Sheaf Theory

Note that a locally constant sheaf is nothing more than a covering space. The definitions of constant and locally constant sheaf are sometimes given in a confusing and misleading way.  Correct category theoretic notions should be given up to isomorphism. Given a space $U$ and an object $A$ there is a presheaf $\mathcal{F'}^A_U$ which yields $A$ on every open set and whose restriction map is $id_A$.  This generates a sheaf $\mathcal{F}^A_U$.  We say that a sheaf $F$ on a space $X$ is a locally constant sheaf if $X$ has a cover such that on each component $U$ of the cover it is isomorphic to a sheaf of the form $\mathcal{F}^A_U$. Monodromy is an abstract way at looking at the liftings of a closed path to a covering space.

Exercise: A locally constant sheaf on $[0,1]$ is a constant sheaf, that is, is isomorphic to a sheaf of the form $\mathcal{F}^A_{[0,1]}$. More generally: a locally constant sheaf on a simply connected space is a constant sheaf.

For $F$ an abelian sheaf,  we define $sup \, F = \{ x \in X: F_x \neq 0 \}$. Show that $sup\,F$ need not be closed. Consider the sheaf $F$ and fix a point $p$ in a Hausdorff space $X$. Then define $F(U) = 0$ for $p \in U$ and $F(U) = \mathbb{Z}_2$ otherwise for some abelian group $A \neq 0$.  Then clearly $sup\, F$ is not closed.   Do not confuse the support of an abelian sheaf with the support of a section $s$ over $U$, the points of $U$ in which $s_x \neq 0$. $sup\, s$ is indeed closed and this follows essentially from the algebraic constraint on restriction map homomorphisms where the image of $0$ must be $0$. Thus the stalk of a zero section must be zero.

Exercise: Show that sections of locally constant abelian sheaf $F$ have closed support (hence unique extensions).  Hint: suppose that $s_x \neq 0$. Then there is an open set $V(x)$ and a section $s_V$ which represents $s_x$. But since $F$ is locally constant we can assume, by taking intersections, that $F_{V(x)}$ is isomorphic to a constant sheaf for a given group $A$. But then clearly we must have $s_y  = s_x$ for all $y \in V(x)$ and hence the complement of $sup\, s$ is open. 

Thus (locally) constant abelian sheaves are abstract versions of many important properties of holomorphic functions.

A locally constant sheaf on a simply connected topological space is a constant sheaf. This follows from homotopy invariance of unique extensions. This is perhaps the most fundamental result about locally constant sheaves; more generally this would be the result concerning the representation of a the fundamental group based at $x$ by automorphisms of the stalk at $x$.