sheafification definition?

Question

I came across two different definitions of sheafification and I'm not sure how they are equivalent. One of them is here: About the sheafification

Another one is from Tennison's sheaf theory: Given a presheaf $F$ over $X,$ we construct the sheaf space $LF:=\sqcup_{x\in X}\mathscr{F_{x}}$ which is a disjoint union of stalks and the continuous map is the natural projection $p:LF\rightarrow X $. The open sets are $s[U]=${ $s_{x}\in LF: x\in U$}. Then the sheafification is $\Gamma LF,$ where we construct the sheaf of sections on the sheaf space $LF.$ $\Gamma LF(U)=${$\text{continuous maps} \ \ \sigma :U\rightarrow LF \ \ \text{s.t.} \ \ p(\sigma)=id_{U} $}. So for example, the map $\hat{s}:U\rightarrow LF(U)$ where $x \mapsto s_{x}\in \mathscr{F_{x}}$ would be in $\Gamma LF(U)$. Can someone explain how they are equivalent?

Answer 1

Here's a concrete way to go back and forth between the descriptions.

A continuous function $\sigma:U \rightarrow LF$ satisfying $p\circ \sigma = \mathbb 1$ corresponds to the element $(\sigma (x)) \in \prod_{x\in U} F_x$. To show that $(\sigma(x)) \in F^{\#}(U)$, let $a \in U$ and let $U'\subset LF$ be so that $\sigma(a) \in U'$. Since basic open sets of $LF$ are of the form $s[V]$ for $s \in F(V)$, we may assume $U' = s[V]$. By continuity of $\sigma$, there exists a neighborhood $W$ of $a$ so that $\sigma(W) \subset s[V]$. But this implies $\sigma(b)=s_b$ for all $b \in W$. Since this holds for all $a \in U$, $(\sigma(a))$ satisfies $(*)$.

For the other direction, take $(s_x)\in F^{\#}(U)$ and let $\sigma$ be the section of $p$ over $U$ defined by $x \mapsto s_x$. We must verify $\sigma$ is continuous. For any $x \in U$ and any basic open set $t[V]\subset LF$ with $s_x \in t[V]$, $(*)$ implies that there exists $W \subset V$ so that $s_y = t_y$ for all $y \in W$. This shows that $x\in\sigma(W) \subset t[V]$, and hence $\sigma$ is continuous.

To completely nail down the equivalence of the definitions, you'll want to check that these identifications of $\Gamma LF(U)$ and $F^{\#}(U)$ for $U \subset X$ open establish an isomorphism of sheaves (i.e. that the maps $\Gamma LF(U) \rightarrow F^{\#}(U)$ are morphisms and the appropriate diagrams involving restrictions commute).