I am not sure how Ehresmann dealt with it, but generally the French replaced conditions on limits by conditions on the representable functors. So to say that c is a category object in some category C is to say that the functor Hom(c,-): C --> Set factors through Cat. More precisely, to give a category structure on c is to prescribe a factorization of Hom(C,-) through Cat. It is easy to see (using Yoneda) that in the presence of pullbacks the two definitions agree. They were quite consistent about this. They would not have said that pullbacks exist (the existence in their sense being automatic since Set-valued functor categories have pullbacks) but rather that pullbacks were representable. In other words to them the pullback existed at the level of hom functors and if the pullback (functor) was representable then it exists in our sense. I have always thought that this usage went back to Grothendieck, but perhaps it was originally Ehresmann's. I have the impression that Ehresmann got interested in categories before Grothendieck. This process is more or less the same as completing the category using the Yoneda embedding. Michael

Andree C. Ehresmann wrote:

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Already in the fifties, Charles Ehresmann has introduced and extensively studied categories internal to Diff (which he called "categories differentiables") in his development of Differential Geometry.

I'm sorry to have not yet gotten ahold of the references you provided - I got distracted by other matters - but if you will forgive my laziness, I'll ask another question:

Could someone please explain how Charles Ehresmann dealt with the fact that Diff does not have all pullbacks? In the definition of internal category we need a pullback to exist: the "object of composable pairs of morphisms", C_1 x_{C_0} C_1. These days, people working on Lie groupoids usually impose some requirements on the source and target maps s,t: C_1 -> C_0 that ensure that this pullback exists. Personally I lean towards a simpler approach, which is simply to demand that the pullback exists. This makes it harder to prove certain theorems, but has a certain charm. I'm wondering what Charles Ehresmann thought about this issue.

Best,

The category Diff has pull-backs of transversal maps, and furthermore these are the only "correct" pull-backs. A natural (and "necessary") requirement for the existence of a given pull-back is transversality, which has a lot of charm. In the well adapted models of SDG all pull-backs of differential manifolds exists, but they will be objects in the topos (actually C-infinity Schemes) which are not differential manifolds unless the pull-back is transversal. The work of Charles Ehresmann and Andre Weil is very much related to this fact. Best, edubuc.