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”.
(…)
Component-definitions are due to:
On $\infty$-local systems in the sense of $(\infty,1)$-vector bundles with flat $\infty$-connections:
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:
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.