In the CRM Proceedings & Lecture Notes Volume 50 "A Celebration of the Mathematical Legacy of Raoul Bott" Herbert Shulman writes (p. 48):
"[Bott] taught many of us to think functorially, like thinking of a group as a category with one object and a morphism for each element, a manifold as a category of pairs (open set, point in the open set), and a bundle as an equivalence class of functors. When someone asked him who invented functors, he said 'functors are prehistoric!'. He talked about 'folk' theorems... theorems everyone knew, but were never written down."
As I've highlighted in bold the latter two everyday examples of categories seem less well-known to me.
The manifold-as-category seems clear enough, the open set is meant to be a coordinate neighborhood of the point. A morphism between two objects is just the transition maps between coordinate neighborhoods. What seems more natural to me would be to associate a category to a given atlas on a manifold. This makes me wonder what the classifying space of these categories looks like. How do they behave as you pass to the maximal atlas? Does anyone know? Since the transition maps (the morphisms) are homeomorphisms/ diffeomorphisms/ biholomorphisms (depending on what type manifold we have) the morphisms in this category are all invertible and so our category is a groupoid. For a connected manifold the classifying space should be a $BG$ or $K(G,1)$. What is $G$?
Finally, Can someone explain why a bundle is an equivalence class of functors? The functor part seems sort of clear, because the projection map is an open map. More explanation would be nice.
