TITLE: Generalized smooth spaces and infinite dimensional manifolds
QUESTION [11 upvotes]: There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fully-faithfully 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 Cartesian-closed. Froelicher spaces are also complete, cocomplete and Cartesian-closed: http://ncatlab.org/nlab/show/Froelicher+space#hausdorff. Do Frechet manifolds also embedd fully-faithfully into Froelicher spaces? If so, if we "cut out a submanifold" of a Frechet space, does it correspond to the sub-Froelicher space when embedded? How about for diffeological spaces?

REPLY [6 votes]: 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 Kriegl-Michor), then Fréchet manifolds embed fully faithfully in Frölicher spaces.