nLab
flat infinity-connection

Contents

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

The generalization of the notion of flat connection from differential geometry to higher differential geometry and generally to higher geometry.

Definition

Given a cohesive (∞,1)-topos (Π)(\Pi \dashv \flat \dashv \sharp) with shape modality Π\Pi and flat modality \flat, a flat \infty-connection an an object XX with coefficients in an object AA is a morphism

:XA \nabla \;\colon\; X \to \flat A

or equivalently a morphism

:Π(X)A. \nabla \;\colon\; \Pi(X) \to A \,.

This is also sometimes called a local system on XX with coefficients in AA, or a cocycle in nonabelian cohomology of XX with constant coefficients AA.

For A=BGA = \mathbf{B}G the delooping of an ∞-group, flat \infty-connections with coefficients in AA are a special case of GG-principal ∞-connections.

For more see at structures in a cohesive (∞,1)-topos – flat ∞-connections.

Last revised on September 29, 2017 at 16:28:32. See the history of this page for a list of all contributions to it.