorization of ${G}_{\Gamma }:\left[0,1{\right]}^{3}\to 2\mathrm{Mor}\left({P}_{2}\left(X\right)\right)$ into

${G}_{\Gamma }:\left[0,1{\right]}^{3}\stackrel{f}{\to }Z\to 2\mathrm{Mor}\left({P}_{2}\left(X\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$G_\Gamma : [0,1]^3 \stackrel{f}{\to} Z \to 2 Mor(\mathbf{P}_2(X)) \,.

By the above construction we have the the push-forwards

${f}_{*}:\frac{\partial }{\partial x}\left(x=0,y=0,z\right)↦\frac{\partial }{\partial x}\left(x=0,y=0,w=0\right)$f_* : \frac{\partial}{\partial x}(x=0,y=0,z) \mapsto \frac{\partial}{\partial x}(x= 0, y = 0, w = 0)

and similarly for $\frac{\partial }{\partial y}$ are indendent of $z$. It follows by the chain rule that also

$\frac{{\partial }^{2}{G}_{\Gamma }}{\partial x\partial y}{\mid }_{\left(x=0,y=0\right)}$\frac{\partial^2 G_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}

is independent of $z$. But at $z=0$ this equals $\frac{{\partial }^{2}{F}_{\Gamma }}{\partial x\partial y}{\mid }_{\left(x=0,y=0\right)}$, while at $z=1$ it equals $\frac{{\partial }^{2}{F}_{\Gamma \prime }}{\partial x\partial y}{\mid }_{\left(x=0,y=0\right)}$. Therefore these two are equal.

### Flat $\infty$-parallel transport in $\mathrm{Top}$

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 $H=$ Top lots of structure of cohesive $\left(\infty ,1\right)$-topos theory degenerates, since by the homotopy hypothesis-theorem here the global section (∞,1)-geometric morphism

$\left(\Pi ⊣\Delta ⊣\Gamma \right):\mathrm{Top}\stackrel{\stackrel{\Pi }{←}}{\stackrel{\stackrel{\Delta }{←}}{\underset{\Gamma }{\to }}}\in \infty \mathrm{Grpd}$(\Pi \dashv \Delta \dashv \Gamma) : Top \stackrel{\overset{\Pi}{\leftarrow}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \in \infty Grpd

an equivalence. The abstract fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\Pi$ is here the ordinary fundamental ∞-groupoid

$\Pi :\mathrm{Top}\stackrel{\simeq }{\to }\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$\Pi : Top \stackrel{\simeq}{\to} \infty Grpd \,.

If both (∞,1)-toposes here are 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

$\left(\mid -\mid ⊣\mathrm{Sing}\right):\mathrm{Top}\stackrel{←}{\stackrel{{\simeq }_{\mathrm{Quillen}}}{\to }}\mathrm{Top}\phantom{\rule{thinmathspace}{0ex}}.$(|-| \dashv Sing) : Top \stackrel{\leftarrow}{\overset{\simeq_{Quillen}}{\to}} Top \,.

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 $F\in \mathrm{Top}$ and $\mathrm{Aut}\left(F\right)\in \mathrm{Top}$ its automorphism ∞-group, $F$-fibrations over a base space $X\in \mathrm{Top}$ are classfied by morphisms

$g:X\to B\mathrm{Aut}\left(F\right)$g : X \to B Aut(F)

into the delooping of $\mathrm{Aut}\left(F\right)$. The corresponding fibration $P\to X$ itself is the homotopy fiber of this cocycles, given by the homotopy pullback

$\begin{array}{ccc}P& \to & *\\ ↓& & ↓\\ X& \stackrel{g}{\to }& B\mathrm{Aut}\left(F\right)\end{array}$\array{ P &\to& * \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& B Aut(F) }

in Top, as described at principal ∞-bundle.

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

$\Pi \left(P\right)\to \Pi \left(X\right)\to B\mathrm{Aut}\left(\Pi \left(F\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\Pi(P) \to \Pi(X) \to B Aut(\Pi(F)) \,.

One may think of the morphism $\Pi \left(X\right)\to B\mathrm{Aut}\left(\Pi \left(F\right)\right)$ now as the $\infty$-parallel transport coresponding to the original fibration:

• to each point in $X$ it assigns the unique object of $B\mathrm{Aut}\left(\Pi \left(F\right)\right)$, which is the fiber $F$ itself;

• to each path $\left(x\to y\right)$ in $X$ it assigns an equivalence between the fibers ${F}_{x}\mathrm{to}{F}_{y}$ etc.

