Idea

An Ehresmann $\infty$-connection is a reformulation of a cocycle in differential cohomology in terms of abstract deRham classes on the total space of the underlying principal ∞-bundle. It generalizes the ordinary notion of an Ehresmann connection on an ordinray principal bundle.

More in detail, recall that, as discussed at differential cohomology,

$\begin{array}{cccccc}Y& \stackrel{g}{\to }& A& & & \mathrm{underlying}\mathrm{cocycle}\\ ↓& & ↓& & & \\ \Pi \left(Y\right)& \stackrel{\nabla }{\to }& EA& & & \mathrm{connection}\\ ↓& & ↓& & & \\ \Pi \left(X\right)& \stackrel{P\simeq \mathrm{char}\left(\nabla \right)}{\to }& BA& & & \mathrm{curvature}\mathrm{characteristic}\mathrm{forms}\end{array}$\array{ Y &\stackrel{g}{\to}& A &&& underlying cocycle \\ \downarrow && \downarrow &&& \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection \\ \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{P \simeq char(\nabla)}{\to} & \mathbf{B}A &&& curvature characteristic forms }

where $Y\to X$ is a hypercover, i.e. an acyclic fibration in $\mathrm{SPSh}\left(C{\right)}^{\mathrm{loc}}$.

Here the top horizontal morphism $g:Y\to A$ is the underlying $A$-valued cocycle that classifies an $A$-principal ∞-bundle $p:P\to X$.

We show below how the entire diagram above may be pulled back along $p$ to the total space of this principal ∞-bundle where it gives rise to $A$-valued differential form data on $P$ that satisfies two constraints. This datum and these constraints are analogous to and generalize the notion of an Ehresmann connection on an ordinary principal bundle.

It is exhibited by a diagram

$\begin{array}{cccccc}{\Pi }_{\mathrm{vert}}\left(P\right)& \stackrel{{\nabla }_{\mathrm{vert}}}{\to }& A& & & \mathrm{connection}\mathrm{restricted}\mathrm{to}\mathrm{fibers}\\ ↓& & ↓& & & \mathrm{first}\mathrm{Ehresmann}\mathrm{condition}\\ \Pi \left(Y\right)& \stackrel{\nabla }{\to }& EA& & & \mathrm{connection}\mathrm{on}\mathrm{total}\mathrm{space}\\ ↓& & ↓& & & \mathrm{second}\mathrm{Ehresmann}\mathrm{condition}\\ \Pi \left(X\right)& \stackrel{P\simeq \mathrm{char}\left(\nabla \right)}{\to }& BA& & & \mathrm{curvature}\mathrm{characteristic}\mathrm{form}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \Pi_{vert}(P) &\stackrel{\nabla_{vert}}{\to}& A &&& connection restricted to fibers \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ \Pi(X) &\stackrel{P \simeq char(\nabla)}{\to} & \mathbf{B}A &&& curvature characteristic form } \,.

Here ${\Pi }_{\mathrm{vert}}\left(P\right)$ denotes the vertical path ∞-groupoid of $P$, described in detail below.

All three horizontal morphisms depicted trivialize when restricted along the canonical inclusion $Y\to \Pi \left(Y\right)$ and $P\to \Pi \left(P\right)$, respectively. This means these are cocycles in flat differential $A$-cohomology whose underlying $A$-class vanishes. This is the characterization of nonabelian deRham cohomology.

In a next step we may express the abstract (∞,1)-topos-theoretic nonabelian deRham cocycle appearing here in terms of Lie-∞-algebra connection data. This turns the Ehresmann ∞-connection into a

Definition

Let $A\in \mathrm{SPSh}\left(C\right)$ be a pointed object with point ${\mathrm{pt}}_{A}:*\to A$.

Recall from principal ∞-bundle that iven an $A$ cocycle $X\stackrel{\simeq }{←}Y\stackrel{g}{\to }A$ the principal ∞-bundle $P\to X$ classified by it is the homotopy fiber of this morphism

$\begin{array}{ccc}P& \to & *\\ ↓& & {↓}^{{\mathrm{pt}}_{A}}\\ X& \stackrel{g}{\to }& A\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ P &\to& {*} \\ \downarrow && \downarrow^{pt_A} \\ X &\stackrel{g}{\to}& A } \,.

