Local structures and intrinsic manifolds

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.

Technical approach and related theories

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.