Manifolds And Local Structures: A General Theory
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A01=Marco Grandis
Author_Marco Grandis
Category=PBMH
Directed Space
Enriched Category
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Fibre Bundle
Inverse Category
Inverse Semigroup
Manifold
Product details
- ISBN 9789811233999
- Publication Date: 10 Mar 2021
- Publisher: World Scientific Publishing Co Pte Ltd
- Publication City/Country: SG
- Product Form: Hardback
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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.
