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

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

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).

Related concepts

• flat connection

• Galois theory

• local system

• constant infinity-stack

• Riemann-Hilbert correspondence

References

General

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

• Bertrand Toën, Vers une interprétation galoisienne de la théorie de l’homotopie, Cahiers de Topologie et Géométrie Différentielle Catégoriques 43 4 (2002) 257-312 [numdam:CTGDC_2002__43_4_257_0]

• Pietro Polesello, Ingo Waschkies, Higher monodromy, Homology, Homotopy and Applications 7 1 (2005) 109-150 [arXiv:0407507, eudml:51918]

In view of cohesive homotopy theory:

• dcct §3.8.5

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

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

Component-definitions are due to:

• Camilo Arias Abad, Florian Schätz: The $A_\infty$ de Rham theorem and integration of representations up to homotopy, International Mathematics Research Notices, 2013 16 (2013) 3790–3855 [arXiv:1011.4693, doi:10.1093/imrn/rns166]

• Jonathan Block, Aaron M. Smith, The higher Riemann–Hilbert correspondence, Advances in Mathematics 252 (2014) 382-405 [arXiv:0908.2843, doi:10.1016/j.aim.2013.11.001]

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

• Block & Smith (2014), below Def. 2.3 (§2.2 in the preprint)

• Manuel Rivera, Mahmoud Zeinalian, §5 of: The colimit of an $\infty$-local system as a twisted tensor product, Higher Structures 4 1 (2020) 33-56 [arXiv:1805.01264, higher-structures:Vol4Iss1]

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$]$