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 on an object $X$ with coefficients in an object $A$ is a morphism

or equivalently a morphism

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

• connection on a smooth principal infinity-bundle

• infinity-local system, local system

• infinity-group infinity-algebra

• flat connection

• Galois theory

• constant infinity-stack

• Riemann-Hilbert correspondence

References

General

On higher version of Galois theory via automorphisms of locally constant $\infty$-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](http://www.numdam.org/item/?id=CTGDC_2002__43_4_257_0)]

• Pietro Polesello, Ingo Waschkies, Higher monodromy, Homology, Homotopy and Applications 7 1 (2005) 109-150 [[arXiv:0407507](http://arxiv.org/abs/math/0407507), eudml:51918]

In view of cohesive homotopy theory:

• dcct §3.8.5

In the generality of $(\infty,1)$-module bundles:

• Michael Hopkins, Jacob Lurie: Local Systems, section 4.3 of: Ambidexterity in K(n)-Local Stable Homotopy Theory (2013) [[pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf), pdf]

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](https://arxiv.org/abs/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](https://arxiv.org/abs/0908.2843), doi:10.1016/j.aim.2013.11.001]

• Camilo Arias Abad, Florian Schätz: Flat $\mathbb{Z}$-graded connections and loop spaces, International Mathematics Research Notices 2018 4 (2018) 961–1008 [[doi:10.1093/imrn/rnw258](https://doi.org/10.1093/imrn/rnw258), arXiv:1510.04975]

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](https://arxiv.org/abs/1805.01264), higher-structures:Vol4Iss1]

and construction of a model category of $\infty$-local systems:

• Hisham Sati, Urs Schreiber , : §3 of: Entanglement A of Global Sections Model Structure for $\mathbb{K}$-Linear $\infty$-Local Systems [[arXiv:2309.07245](https://arxiv.org/abs/2309.07245)] [[arXiv:2604.05671](https://arxiv.org/abs/2604.05671)]

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, Trans. Amer. Math. Soc. 377 (2024) 3129-3171 [[arXiv:2105.00461](https://arxiv.org/abs/2105.00461), doi:10.1090/tran/9068]

$\infty$-Local systems as topological quantum state spaces

On constructions of (extended, Dijkgraaf-Witten-type) functorial topological quantum field theories whose (relative) quantum state spaces are $\infty$-local systems (possibly in more general $(\infty,1)$-categories that of chain complexes):

• Daniel Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman: Topological Quantum Field Theories from Compact Lie Groups, in P. R. Kotiuga (ed.), A celebration of the mathematical legacy of Raoul Bott, AMS (2010) 367-403 [[arXiv:0905.0731](http://arxiv.org/abs/0905.0731), doi:10.1090/crmp/050, ISBN:978-0-8218-4777-0]

• Jeffrey C. Morton: Extended TQFT, Gauge Theory, And 2-Linearization, J. Homotopy Relat. Struct. 10 (2015) 127–187 [[arXiv:1003.5603](https://arxiv.org/abs/1003.5603), doi:10.1007/s40062-013-0047-2]

• Urs Schreiber: Quantization via Linear homotopy types, talk notes (2014) [[arXiv:1402.7041](https://arxiv.org/abs/1402.7041)]

• Daniel S. Freed, Gregory W. Moore, Constantin Teleman, §A.2 in: Topological symmetry in quantum field theory, Quantum Topology, 15 3/4 (2023) 779–869 [[arXiv:2209.07471](https://arxiv.org/abs/2209.07471), doi:10.4171/qt/223]