Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations

Abstract: This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment, we laid out the natural formulation of bosonic variational field theory via the functorial geometry of smooth sets. Here, we extend this to the category ThickenedSmoothSets of infinitesimally thickened smooth sets. We first describe the Cahiers topos in a simplified, but fully rigorous, $\mathbb{R}$-algebraic setting -- which should serve as a more accessible introduction to the theory of Synthetic Differential Geometry to both physicists and mathematicians. Then, we formulate local Lagrangian field theory in this rigorous setting in which infinitesimal spaces exist and interact correctly with the field-theoretic spaces of infinite jet bundles, off-shell and on-shell spaces of fields etc. This setting subsumes all previous constructions and further recovers all the relevant tangent bundles of traditional (off-shell and on-shell) field theory considerations via the synthetic tangent bundle construction, i.e., as ``infinitesimal curves'' in those spaces, which were previously defined only in an ad-hoc manner. Beyond finally establishing a firm foundation for such aspects of the theory, this approach recognizes the variational principle of local Lagrangian field theory, equivalently, as the intersection of thickened smooth sets. It also suggests the rigorous formalization of perturbative field theory as the restriction to a (synthetic) infinitesimal neighborhood around a field configuration. Furthermore, our context naturally accommodates more general, rigorous considerations, in which the manifolds may have boundaries and corners, a situation that has recently been attracting greater attention in the field-theoretical literature.

• The paper presents a synthetic topos-theoretic framework using thickened smooth sets, rigorously formalizing Lagrangian field theory and infinitesimal geometry.
• It systematically constructs synthetic tangent and higher jet bundles via internal homs, ensuring a coordinate-free recovery of classical geometrical structures.
• The approach extends to spaces with boundaries and corners, offering a robust foundation for perturbative expansions and formal field manipulations.

Synthetic Foundations for Field Theory: Thickened Smooth Sets and Higher Geometry

Motivation and Context

"Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations" (2512.22816) systematically extends the synthetic geometric formalism for Lagrangian field theory. Building on the functorial paradigm initiated in the first installment, which treated variational problems within the topos-theoretic framework of smooth sets, this work introduces and develops the site of infinitesimally thickened smooth sets. The formalism leverages the Cahiers topos (the topos of sheaves over the site of thickened Cartesian spaces, i.e., products of $\mathbb{R}^k$ with infinitesimal ‘test spaces’) as a more comprehensive and robust synthetic foundation, naturally accommodating both smooth structure and nilpotent infinitesimal extensions. This setting subsumes all previous approaches and rigorously codifies the infinitesimal constructions that were previously accessible only in a heuristic or coordinate-dependent way.

Rigorous Construction of Thickened Probes and the Cahiers Topos

The authors establish that the site for the synthetic foundation should be the category $Th$ whose objects are products $R^k\times D$, with $D$ a formal infinitesimal point (e.g., given by a finite-dimensional nilpotent $\mathbb{R}$-algebra, or more generally a Weil algebra). Morphisms are defined via pullback maps on function algebras. The coverage employed is that of ‘good open covers’ extended trivially in the nilpotent directions, and sheaves over this site define the category of infinitesimally thickened smooth sets.

A key technical contribution is the systematic presentation of the Cahiers topos using only $R$-algebraic tools rather than the more technical machinery of $C^\infty$-algebras. The authors prove that all function algebras arising in this context are quotients of smooth function algebras by ideals generated by nilpotents; in particular, they are always finitely generated as $C^\infty$-algebras. This identification makes many constructions more accessible and transparent, and is an essential step for applications to field-theoretic models.

Synthetic Tangent and Jet Bundles

A central pillar of the framework is the synthetic construction of tangent and jet bundles through internal homs. The assignment $TF := [D^1(1), F]$, where $D^1(1)$ is the first-order infinitesimal (the spectrum of the dual numbers), recovers the tangent bundle in a universal, coordinate-free way for any thickened smooth set $F$. The authors prove that, for classical manifolds, this construction yields the standard tangent bundle, while for generalized smooth spaces (including spaces of fields, mapping spaces, and infinite jet spaces) it provides an internally consistent and functorial geometric object, capturing true infinitesimal directions not always accessible via classical geometric methods, especially for spaces with boundary or corners.

