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?
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