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Structures on categories
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3—manifold adjoint admits atlas b—boundary b—completion Brickell and Clark Cauchy completion chart closed timelike curves commuting components construction continuous map Corollary defined determined diagram diffeomorphism differential Dodson and Poston Ellis 43 equivalence Example exists fibre bundle fibre bundle P,G,M finite functor geometry Geroch given Hawking and Ellis Hence Hirsch 49 holonomy group homeomorphic horizontal lift isomorphism Kobayashi and Nomizu left limit Lie algebra Lie group line element field Lorentz group Lorentz structure map f metric tensor metric tensor field morphisms neighbourhood Nomizu 57 non—compact nowhere—zero vector field orientable orthonormal paracompact parallel transport parallelizable partition of unity physical principal fibre bundle projective limit Proof Riemannian structure right limit second countable smooth spacelike spacetime M,g spinor structure structure group submanifold subset tangent vector tensor field theory time—orientable topological space topology trivial unique vector bundle vector field vector space