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

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*.

Last revised on May 26, 2022 at 02:18:09. See the history of this page for a list of all contributions to it.