Proposition 9

Let B be a topological space and let $(E, p)$ be a covering of B. Let T be a simply connected, locally connected and locally compact topological space. Let us denote by $\widetilde{p}:\mathscr{C}_c(T; E)\rightarrow \mathscr{C}_c(T; B)$ the mapping $g\mapsto p\circ g$. Let $t$ be a point of T. Let us denote by $e_E:\mathscr{C}_c(T; E)\rightarrow E$ the mapping which to $g\in \mathscr{C}_c(T; E)$ associates $g(t)$, and by $e_B:\mathscr{C}_c(T; B)\rightarrow B$ the mapping which to $f\in \mathscr{C}_c(T; B)$ associates $f(t)$.

The square

$\mathscr{C}_c(T; E)^{e_E}$ E

$\widetilde{p}p$

$\mathscr{C}_c(T; B)^{e_B}$ B is cartesian.

Let us first prove a lemma.