nLab evaluation fibration of mapping spaces

Idea

In the context of a mapping space, an important family of smooth maps are the evaluation maps. That is, for a sequentially compact Frölicher space $S$ and a manifold $M$, we pick a point $p \in S$ and consider the map $ev_p \colon C^\infty(S,M) \to M$ given by $ev_p(\alpha) = \alpha(p)$. This is smooth (see smooth maps of mapping spaces?). At $q \in M$, the fibre is the space of smooth maps $S \to M$ which take $p$ to $q$. This is again a smooth manifold (as shown in manifold structure of mapping spaces). Providing $M$ has enough diffeomorphisms, this is the projection of a fibre bundle. That is to say, the sequence:

$$C^\infty(S,p;M,q) \to C^\infty(S,M) \to M$$

is a fibre bundle.

The remark about “enough diffeomorphisms” is the key to proving this. To prove that this is a fibre bundle, we need to show that if $\beta \colon S \to M$ nearly takes $p$ to $q$ then we can deform $\beta$ to a map which takes $p$ to $q$ on the nose. Knowing that $M$ has the structure of a manifold, we can interpret the statement “$\beta(p)$ is close to $q$” as meaning that we have fixed a chart near $q$ and require that $\beta(p)$ be in the codomain of that chart. We therefore have a good choice of deformation for $\beta(p)$ itself: deform it along the “straight line” from $q$ to $\beta(p)$ as defined by the chart. The problem with this is that it only tells us what to do with $\beta(p)$, not with the rest of $\beta$. So we need to drag the rest of $\beta$ along with $\beta(p)$. This is where the diffeomorphisms come in: instead of moving $\beta(p)$ along a path, we deform the entire manifold using a diffeomorphism so that $\beta(p)$ is taken to $q$. We do this using the methods of propagating flows.