Proposition 4

If the space B is paracompact and if the space $E'$ is homotopic to a point, the sheaf on B of morphisms of principal fibre spaces with group G from E into $E'$ is a soft sheaf.

There exists an open covering $(U_j)_{j\in J}$ of B such that, for every $j\in J$, the fibre space $E_{U_j}$ is trivialisable. Since the space B is paracompact, we may suppose that the covering $(U_j)_{j\in J}$ is locally finite (TG, IX, p. 49) and choose a covering $(A_j)_{j\in J}$ of B where, for every $j\in J$, the set $A_j$ is closed in B and contained in $U_j$ (TG, IX, p. 49, prop. 4 and p. 48, cor. 1).

By I, p. 65, cor. 2 of prop. 6, it is enough, in order to prove the proposition, to establish the following assertion: let U be an open subset of B such that the principal fibre space $(E_U, p_U)$ is trivialisable, let A be a closed subset of B contained in U, let V be an open neighbourhood of A contained in U, and let $f$ be an element of $\mathscr{F}(V)$; then there exists an open neighbourhood W of A in V and an element $f'$ of $\mathscr{F}(U)$ such that $r_{WU}(f') =r_{WV}(f)$. Let us prove this assertion.

Let W be an open subset of B such that $A\subset W\subset \overline{W}\subset V$. Let $s: U\rightarrow E_U$ be a section of $(E_U, p_U)$. Apply Corollary 3 (IV, p. 438) to the space B, the closed subset $\overline{W}$ and the neighbourhood V of $\overline{W}$, and to the mapping $g=f\circ (s|_V)$ of $E_V$ into $E'$. There then exists a continuous mapping $\widetilde{g}: B\rightarrow E'$ which coincides with $g$ on $\overline{W}$. Put $h=\widetilde{g}|_U$. We have $h|_W=g|_W=f\circ (s|_W)$.

The mapping $H: U\times G\rightarrow E'$ defined by $H(x, g) =h(x)\cdot g$ for $(x, g)\in U\times G$ is continuous and compatible with the operations of G in $U\times G$ and $E'$. Put $f'= H\circ s^{-1}$; this is an element of $\mathscr{F}(U)$. For every $x\in W$, the mappings $f$ and $f'$ coincide at the point $s(x)$, hence at every point of $\overset{-1}{p}(x)$, since they are morphisms of principal fibre spaces. The proposition follows.