hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
The result of evaluation fibration of mapping spaces extends to more general evaluation maps between mapping spaces. One way to interpret that result is that the inclusion $C^\infty(S,p;M,q) \to C^\infty(S,M)$ has a tubular neighbourhood. Providing $M$ has enough diffeomorphisms, this is true of more general inclusions where they are defined by “coincidences”. That is to say, if $P$ is a condition on maps $S \to M$ that prescribes where certain points “coincide”, then the submanifold of $C^\infty(S,M)$ of smooth maps satisfying this condition will have a tubular neighbourhood in the manifold of all smooth maps.