# 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 $(\esh \dashv \flat \dashv \sharp)$ with shape modality $\esh$ 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

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

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

## Examples

### Flat principal $\infty$-bundles

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.

### Flat $(\infty,1)$-vector bundles ($\infty$-local systems)

For $A = Core(Ch_k)$ the core of an (infinity,1)-category of chain complexes, functors $\esh X \longrightarrow A$ are $(\infty,1)$-vector bundles with flat $\infty$-connections.

In parts of the literature this case is understood by default when speaking of “$\infty$-local systems”. Other parts refer to this as “representations up to homotopy” (really: up to coherent higher homotopy).

### General

On higher version of Galois theory via automorphisms of locally constant $\infy$-stacks:

In view of cohesive homotopy theory:

### Flat $(\infty,1)$-vector bundles ($\infty$-local systems)

Component-definitions are due to:

Identification with $(\infty,1)$-functors is made explicit in:

Enhancement of the Chern-Weil homomorphism from ordinary cohomology-groups to dg-categories of $\infty$-local systems:

• Camilo Arias Abad, Santiago Pineda Montoya, Alexander Quintero Velez, Chern-Weil theory for $\infty$-local systems $[$arXiv:2105.00461$]$