Mapping out of higher-order or higher-dimensional infinitesimal disks $D^m(l)$ is shown to recover higher-order tangent bundles, higher jets, or more generalized variations, with precise functoriality and limits corresponding to projective systems of finite jets.

Field Theory Structures in the Infinitesimal Topos

All the principal constructs of local Lagrangian field theory are rigorously encoded in this synthetic language:

• Infinite jet bundles and their tangent bundles are defined as limits of representable objects, and their local structure and split exact sequences are recovered functorially from their synthetic definition.
• The Euler–Lagrange operator is constructed entirely in the topos, and its critical locus (the on-shell set of field configurations) is rigorously the fibered intersection of the field configuration space and the zero section of the cotangent bundle in $Th$.
• The synthetic formalism proves that the variational principle for local functionals is realized as an intersection in the topos, with all equations and spaces being naturally thickened and encoding both smooth and infinitesimal variation.

A strong theoretical result is that perturbative field theory expansions are realized as restrictions to synthetic infinitesimal neighborhoods around specific ‘background’ field configurations—making rigorous the R-linear nilpotent Taylor expansions familiar in physics, naturally up to isomorphism.

Embedding Manifolds with Boundaries and Corners

The paper extends previously established functorial geometry embeddings to manifolds with boundary and corners, and smooth closed subspaces, by a careful analysis of the algebraic structure underlying these spaces. All such embeddings are compatible with the infinitesimal structure introduced in the synthetic setting and show that the formalism handles initial-value problems and boundaries on equal footing with the interior theory.

Additional Technical Contributions

The appendix contains novel proofs and expository results, including:

• A streamlined $R$-algebraic presentation of the Cahiers topos as a well-adapted model for synthetic differential geometry.
• Local expressions and quotient algebra characterizations for thickened smooth sets and their mapping spaces.
• A careful generalization of Hadamard’s lemma, serving as the algebraic engine for all explicit expressions involving infinitesimal extensions.
• Explicit constructions and functorial properties (and limitations) of the synthetic tangent, Weil, and jet bundles for spaces with corners.

Implications and Outlook

This work provides a comprehensive and practical foundation for synthetic approaches to Lagrangian field theory. The functorial, algebraic, and topos-theoretic perspective not only recovers all classical results, but makes conceptual sense of physicists’ infinitesimal manipulations—without ad hoc truncations or unexplained nilpotency. The approach elucidates the true geometric content of variational problems on arbitrary smooth spaces, and clarifies perturbative expansions as intrinsic, coordinate-free constructions.

The framework is shown to be robust under the inclusion of more general geometric features (boundaries, corners, closed subspaces) and is fully compatible with differential cohomological and higher-categorical enhancements (to be addressed in subsequent works).

From a practical standpoint, the framework allows for:

• Systematic computation of tangent and cotangent spaces for spaces of mappings, field configurations, jet spaces, and critical loci.
• A clear prescription for the rigorization of formal physics manipulations (e.g., variation, perturbation, path integral localization).
• Infinitesimal and synthetic (i.e., genuinely coordinate-free) descriptions of all constructions, sidestepping functional-analytic pathologies.

Conclusion

The paper achieves a rigorous and accessible synthetic foundation for the differential geometry underlying Lagrangian field theory using the Cahiers topos of thickened smooth sets. By bridging the gap between the heuristic use of infinitesimals and the formal requirements of modern geometry and topos theory, it enables precise definitions, functorial constructions, and a fully controllable calculus for both classical and future higher-categorical generalizations of field theory. This foundation is suitable for the extension to supergeometry (fermionic fields), higher toposes, and cohesive/differentially cohomological models, which are identified as natural future directions.