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

We may embed this example into the smooth context by regarding $\mathrm{Aut}\left(F\right)$ as a discrete ∞-Lie groupoid as discussed in the section Flat ∞-Parallel transport in ∞LieGrpd.

For that purpose let

$\left({\Pi }_{\mathrm{smooth}}⊣{\mathrm{Disc}}_{\mathrm{smooth}}⊣{\Gamma }_{\mathrm{smooth}}\right):\infty \mathrm{LieGrpd}\stackrel{\stackrel{{\Pi }_{\mathrm{smooth}}}{\to }}{\stackrel{\stackrel{{\mathrm{Disc}}_{\mathrm{smooth}}}{←}}{\underset{{\Gamma }_{\mathrm{smooth}}}{\to }}}\infty \mathrm{Grpd}\simeq \mathrm{Top}$(\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

We may reflect the ∞-group $\mathrm{Aut}\left(F\right)$ into this using the constant ∞-stack-functor $\mathrm{Disc}$ to get the discrete ∞-Lie group $\mathrm{Disc}\mathrm{Aut}\left(F\right)$. Let then $X$ be a paracompact smooth manifold, regarded naturally as an object of ∞LieGrpd. Then we can consider cocycles/classifying morphisms

$X\to B\mathrm{Disc}\mathrm{Aut}\left(F\right)\phantom{\rule{thinmathspace}{0ex}},$X \to \mathbf{B} Disc Aut(F) \,,

now in the smooth context of $\infty \mathrm{LieGrpd}$.

###### Proposition

The ∞-groupoid of $F$-fibrations in Top is equivalent to the $\infty$-groupoid of $\mathrm{Disc}\mathrm{Aut}\left(F\right)$-principal ∞-bundles in ∞LieGrpd:

$\infty \mathrm{LieGrpd}\left(X,B\mathrm{Disc}\mathrm{Aut}\left(F\right)\right)\simeq \mathrm{Top}\left(X,B\mathrm{Aut}\left(F\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\infty LieGrpd(X, \mathbf{B} Disc Aut(F)) \simeq Top(X, B Aut(F)) \,.

Moreover, all the principal ∞-bundles classified by the morphisms on the left have canonical extensions to Flat differential cohomology in $\infty \mathrm{LieGrpd}$, in that the flat parallel $\infty$-transport ${\nabla }_{\mathrm{flat}}$ in

$\begin{array}{ccc}X& \stackrel{g}{\to }& B\mathrm{Disc}\mathrm{Aut}\left(F\right)\\ ↓& {↗}_{{\nabla }_{\mathrm{flat}}}\\ \Pi \left(X\right)\end{array}$\array{ X &\stackrel{g}{\to}& \mathbf{B} Disc Aut(F) \\ \downarrow & \nearrow_{\nabla_{flat}} \\ \mathbf{\Pi}(X) }

always exists.

###### 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 locally ∞-connected (∞,1)-topos the functor $\mathrm{Disc}$ is a full and faithful (∞,1)-functor.

(…)

### $\infty$-Parallel transport from flat differential forms with values in chain complexes

A typical choice for an (∞,1)-category of ”$\infty$-vector spaces” is that 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 $\mathrm{Mod}$ for this stack, then the $\infty$-parallel transport for a flat $\infty$-vector bundle on some $X$ is a morphism

$\Pi \left(X\right)\to \mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{\Pi}(X) \to Mod \,.

This is typically given by differential form data with values in $\mathrm{Mod}$.

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 $\Pi \left(X\right)\to \mathrm{Mod}$ is in (AbadSchaetz), building on a construciton in (Igusa).

## Applications

### In physics

In physics various action functionals for quantum field theories are nothing but higher parallel transport.

## 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

The description of connections on a 2-bundle in terms of such parallel 2-transport

Parallel transport for circle n-bundles with connection is discussed generally in

• Kiyonori Gomi and Yuji Terashima, Higher dimensional parallel transport Mathematical Research Letters 8, 25–33 (2001) (pdf)

and

• David Lipsky, Cocycle constructions for topological field theories (2010) (pdf)

The integration of flat differential forms with values in chain complexes toflat $\infty$-parallel transport on $\infty$-vector bundles is in
Remarks on $\infty$-parallel transport in Top are in