# 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,

$\array{ Y &\stackrel{g}{\to}& A &&& underlying cocycle \\ \downarrow && \downarrow &&& \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection \\ \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{P \simeq char(\nabla)}{\to} & \mathbf{B}A &&& curvature characteristic forms }$

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

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

$\array{ \Pi_{vert}(P) &\stackrel{\nabla_{vert}}{\to}& A &&& connection restricted to fibers \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ \Pi(X) &\stackrel{P \simeq char(\nabla)}{\to} & \mathbf{B}A &&& curvature characteristic form } \,.$

Here $\Pi_vert(P)$ denotes the vertical path ∞-groupoid of $P$, described in detail below.

All three horizontal morphisms depicted trivialize when restricted along the canonical inclusion $Y \to \Pi(Y)$ and $P \to \Pi(P)$, 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 $A \in SPSh(C)$ be a pointed object with point $pt_A : {*} \to A$.

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

$\array{ P &\to& {*} \\ \downarrow && \downarrow^{pt_A} \\ X &\stackrel{g}{\to}& A } \,.$

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

$\array{ P &\to& \mathbf{E}_{pt} A \\ \downarrow && \downarrow \\ Y &\to& A \\ \downarrow \\ X } \,.$
###### Definition (vertical path ∞-groupoid)

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

$\array{ \Pi_{vert}(P) &\to& Y \\ \downarrow && \downarrow \\ \Pi(P) &\to& \Pi(Y) } \,.$
###### Remark

By the universal property of the pullback the commutativity of

$\array{ P &\to& Y \\ \downarrow && \downarrow \\ \Pi(P) &\to& \Pi(Y) }$

induces a universal morphism

$P \to \Pi_vert(P)$

into the vertical path ∞-groupoid.

Given a strictly commuting diagram

$\array{ Y &\stackrel{g}{\to}& A \\ \downarrow && \downarrow &&& \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A \\ \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{char(\nabla)}{\to} & \mathbf{B}A }$

in $SPSh(C)$ 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

$\array{ \Pi_{vert}(Y) &\to& Y &\stackrel{g}{\to}& A \\ \downarrow && \downarrow && \downarrow &&& \\ \Pi(P) &\to& \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}_{pt}A \\ \downarrow && \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{=}{\to}& \Pi(X) &\stackrel{char(\nabla)}{\to} & \mathbf{B}A }$
###### Lemma

All three horizontal morphisms of this pasted diagram trivialize when restricted to constant paths along $P \to \Pi_{vert}(P)$, along $P \to \Pi(P)$ and $Y \to \Pi(Y)$, 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

$\array{ && P &\to& \mathbf{E}_{pt}A \\ &\swarrow& \downarrow && \downarrow \\ \Pi_{vert}(P) &\to& Y &\stackrel{g}{\to}& A } \,.$

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

$\array{ P &\to& \mathbf{E}_{pt}A &\to& {*} \\ \downarrow && \downarrow && \downarrow^{pt_A} \\ && A^I &\stackrel{d_0}{\to}& A \\ \downarrow && \downarrow^{d_1} \\ Y &\stackrel{g}{\to}& A }$

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

Revised on September 17, 2009 12:11:25 by Urs Schreiber (195.37.209.182)