Schreiber
Ehresmann ∞-connection

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,

Y g A underlyingcocycle Π(Y) EA connection Π(X) Pchar() BA curvaturecharacteristicforms \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 YXY \to X is a hypercover, i.e. an acyclic fibration in SPSh(C) locSPSh(C)^{loc}.

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

We show below how the entire diagram above may be pulled back along pp to the total space of this principal ∞-bundle where it gives rise to AA-valued differential form data on PP 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

Π vert(P) vert A connectionrestrictedtofibers firstEhresmanncondition Π(Y) EA connectionontotalspace secondEhresmanncondition Π(X) Pchar() BA curvaturecharacteristicform. \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 Π vert(P)\Pi_vert(P) denotes the vertical path ∞-groupoid of PP, described in detail below.

All three horizontal morphisms depicted trivialize when restricted along the canonical inclusion YΠ(Y)Y \to \Pi(Y) and PΠ(P)P \to \Pi(P), respectively. This means these are cocycles in flat differential AA-cohomology whose underlying AA-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 ASPSh(C)A \in SPSh(C) be a pointed object with point pt A:*Apt_A : {*} \to A.

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

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

If the cocycle g:XAg : X \to A is modeled in the model structure on simplicial presheaves as a span XYAX \stackrel{\simeq}{\leftarrow} Y \stackrel{}{\to} A with AA fibrant then this PP here is modeled by the ordinary pullback of the object E ptA\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

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

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

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

By the universal property of the pullback the commutativity of

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

induces a universal morphism

PΠ vert(P) P \to \Pi_vert(P)

into the vertical path ∞-groupoid.

Given a strictly commuting diagram

Y g A Π(Y) EA Π(X) char() BA \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)SPSh(C) represrenting a [[differential cohomology|diffential cocycle], let p:PYp : P \to Y be the morphism classified by g:YAg : Y \to A as recalled above.

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

Π vert(Y) Y g A Π(P) Π(Y) E ptA Π(X) = Π(X) char() BA \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 PΠ vert(P)P \to \Pi_{vert}(P), along PΠ(P)P \to \Pi(P) and YΠ(Y)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 PP reduces to the trivialization of any AA-cocycle when pulled back along the total space of the bundle it classifies: we have a commuting diagram

P E ptA Π vert(P) Y g A. \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 ptA\mathbf{E}_{pt}A in terms of the pullback diagram this extends to a diagram

P E ptA * pt A A I d 0 A d 1 Y g A \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)