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 $(\esh \dashv \flat \dashv \sharp)$ 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
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.
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.
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).
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:
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:
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:
Last revised on March 30, 2024 at 20:02:48. See the history of this page for a list of all contributions to it.