Manifolds and Local Structures

Manifolds and Local Structures

A General Theory

Marco Grandis


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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.Contents:

  • Preface
  • Introduction
  • Order, Semigroups and Categories
  • Inverse Categories and Topological Background
  • Cohesive Categories and Manifolds
  • From Topological Manifolds to Fibre Bundles
  • Complements on Category Theory
  • Enriched Categories and Cauchy Completion
  • Solutions of the Exercises
  • References
  • Index

Readership: Graduate students, PhD students and researchers in mathematics, physics & computer science.Manifold;Fibre Bundle;Enriched Category;Inverse Semigroup;Inverse Category;Directed Space0Key Features:
  • This textbook presents a unified approach to local structures, a wide class of mathematical structures ranging from differentiable manifolds to fibre bundles and simplicial complexes
  • Notions are presented in a concrete way, starting from elementary examples. There are some 250 exercises; the solution is generally deferred to the last chapter
  • Many references for further reading or study are given