Synthetic Differential Jet Bundles are Reduced

TL;DR

This paper proves that the passage to the Cahiers topos preserves the projective limits defining infinite jet bundles, strengthening the theoretical foundation of synthetic differential geometry applied to PDEs.

Contribution

It provides a detailed proof that the Cahiers topos preserves projective limits of jet bundles, confirming a key assumption in the synthetic differential geometry approach.

Findings

The passage to the Cahiers topos preserves infinite jet bundle structures.

Supports the use of synthetic differential geometry in PDE solution theory.

Strengthens the theoretical framework connecting diffieties and comonadic formulations.

Abstract

We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fr\'echet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof.