Proposition 12

Suppose that the space B is locally connected and that the B-spaces E and $E'$ are coverings. Then the sheaf $\mathscr{M}$or$_B(E; E')$ is locally constant and, for every $b\in B$, the mapping $\theta_b$ is a bijection of its stalk at $b$ onto the set of mappings of $E_b$ into $E'_b$.

Analogously, the sheaf $\mathscr{I}$som$_B(E; E')$ is locally constant and, for every point $b$ of B, the mapping $\theta '_b$ is a bijection of its stalk at $b$ onto the set of bijections of $E_b$ onto $E'_b$.

We shall denote by $\mathscr{M}=\mathscr{M}$or$_B(E; E')$ and $\mathscr{I}=\mathscr{I}$som$_B(E; E')$. The assertions to be proved are of a local nature on B; we may therefore suppose that $E = B\times F$ and $E'= B\times F'$ are trivial coverings, where F and $F'$ are discrete topological spaces.

Let us first prove that, for every connected open set U of B and every point $b$ of U, the mapping $\theta_{b,U}$ is a bijection. Let $f: U\times F\rightarrow$ $U\times F'$ be a morphism of U-spaces; then $\theta_{b,U}(f)$ is the mapping $y\mapsto$ pr$_2(f(b, y))$ of F into $F'$. For every $y\in F$, the mapping $x\mapsto$ pr$_2(f(x, y))$ is a continuous mapping of the connected space U into the discrete space $F'$, and is therefore constant, equal to pr$_2(f(b, y))$. Thus $f(x, y) = (x, \theta_{b,U}(f)(y))$. It follows that $\theta_{b,U}$ is a bijection; its inverse bijection associates with every mapping $\varphi : F\rightarrow F'$ the U-morphism of $U\times F$ into $U\times F'$ given by $(x, y)\mapsto (x, \varphi (y))$.

By hypothesis, every point $b\in B$ admits a basis of connected open neighbourhoods; the mapping $\theta_b$ is therefore a bijection. The mappings $\theta^{-1}_{b,U}$, for U a nonempty connected open set of B and $b$ any point of $b$, define a morphism of presheaves (relative to the basis of connected open sets of B) of the constant sheaf of stalk-type ${F'}^F$ into the sheaf $\mathscr{M}$. From what precedes, this morphism induces a bijection on the stalks; it is therefore an isomorphism.

The assertions relative to the sheaf $\mathscr{I}$ are proved analogously.