If the cocycle $g:X\to A$ is modeled in the model structure on simplicial presheaves as a span $X\stackrel{\simeq }{←}Y\stackrel{}{\to }A$ with $A$ fibrant then this $P$ here is modeled by the ordinary pullback of the object ${E}_{\mathrm{pt}}A$ that is described at list of notation and constructions in categories of fibrant objects i.e. by the ordinary pullback diagram

$\begin{array}{ccc}P& \to & {E}_{\mathrm{pt}}A\\ ↓& & ↓\\ Y& \to & A\\ ↓\\ X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ P &\to& \mathbf{E}_{pt} A \\ \downarrow && \downarrow \\ Y &\to& A \\ \downarrow \\ X } \,.
Definition (vertical path ∞-groupoid)

Given any morphism $p:P\to Y$ in $\mathrm{SPSh}\left(C\right)$ we say that the the vertical path ∞-groupoid with respect to $p$ is the objec ${\Pi }_{\mathrm{vert}}\left(P\right)$ given by the ordinary pullback

$\begin{array}{ccc}{\Pi }_{\mathrm{vert}}\left(P\right)& \to & Y\\ ↓& & ↓\\ \Pi \left(P\right)& \to & \Pi \left(Y\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \Pi_{vert}(P) &\to& Y \\ \downarrow && \downarrow \\ \Pi(P) &\to& \Pi(Y) } \,.
Remark

By the universal property of the pullback the commutativity of

$\begin{array}{ccc}P& \to & Y\\ ↓& & ↓\\ \Pi \left(P\right)& \to & \Pi \left(Y\right)\end{array}$\array{ P &\to& Y \\ \downarrow && \downarrow \\ \Pi(P) &\to& \Pi(Y) }

induces a universal morphism

$P\to {\Pi }_{\mathrm{vert}}\left(P\right)$P \to \Pi_vert(P)

into the vertical path ∞-groupoid.

Given a strictly commuting diagram

$\begin{array}{ccc}Y& \stackrel{g}{\to }& A\\ ↓& & ↓& & & \\ \Pi \left(Y\right)& \stackrel{\nabla }{\to }& EA\\ ↓& & ↓& & & \\ \Pi \left(X\right)& \stackrel{\mathrm{char}\left(\nabla \right)}{\to }& BA\end{array}$\array{ Y &\stackrel{g}{\to}& A \\ \downarrow && \downarrow &&& \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A \\ \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{char(\nabla)}{\to} & \mathbf{B}A }

in $\mathrm{SPSh}\left(C\right)$ represrenting a [[differential cohomology|diffential cocycle], let $p:P\to Y$ be the morphism classified by $g:Y\to A$ as recalled above.

We may paste to the diagra the square that defines the vertical path ∞-groupoid of $p:P\to Y$ to obtain

$\begin{array}{ccccc}{\Pi }_{\mathrm{vert}}\left(Y\right)& \to & Y& \stackrel{g}{\to }& A\\ ↓& & ↓& & ↓& & & \\ \Pi \left(P\right)& \to & \Pi \left(Y\right)& \stackrel{\nabla }{\to }& {E}_{\mathrm{pt}}A\\ ↓& & ↓& & ↓& & & \\ \Pi \left(X\right)& \stackrel{=}{\to }& \Pi \left(X\right)& \stackrel{\mathrm{char}\left(\nabla \right)}{\to }& BA\end{array}$\array{ \Pi_{vert}(Y) &\to& Y &\stackrel{g}{\to}& A \\ \downarrow && \downarrow && \downarrow &&& \\ \Pi(P) &\to& \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}_{pt}A \\ \downarrow && \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{=}{\to}& \Pi(X) &\stackrel{char(\nabla)}{\to} & \mathbf{B}A }
Lemma

All three horizontal morphisms of this pasted diagram trivialize when restricted to constant paths along $P\to {\Pi }_{\mathrm{vert}}\left(P\right)$, along $P\to \Pi \left(P\right)$ and $Y\to \Pi \left(Y\right)$, respectively.

Therefore all three morphism represent cocycles in (nonabelian) deRham cohomology.

Proof

The bottom horizontal morphism trivializes by assumption that the diagram representes a differential cocycle.

The trivialization of the top horizontal morphism when restricted to $P$ reduces to the trivialization of any $A$-cocycle when pulled back along the total space of the bundle it classifies: we have a commuting diagram

$\begin{array}{ccccc}& & P& \to & {E}_{\mathrm{pt}}A\\ & ↙& ↓& & ↓\\ {\Pi }_{\mathrm{vert}}\left(P\right)& \to & Y& \stackrel{g}{\to }& A\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && P &\to& \mathbf{E}_{pt}A \\ &\swarrow& \downarrow && \downarrow \\ \Pi_{vert}(P) &\to& Y &\stackrel{g}{\to}& A } \,.

By the very definition of ${E}_{\mathrm{pt}}A$ in terms of the pullback diagram this extends to a diagram

$\begin{array}{ccccc}P& \to & {E}_{\mathrm{pt}}A& \to & *\\ ↓& & ↓& & {↓}^{{\mathrm{pt}}_{A}}\\ & & {A}^{I}& \stackrel{{d}_{0}}{\to }& A\\ ↓& & {↓}^{{d}_{1}}\\ Y& \stackrel{g}{\to }& A\end{array}$\array{ P &\to& \mathbf{E}_{pt}A &\to& {*} \\ \downarrow && \downarrow && \downarrow^{pt_A} \\ && A^I &\stackrel{d_0}{\to}& A \\ \downarrow && \downarrow^{d_1} \\ Y &\stackrel{g}{\to}& A }

which exhibits the homotopy from the pulled back cocycle to the trivial cocycle.