There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fullyfaithfully into diffeological spaces. (Diffeological spaces are concrete sheaves on the site of (Euclidean) manifolds http://ncatlab.org/nlab/show/diffeological+space). Diffeological spaces are a complete and cocomplete quasitopos, so, in particular are Cartesianclosed. Froelicher spaces are also complete, cocomplete and Cartesianclosed: http://ncatlab.org/nlab/show/Froelicher+space#hausdorff. Do Frechet manifolds also embedd fullyfaithfully into Froelicher spaces? If so, if we "cut out a submanifold" of a Frechet space, does it correspond to the subFroelicher space when embedded? How about for diffeological spaces?

2$\begingroup$ Can you provide a reference to the Losik result? In particular, I'd want to know what definition of "Frechet manifold" is used since the term "smooth" is ambiguous for Frechet spaces. $\endgroup$– Andrew StaceyJul 19 '10 at 8:03

1$\begingroup$ mathnet.ru/php/… $\endgroup$– David CarchediJul 19 '10 at 23:24
In Jordan Watt's thesis it is shown that Frölicher spaces form a full subcategory of diffeological spaces, so if the functor from Fréchet manifolds to diffeological spaces is fully faithful (as seems to be the case by the result of Losik), and if this functor factors through Frölicher spaces (as shown in KrieglMichor), then Fréchet manifolds embed fully faithfully in Frölicher spaces.
The category of Frechet manifolds and smooth mappings (note that $C^\infty_c$ equals smooth in the setting of convenient calculus here) embeds faithfully into the category of Frölicher spaces: The Frölicherstructure is generated by the smooth curves in the Frechet manifold.
The embedding might be not full: Namely, the question is whether there are "enough" smooth functions on the manifold (which might not be smoothly paracompact); if not, then we might get more smooth curves in the Frölicher space; thus $\mathbb R\to M$ has more morphisms in the category of Frölicher spaces that in the category of Frechet manifolds. We (Andreas Kriegl and me) do not know an example of this. The nearest is in 27.6 in
By lemma 27.5 there, smoothly regular Frechet manifolds embed fully into Frölicher spaces.
The answer to the second question is YES: A Frechet submanifold of a linear Frechet space is uniquely a Frölicher space: Restrictions of linear functionals on the Frechet space suffice to recognize smooth curves.

$\begingroup$ $C^\infty_c$ as in Keller's classification of smoothness (that is, smooth in the sense of Michal and Bastiani)? $\endgroup$ May 26 '16 at 23:43