# nLab flat infinity-connection

Contents

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

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 $(\Pi \dashv \flat \dashv \sharp)$ with shape modality $\Pi$ and flat modality $\flat$, a flat $\infty$-connection an an object $X$ with coefficients in an object $A$ is a morphism

$\nabla \;\colon\; X \to \flat A$

or equivalently a morphism

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

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

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