+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### $\infty$-Chern-Weil theory +--{: .hide} [[!include infinity-Chern-Weil theory - contents]] =-- #### Differential cohomology +--{: .hide} [[!include differential cohomology - contents]] =-- =-- =-- \tableofcontents ## 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 $$ \nabla \;\colon\; X \to \flat A $$ 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](cohesive+%28infinity,1%29-topos+--+structures#FlatDifferentialCohomology)_. ## 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 [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|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 [[coherence|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 infinity-stacks|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)] * {#PoleselloWaschkies05} [[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](https://eudml.org/doc/51918)] In view of [[cohesive homotopy theory]]: * [[schreiber:differential cohomology in a cohesive topos|dcct §3.8.5]] In the generality of [[(infinity,1)-module bundles|$(\infty,1)$-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), [[HopkinsLurie-Ambidexterity.pdf:file]]] ### Flat $(\infty,1)$-vector bundles ($\infty$-local systems) {#ReferencesFlatInfinityVectorBundles} On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|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](https://doi.org/10.1093/imrn/rns166)] * {#BlockSmith14} [[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](https://doi.org/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](https://arxiv.org/abs/1510.04975)] Identification with [[(infinity,1)-functors|$(\infty,1)$-functors]] is made explicit in: * [Block & Smith (2014), below Def. 2.3 (§2.2 in the preprint)](#BlockSmith14) * [[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](https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/RiveraZeinalian)] and construction of a [[model category]] of $\infty$-local systems: * [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Model Structure on K-Linear Infinity-Local Systems|A Global Model Structure for $\mathbb{K}$-Linear $\infty$-Local Systems]]* [[arXiv:2604.05671](https://arxiv.org/abs/2604.05671)] Enhancement of the [[Chern-Weil homomorphism]] from [[ordinary cohomology]]-groups to [[dg-categories]] of [[infinity-local systems|$\infty$-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](https://doi.org/10.1090/tran/9068)] ### $\infty$-Local systems as topological quantum state spaces On constructions of ([[extended TQFT|extended]], [[Dijkgraaf-Witten theory|Dijkgraaf-Witten-type]]) [[functorial field theory|functorial]] [[topological quantum field theories]] whose (relative) [[quantum state spaces]] are $\infty$-local systems (possibly in more general [[(infinity,1)-category|$(\infty,1)$-categories]] that [[(infinity,1)-category of chain complexes|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](https://doi.org/10.1090/crmp/050), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/view?ProductCode=CRMP/50)] * [[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](https://doi.org/10.1007/s40062-013-0047-2)] * [[Urs Schreiber]]: *[[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](https://doi.org/10.4171/qt/223)] [[!redirects flat infinity-connections]] [[!redirects flat ∞-connection]] [[!redirects flat ∞-connections]] [[!redirects flat infinity-vector bundle]] [[!redirects flat infinity-vector bundles]] [[!redirects flat ∞-vector bundle]] [[!redirects flat ∞-vector bundles]] [[!redirects infinity-local system]] [[!redirects infinity-local systems]] [[!redirects ∞-local system]] [[!redirects ∞-local systems]] [[!redirects flat (∞,1)-bundle]] [[!redirects flat (∞,1)-bundles]] [[!redirects flat (infinity,1)-bundle]] [[!redirects flat (infinity,1)-bundles]] [[!redirects flat principal ∞-connection]] [[!redirects flat principal ∞-connections]] [[!redirects flat principal infinity-connection]] [[!redirects flat principal infinity-connections]] [[!redirects flat ∞-bundle]] [[!redirects flat ∞-bundles]] [[!redirects flat infinity-bundle]] [[!redirects flat infinity-bundles]]