nLab flat infinity-connection

Contents

Context

\infty-Chern-Weil theory

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 (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

:ʃ(X)A. \nabla \;\colon\; \esh(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 more see at structures in a cohesive (∞,1)-topos – flat ∞-connections.

Examples

Flat principal \infty-bundles

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.

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

For A=Core(Ch k)A = Core(Ch_k) the core of an (infinity,1)-category of chain complexes, functors ʃXA\esh X \longrightarrow A are ( , 1 ) (\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”.

References

General

(…)

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

Component-definitions are due to:

On \infty -local systems in the sense of ( , 1 ) (\infty,1) -vector bundles with flat \infty -connections:

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

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

Last revised on September 20, 2022 at 02:26:01. See the history of this page for a list of all contributions to it.