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I created Michal-Bastiani smooth map, and linked to it from diffeological space, Andree Ehresmann and locally convex topological vector space.
I’ve added an example to Michal-Bastiani smooth map that shows that $lctvs_{MB} \to lctvs_{convenient}$ is not even full on isomorphisms, supplied in an answer to by MO question.
This is kinda interesting in that work on infinite-dimensional manifolds by Gloeckner and his circle is not quite compatible (in general) with treating generalised smooth spaces via sheaves on $Cart$.
One thing I’d like to know is if we know a lctvs is such that it is faithfully represented (in some way: detecting smooth maps out or in by behaviour on smooth probes or co-probes from $Cart$) in $DiffeologicalSpace$, then it is from some particular class of spaces. This includes all Frechet spaces, but I don’t know this class is bigger, or if we can even get an independent set of conditions…
work on infinite-dimensional manifolds by Gloeckner and his circle is not quite compatible (in general) with treating generalised smooth spaces via sheaves on Cart.
As long as the functor is faithful there may be little harm: it means that everything that Gloeckner and his circle do holds true also in $Sh(CartSp)$, only that in the latter context more holds true, even (more maps are available).
it means that everything that Gloeckner and his circle do holds true also in Sh(CartSp)
Sure :-)
I guess another question one might ask (and this is just following an idle thread) is whether there are MB-smooth bijections whose inverse is only conveniently smooth.
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