The Cartesian closed structure on the category of diffeological spaces in particular gives, for any smooth manifolds (or even diffeological spaces) $M,M'$ a diffeological space $\mathrm{SmthMaps}(M,M')$ of smooth maps $M\rightarrow M'$.
I was wondering:
Given smooth manifolds $M,M'$ with $\mathrm{dim}M = m$, and $M$ oriented, for each compactly-supported $m$-form $\omega$ on $M'$, does the integration map $$ \begin{matrix} \mathrm{SmthMaps}(M,M') &\rightarrow & \mathbb{R} \\ f &\mapsto& \int_M f^* \omega \end{matrix} $$ lift to a morphism of diffeological spaces? (If the lift exists, it is unique, since the forgetful functor from diffeological spaces to topological spaces is faithful.)
More generally, does $\Omega^m_c(M')$ have a diffeology, with the above giving furthermore a morphism $\mathrm{SmthMaps}(M,M') \times \Omega^m_c(M')\rightarrow \mathbb{R} : (f,\omega) \mapsto \int_M f^* \omega$?
If $M'$ is an oriented Riemannian manifold with induced volume form $\omega$, and $\mathrm{dim}M' = \mathrm{dim}M+1$, and $M$ is an oriented smooth manifold, and we let $F \subset \mathrm{SmthMaps}(M,M')$ be the subset consisting only of smooth embeddings (or more generally immersions), then does $F$ "inherit" a diffeology, and does $$ \begin{matrix} F &\rightarrow & \mathbb{R} \\ f &\mapsto& \int_M f^* (\iota_{N_f}\omega) \end{matrix} $$ lift to a diffeological space morphism? Here $N_f$ denotes the unit normal vector field to $f$, its direction determined by the orientations on $M,M'$ and by the immersion $f$.