I have come accross this brief description of Charles Ehresmann's life given by his wife: http://www.cs.le.ac.uk/people/ah83/cat-myths/myth0002.html
I quote the part from the text relevant to my question:
Indeed, groupoids intervene in fibre bundle theory in two different ways:
A. Actions of a topological groupoid. Denote by E a fibre bundle. The isomorphisms from fibre to fibre form a groupoid, which is equipped with a topology compatible with the maps domain, codomain, composition and inversion. This gives a topological groupoid (in the sense: internal groupoid in the category Top of topological spaces), which acts continuously on the topological space E. This topological groupoid G satisfies the axiom:
(LT) For each object, say x, of G (identified with a point of the base B of E ), there exists a local section s: U -> G of the codomain map on a neighborhood U of x such that s(y): x -> y for each y in U.
Conversely, to a topological groupoid satisfying (LT) (called a locally trivial groupoid) naturally corresponds a principal fibre bundle, and to its actions, the associated fibre bundles. This defines an equivalence from the category of fibre bundles to the category of actions of locally trivial groupoids. In this setting, connections, prolongations of manifolds,... are very easily defined.
More generally, the jets between all germs of manifolds form a (big) differentiable category (i.e., internal category in the category Diff of differentiable maps) and Charles described Differential Geometry as the study of this category and of the actions of its subcategories.
This 'categorical' point of view is indicated in a series of very concise papers from 1958 to 1969 (CE I). Charles always thought of writing a book on this subject, and he regretted to have spent so much time in Bourbaki's team in the forties instead of developing his own ideas.
I find this viewpoint of Differential Geometry as expressed in the quote above very interesting. As stated above, the approach to DG, starting from the category equivalence between actions of locally trivial groupoids and fiber bundles, has never been written up in textbook form by Ehresmann himself and is only to be found in "concise" papers. I wonder if this very natural, transparent and easy approach to differential geometry has been developed in the meanwhile by other authors. I think it would be especially interesting for students as a first introduction to Differential Geometry. Are there papers or textbooks in which connections and other differential geometric structures are developed just starting from the notion of locally trivial categories / groupoids and other structured categories internal to the category of smooth maps? The category equivalence itself is easy to prove, but I find the remark in the quote that the other structures are "very easily defined" - within this framework - not to be really entirely true, at least judging by my own first attempts. I have not yet tried to read Ehresmann's voluminous collected works (which are available online: http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/C.E_Works.htm).
