Exercise 4, § 3

Let X be a topological space and let $\mathscr{F}$ be a presheaf on X.

For every open set U of X, denote by $S_U(\mathscr{F})$ the set of pairs $(\mathscr{V},(s_V)_{V\in\mathscr{V}})$, satisfying the following properties:

(i) The elements of $\mathscr{V}$ are open subsets of X and one has U = $\bigcup_{V\in\mathscr{V}}V$ ;

(ii) For every $V\in \mathscr{V}$, one has $s_V\in \mathscr{F}(V)$;

(iii) For every pair $(V,W)$ of elements of $\mathscr{V}$, one has $s_V|(V\cap W) =$ $s_W|(V\cap W)$.

The presheaf $\mathscr{F}$ is said to be separated if, for every element $(\mathscr{V},(s_V))$ of $S_U(\mathscr{F})$, there exists at most one element $s\in \mathscr{F}(U)$ such that $s|V =s_V$ for every $V\in \mathscr{V}$.

a) Prove that a sheaf is a separated presheaf.

b) For every open set U of X, equip the set $S_U(\mathscr{F})$ with the least fine equivalence relation for which, given two families $s=$ $(\mathscr{V},(s_V)_{V\in\mathscr{V}})$ and $t= (\mathscr{W},(t_W)_{W\in\mathscr{W}})$ of $S_U(\mathscr{F})$, one has $R(s, t)$ if, for every open set $V\in \mathscr{V}$, there exists an open set $W\in \mathscr{W}$ such that $W\subset V$ and $s_V|_W=t_W$. We denote by $\mathscr{F}^+(U)$ the set of equivalence classes.

Let U and V be open sets of X such that $V\subset U$. Prove that there exists a unique mapping $\sigma_{UV}:\mathscr{F}^+(U)\rightarrow \mathscr{F}^+(V)$ such that the image of the class of an element $(\mathscr{W},(t_W)_{W\in\mathscr{W}})$ of $S_U(\mathscr{F})$ is the class of the element $(\mathscr{W}_V,(t_W|_{W\cap V}))$, where $\mathscr{W}_V$ denotes the family of the sets $W\cap V$, for $W\in \mathscr{W}$.

c) Prove that $(\mathscr{F}^+(U), \sigma_{UV})$ is a separated presheaf.

d) Prove that there exists a unique morphism of presheaves $i_{\mathscr{F}}:\mathscr{F}\rightarrow \mathscr{F}^+$ such that, for every open set U of X and every element $s\in \mathscr{F}(U)$, the section $i_{\mathscr{F}}(s)$ is the class of the element $(\{U\},(s_U)_{U\in \{U\}})$ of $S_U(\mathscr{F})$.

e) Prove that for every separated presheaf $\mathscr{G}$ on X and every morphism $f:\mathscr{F}\rightarrow \mathscr{G}$ of presheaves, there exists a unique morphism of presheaves $f^+:\mathscr{F}^+\rightarrow \mathscr{G}$ such that $f=f^+\circ i_{\mathscr{F}}$.

f) Prove that $i_{\mathscr{F}}$ is an isomorphism if and only if $\mathscr{F}$ is a sheaf.

g) If $\mathscr{F}$ is a separated presheaf, prove that the presheaf $\mathscr{F}^+$ is a sheaf.

h) Prove that for every sheaf $\mathscr{G}$ and every morphism $f:\mathscr{F}\rightarrow \mathscr{G}$ of presheaves, there exists a unique morphism $f^{++}:\mathscr{F}^{++}\rightarrow \mathscr{G}$ such that $f=f^{++}\circ i_{\mathscr{F}^+}\circ i_{\mathscr{F}}$. Deduce that the sheaf $\mathscr{F}^{++}$ is isomorphic to the sheaf associated to $\mathscr{F}$.

i) Let A be a set of cardinal $\geqslant 2$ and let $\mathscr{F}$ be the presheaf on X given by $\mathscr{F}(U) = A$ for every open set U of X, the restriction mappings being the identity mapping of A. Prove that $\mathscr{F}$ is not a sheaf. Calculate $\mathscr{F}^+(U)$ for every open set U of X. Verify in particular that $\mathscr{F}^+$ is not a sheaf if there exists an open set of X which is not connected.