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I have difficulties with the notions of set and category theory.

  1. I don't understand the difference between: a set, a collection and a universe! Although I understand why the collection of all sets is not a set (because of a well known paradox), but what about the collection of all groups (resp.abelian groups), all smooth atlases on a given manifold...I guess this type of collection is not a set, but how to prove it?

  2. Natural transformations is an important notion to ditinguish natural notions: isomorphism between a vector space and its bidual, the symplectic structure of the cotangent bundle. But what about the vector space structure of an affine space? the linear structure of the tangent space of a manifold, how to prove these are intrinsic notions?

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    $\begingroup$ Your two questions are very different and should be asked separately. Some brief comments: (1) You should specify the axioms of set theory you are considering. (2) The vector space structure on an affine space is not natural. $\endgroup$ – Zhen Lin Nov 28 '14 at 12:21

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