How does a section of a stack give a sheaf?

Question

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a locally constant sheaf, and on up the categorical ladder.

I know that one motivating example of a stack is the functor on a site which assigns to each object the category of sheaves over that object. For this stack, by construction each section is a sheaf. But how do we see that in general? If $F$ is a stack and $c\in F(U)$ I know that $V\mapsto \hom(c|V,c|V)$ is a sheaf, so that’s one way, but I don’t think that is what is meant here.

Answer 1

See the $n$lab article on locally constant sheaves:

A locally constant sheaf $A$ over a topological space is a sheaf of sections of a covering space of $X$: there is a cover of $X$ by open subsets $\{U_i\}$ such that the restrictions $A|_{U_i}$ are constant sheaves.

More elegantly said: locally constant sheaves are the sections of constant stacks:

Let $C = \mathrm{Core(FinSet)}\in $$\mathrm{Grpd}$ be the core of the category $\mathrm{FinSet}$ of finite set, let $\mathrm{const}_C : \mathrm{Op}(X)^{op} \to \mathrm{Grpd}$ the presheaf constant on $C$, i.e. the functor on the opposite category of the category of open subsets of $X$ that sends everything to (the identity on) $C$. Then the constant stack on $C$ is the stackification $\bar{\mathrm{const}_C}: \mathrm{Op}(X)^{op} \to \mathrm{Grpd}$.

Write then $X$ for the space $X$ regarded as a sheaf or trivial covering space over itself, i.e. the terminal object $X$ in sheaves and hence in stacks over $X$. Then by definition of stackification morphisms

$$ X \to \bar{\mathrm{const}_C} $$

are represented by

• an open cover $\{U_i\}$ of $X$;

• over each $U_i$ a choice $F_i \in C$ of object in $C$, hence a finite set in $C$;

• over each double overlap $U_{i j} = U_i \cap U_j$ an morphism $g_{i j} : F_i|_{I_{i j}} \stackrel{\simeq}{\to} F_j|_{U_{i j}}$, hence a bijection of finite sets;

• such that on triple overlaps we have $g_{i k}|_{U_{i j k}}= g_{j k}|_{U_{i j k}}\circ g_{i j}|_{U_{i j k}}$.

Such data clearly is the local data for a covering space over $X$ with typical fiber any of the $F_i$.