A precise and general notion of manifold

TL;DR

This paper formalizes a universal definition of manifolds using gluing data structured as an equivalence-partially ordered set, and establishes reconstruction theorems linking these data to manifold structures and morphisms.

Contribution

It introduces a novel formalization of general manifolds through gluing data and develops reconstruction theorems, connecting manifold theory with ordered groupoids and category theory.

Findings

Gluing data form an equivalence-partially ordered set (e-pos).

Reconstruction theorems enable building manifolds from gluing data.

Morphisms between manifolds are described via natural relations between groupoids.

Abstract

We give a completely formalized definition of a notion of " general manifold ". It turns out that " gluing data " form an equivalence-partially ordered set (e-pos), which is a special instance of an ordered groupoid. We state and prove reconstruction theorems, allowing to reconstruct general manifolds and their mor-phisms from such gluing data. To describe morphisms between manifolds, the notion of natural relations between groupoids is introduced, which emphasizes the close analogy with natural transformations of general category theory.