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Todd Trimble
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You shouldn't expect vector bundles to form an abelian category because they are projective module objects over local ring objects in a category of sheaves over a space. In other words, there is really no more reason to expect this of vector bundles than you would of a category of finitely generated projective modules over a ring.

Todd Trimble
  • 53.3k
  • 6
  • 205
  • 322