Under suitable conditions (if we have Galois theory) local systems on correspond to functors out of the fundamental groupoid of , or more generally to (∞,1)-functors out of the fundamental ∞-groupoid. These in turn are equivalently flat connections (this relation is known as the Riemann-Hilbert correspondence) or generally flat ∞-connections.
For an (∞,1)-sheaf (∞,1)-topos, write
Write ∞Grpd for the core ∞-groupoid of the (∞,1)-category of finite -groupoids. (We can drop the finiteness condition by making use of a higher universe.) This is canonically a pointed object , with points the terminal groupoid.
in or equivalently the object in the over-(∞,1)-topos
that is classified by under the (∞,1)-Grothendieck construction
we call the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos. In this case, by the adjunction hom-equivalence we have
This is essentially the basic statement around which Galois theory revolves.
The (∞,1)-sheaf (∞,1)-topos over a locally contractible space is locally -connected, and many authors identify local systems on such a topological space with representations of its fundamental groupoid.
where is the homotopy fiber of .
Local systems can also be considered in abelian contexts. One finds the following version of a local system
Regarded as a sheaf with values in abelian groups, such a linear local system serves as the coefficient for abelian sheaf cohomology. As discussed there, this is in degree nothing but the intrinsic cohomology of the -topos with coefficients in the Eilenberg-MacLane object .
On a connected topological space this is the same as a sheaf of sections of a finite-dimensional vector bundle equipped with flat connection on a bundle; and it also corresponds to the representations of the fundamental group in the typical stalk. On an analytic manifold or a variety, there is an equivalence between the category of non-singular coherent -modules and local systems on .
simplicial local system: within Sullivan’s (1977) theory of Infinitesimal computations in topology, he refers to ‘local systems’ several times. This seems to be simplicial in nature. This entry explores some of the uses of that notion based on Halperin’s lecture notes on minimal models
An early version of the definition of local system appears in
A definition appears as an exercise in
on page 58 :
A local system on a space is a covariant functor from the fundamental groupoid of to some category.
A blog exposition of some aspects of linear local system is developed here:
David Speyer, Three ways of looking at a local system
A clear-sighted description of locally constant -stacks / -local systems as sections of constant -stacks is in
See also at function field analogy.