hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
As shown in manifolds of mapping spaces, the space of smooth maps from a sequentially compact Frölicher (or diffeological or Chen) space is again a smooth manifold. As such, its topology can be determined by considering how the charts are glued together.
In some cases, there is an easy method available to study the topology. If the target manifold, say $M$, embeds as a submanifold of some convenient vector space, say $V$, then $C^\infty(S,M)$ embeds as a submanifold of $C^\infty(S,V)$. If the original embedding is as a closed submanifold, so is the embedding of mapping spaces. This means that it is possible to propagate topological results down from $C^\infty(S,V)$ to $C^\infty(S,M)$. Furthermore, if $M$ is a deformation retract of a neighbourhood of its image in $V$, then so is $C^\infty(S,M)$ in $C^\infty(S,V)$.
Let $g \colon M \to N$ be an embedding of smooth manifolds. Let $S$ be a sequentially compact Frölicher space. Then $C^\infty(S,g) \colon C^\infty(S,M) \to C^\infty(S,N)$ is an embedding of smooth manifolds.