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Manifolds And Local Structures: A General Theory (Hardcover)
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Manifolds And Local Structures: A General Theory (Hardcover)
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Total price: R3,483
Discovery Miles: 34 830
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Local structures, like differentiable manifolds, fibre bundles,
vector bundles and foliations, can be obtained by gluing together a
family of suitable 'elementary spaces', by means of partial
homeomorphisms that fix the gluing conditions and form a sort of
'intrinsic atlas', instead of the more usual system of charts
living in an external framework.An 'intrinsic manifold' is defined
here as such an atlas, in a suitable category of elementary spaces:
open euclidean spaces, or trivial bundles, or trivial vector
bundles, and so on.This uniform approach allows us to move from one
basis to another: for instance, the elementary tangent bundle of an
open Euclidean space is automatically extended to the tangent
bundle of any differentiable manifold. The same holds for tensor
calculus.Technically, the goal of this book is to treat these
structures as 'symmetric enriched categories' over a suitable
basis, generally an ordered category of partial mappings.This
approach to gluing structures is related to Ehresmann's one, based
on inductive pseudogroups and inductive categories. A second source
was the theory of enriched categories and Lawvere's unusual view of
interesting mathematical structures as categories enriched over a
suitable basis.
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