I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background:

Diffeological spaces are a Cartesian-closed, complete, and cocomplete category containing all infinite dimensional manifolds, and in fact even form a quasi-topos.

Diffeological spaces, concisely, are nothing more than concrete sheaves on the site of Cartesian manifolds (manifolds of the form $\mathbb{R}^n$):

http://ncatlab.org/nlab/show/concrete+sheaf

However, the category of ALL sheaves on Cartesian manifolds, categorically is even nicer, since it is a genuine topos.

$\textbf{My question is:}$ What can you do with diffeological spaces that you cannot do with general sheaves? Or, more generally, what are the advantages of diffeological spaces over general sheaves?

All of the generalizations of differential geometry concepts to diffeological spaces I have seen so far, actually carry over to genuine topos of sheaves (though sometimes with a little more work).

I'm aware that you gain the ability to work with a set with extra structure and talk about its points etc, but, what does this gain you? It seems that you can always use Grothendieck's functor of points approach instead.

Is it that limits and colimits are more like their counterparts for manifolds?

smoothcotangent complex. – Harry Gindi Dec 7 '10 at 21:46