Synthetic Differential Jet Bundles are Reduced
Grigorios Giotopoulos, Igor Khavkine, Hisham Sati, Urs Schreiber
TL;DR
Problem: In synthetic differential geometry jet bundles are classically defined as projective limits in Fréchet manifolds, and it remained unclear whether the inclusion $i_!$ to formal smooth sets preserves these limits. Method: The authors prove the main theorem $i_! \\varprojlim_k J^k_\\Sigma E \\simeq \\varprojlim_k (i_! J^k_\\Sigma E)$ for finite-dimensional surjective submersions, showing the synthetic limit $\\mathbf{R}^\\infty := \\varprojlim_k i_! \\mathbb{R}^k$ is reduced and developing a staged constructive factorization argument to fill commuting diagrams through Cartesian spaces, then extending from $U = \\ast$, $\\mathbb{D} = \\mathbb{D}^1(1)$ to general $U$ and $\\mathbb{D}$. Contributions: Theorem i!PreservesInfiniteJetBundles provides a canonical isomorphism and the corollary MapsOutOfJetBundleAreLocallyOfFiniteOrder, linking SDG jet theory to maps locally of finite jet order. Significance: This solidifies the SDG formulation of diffieties and jet bundles in the Cahiers topos and has potential impact on variational calculus and Lagrangian field theory by connecting Fréchet-limit constructions with formal smooth-set perspectives.
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 Fréchet 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.