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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?

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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. –  Andrew Stacey Jul 19 '10 at 8:03
    
mathnet.ru/php/… –  David Carchedi Jul 19 '10 at 23:24
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