Gluing Manifolds in the Cahiers Topos

TL;DR

This paper establishes a fully faithful embedding of manifolds with corners into the Cahiers topos, preserving key structures, and develops a theory for gluing such manifolds within this topos, aiming at applications in Field Theory.

Contribution

It introduces a new embedding of manifolds with corners into the Cahiers topos and develops a gluing theory compatible with Synthetic Differential Geometry.

Findings

Embedding preserves open covers and transverse fiber products.

Gluing along a face corresponds to a pushout with an infinitesimal thickening.

Framework supports future applications in Field Theory.

Abstract

We show that there is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos, one of the premier models for Synthetic Differential Geometry. This embedding is shown to have a number of nice properties, such as preservation of open covers and transverse fibre products. We develop a theory for gluing manifolds with corners in the Cahiers topos. In this setting, the result of gluing together manifolds with corners along a common face is shown to coincide with a pushout along an infinitesimally thickened face. Our theory is designed with a view toward future applications in Field Theory within the context of Synthetic Differential Geometry.