Could not include topos theory - contents
More elegantly said: locally constant sheaves are the sections of constant stacks:
Let Grpd be the core of the category FinSet of finite set, let the presheaf constant on , i.e. the functor on the opposite category of the category of open subsets of that sends everything to (the identity on) . Then the constant stack on is the stackification .
are represented by
an open cover of ;
over each a choice of object in , hence a finite set in ;
over each double overlap an morphism , hence a bijection of finite sets;
such that on triple overlaps we have .
Such data clearly is the local data for a covering space over with typical fiber any of the .
Without further assumption on we have the following definition.
for some Set.
An object which is locally constant and -split for some is called locally constant.
If is a locally connected topos there is another characterization of locally constant sheaves.
From the discussion at locally connected topos we have that
In this case the above definition is equivalent to the following one.
This means that on each connected component of a locally constant sheaf is the -associated bundle to an -principal bundle induced by the canonical permutation representation of the automorphism group on .
Locally constant sheaves are sheaves of sections of covering spaces.
This is the content of Galois theory.
In sufficiently highly locally connected cases, we have:
a locally constant sheaf is a section of a constant stack;
a locally constant stack is a section of (… and so on…)
A locally constant sheaf / -stack is also called a local system.
The definition of locally constant sheaf originates in the notion of covering projection
Lecture notes are in
The topos-theoretic definition is reproduced in the context of a discussion of the notion of Galois topos as definition 5.1.1 in
and definition 2.2 in
or as definition 1 in
Discussion of the notions of locally constant sheaves is at