nLab curving

Idea

The differential 2-form-data in a circle 2-bundle with connection is sometimes called the “curving” of the connection data.

This terminology results from thinking of the circle 2-bundle equivalently as a bundle gerbe given by a transition-circle bundle on a suitable Cech groupoid. From this perspective the differential 1-form is an ordinary connection on a bundle for the transition bundles, and the 2-form data looks a bit like a variant of the curvature of these bundles, which not exactly being this curvature. Hence “curving”.

But of course the circle 2-bundle with connection has its own curvature differential 3-form. At least as soon as one passes to circle n-bundles with connection for arbitrary $n \in \mathbb{N}$, the term “curving” becomes inappropriate.

Properties

$(\mathbf{B}U(1))$-principal 2-connections without curving

Circle 2-bundles “with connection but without curving” have as moduli 2-stack the delooping

$\mathbf{B}(\mathbf{B}U(1)_{conn}) \in$ Smooth∞Grpd

of the moduli stack $\mathbf{B}U(1)_{conn}$ or circle bundles with connection.

The higher Atiyah groupoid of such a “pseudo-2-connection” $\nabla_1 \colon X \to \mathbf{B}(\mathbf{B}U(1)_{conn})$ is the corresponding higher Courant groupoid which Lie integrates the corresponding standard Courant Lie 2-algebroid. (See at higher Atiyah groupoid for details.)

Related concepts

• curvature

• differential cohomology