nLab flat infinity-connection

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

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

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.

Related concepts

• flat connection

• Galois theory

• local system

• constant infinity-stack

• Riemann-Hilbert correspondence