In Hirsch's book, there is a wonderful theorem 2.11:
where a structure functor is simply a presheaf, and continuous means it is a sheaf (it has the gluing property). Nontrivial means there is at least one local section.
Locally extendable means, in his words:
As an example, if $X$ is a $C^r$ manifold, and if $\mathfrak{F}(Y), Y\subseteq X$ is the set of compatible $C^s$ differential structures on $Y$, $1\leq r <s \leq \infty$, then $(\mathfrak{F}, \mathfrak{U})$ ($\mathfrak{U}$ the open sets of $X$) form a nontrivial, continuous, locally extendable structure functor.
My question is: how can we use this technique to the problem of Finding a Retraction for a Collar Neighborhood? I copy the relevant excerpt:
