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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.

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I would guess that we are supposed to think of a bundle as a functor from the manifold-as-category to the category of vector spaces (if you want vector bundles) or principal G-spaces (for principal bundles). But I don't know if that is what Bott had in mind here. – Charles Rezk Jul 13 '10 at 18:19
Are you sure that all the morphisms in your category are invertible? IF you want to encode a specific manifold as a category, then I would guess that you want to include morphisms that correspond to inclusions of opens subsets of $M$ into each other. My feeling is that the classifying space of this category would be equivalent to $M$. I could be totally off though. – Gregory Arone Jul 13 '10 at 18:20
I also wonder if we are supposed to think of a discrete category or a topological one. If each point in each open set in the chart corresponds to an object, then the set of objects should have a natural topology. Do we take it into account? – Gregory Arone Jul 13 '10 at 18:30
@Kevin, Gregory: I don't think Open(X) is the category intended in the quote. In particular, the manifoldhood should be captured in the category. @Charles: Nice! Where is the equivalence? – Justin Curry Jul 13 '10 at 20:26
I believe I have found the reference: Bott's Mexico lectures "Characteristic Classes and Foliations" pg. 31-35 defines the topological category associated to an open cover of the manifold and uses continuous functors to prove the classic 1-1 correspondence between rank k vector bundles and [X,BGl_k] – Justin Curry Oct 30 '10 at 19:02

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