nLab flat infinity-connection



\infty-Chern-Weil theory

Differential cohomology



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 (ʃ)(\esh \dashv \flat \dashv \sharp) with shape modality ʃ\esh 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 May 26, 2022 at 02:18:09. See the history of this page for a list of all contributions to it.