nLab tubular neighbourhood of a mapping space

Idea

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.