This entry is a synopsis of the complete 52-page manuscript attached as a PDF. The ZIP includes the manuscript source and executable Rust examples.
Classical field equations as a map of stacks
Let a background be an oriented and time-oriented Lorentzian four-manifold with signature $(-+++)$, and use the unit circle $\mathbb T=\mathbb R/\mathbb Z$. In the chosen normalization the ordinary curvature has integral periods. The vacuum constitutive map is $G=\star_g F$, and the sourced equations are $$ dF=0,\qquad dG=J. $$ Here $J$ is a prescribed closed three-form. With $F=B-dt\wedge E$, $G=D+dt\wedge H$, and $J=\rho-dt\wedge j$, the equations recover both Gauss constraints, Faraday induction, and the Ampère–Maxwell law. Keeping $G$ and $J$ twisted by the orientation line gives the premetric formulation before choosing an orientation.
The absolute smooth higher topos contains the degree-two differential classifying object $\mathbf B\mathbb T_\nabla$. The equation is natural on a site of Lorentzian backgrounds, with a separate factor of smooth parameter manifolds. A family over $(T,M)$ is a circle bundle on $T\times M$ with a connection along the spacetime directions. Differentiation and the Hodge operator act vertically. This is essential because the internal mapping stack of full connections on $T\times M$ also records connection components along $T$.
Write $\mathcal C$ for the vertical connection stack and $Z_v^k$ for the sheaf of closed vertical $k$-forms. The universal equation map is $$ E_g\colon \mathcal C\longrightarrow Z_v^3,\qquad E_g=d_v\star_g\operatorname{curv}_v. $$ The solution object over a current map $J\colon B\to Z_v^3$ is $\mathcal C\times^h_{Z_v^3}B$. Evaluation at an ordinary background gives the groupoid of circle connections solving the prescribed Maxwell equation and their connection-preserving isomorphisms. The real curvature formulation replaces $\mathcal C$ by $Z_v^2$ and $E_g$ by $d_v\star_g$. A constitutive relation can instead be represented by a span whose projections are $F$ and $G$, followed by the joint constraint $(d_vF,d_vG)=(0,J)$. This keeps the material state separate from the differential equations.
Descent, current obstructions, and curvature lifts
The solutions satisfy descent for parameter and spacetime open covers. This follows by gluing bundles and vertical connections, using locality of their curvature and field equation. Base change along a map of source objects is associativity of pullbacks. The fixed-source homotopy fiber retains connection-preserving arrows because the current object is a sheaf of sets. It imposes ordinary equality of currents at each stage.
There is a natural curvature map from the connection solution object to the real solution sheaf. This map is locally surjective and identifies the topos-level sheaf of local components with the real solution sheaf. The map on global components has image precisely the solutions with integral magnetic periods. For a fixed ordinary background the fiber over any integral solution is a torsor for the flat connection groupoid, with component group $H^1(M;\mathbb R/\mathbb Z)$ and automorphism group $H^0(M;\mathbb R/\mathbb Z)$. The distinction is visible in the flat Aharonov–Bohm connection and in the flux through a sphere linking a monopole.
For vertical families there is an obstruction class on $T\times M$. Let $\mathcal R_v$ and $\mathcal T_v$ be the sheaves of smooth real and circle-valued functions locally constant in the spacetime variable. The vertical de Rham complex resolves $\mathcal R_v$, and the exponential sequence is $0\to\underline{\mathbb Z}\to\mathcal R_v\to\mathcal T_v\to0$. A closed vertical curvature lifts exactly when its class in $H^2(T\times M;\mathcal R_v)$ is in the image of integral cohomology. Local vertical primitives and their triple-overlap defect give the lifting proof. On compact fibers, Hodge theory with an auxiliary Riemannian metric supplies a jointly smooth primitive for an exact family and proves sufficiency of fiberwise integral periods for the product family.
The current obstruction is distinct. Closure of $J$ is local conservation. A global excitation satisfying $dG=J$ additionally requires $[J]=0$ in de Rham cohomology. For an exact current, the real vacuum range condition can be written using a primitive $G_0$: the class of $d\star_gG_0$ must vanish in $d\Omega^2/(d\star_g Z^2)$. The class is independent of $G_0$. In the smooth metric site the universal equation map is nonetheless an effective epimorphism. Locally, a cutoff of a primitive extends the closed current to a compactly supported conserved current on a small globally hyperbolic neighborhood. Solving the retarded wave equation on one-forms produces a Lorenz-gauge potential for that source. Continuity of the retarded operator gives smooth dependence on passive parameters with a common spacetime cutoff.
Cauchy data, duality, and illustrative software
On a complete ultrastatic product the solution groupoid is equivalent to the groupoid of an initial circle connection and electric two-form satisfying Gauss's constraint. Temporal parallel transport identifies the bundle with the pullback of its initial bundle. A normally hyperbolic one-form equation gives the evolving spatial potential and a scalar homogeneous wave equation propagates the Gauss constraint. This implies that exactness is sufficient for existence for an unrestricted smooth closed current on this class of backgrounds.
Classical real vacuum fields admit electric–magnetic rotations since $\star_g^2=-1$. A circle connection imposes integrality only on $F$, and a quarter-turn can send it outside the allowable curvature sectors. Keeping separate circle lifts of both $F$ and $\star_g F$ restores a quarter-turn action on their paired stack, including the respective flat choices. The manuscript also treats higher-form gauge degrees, the linear deformation complex, the covariant presymplectic current, and the distinction between configuration descent and injectivity of observable maps.
The attachment contains the full manuscript with historical and technical notes, linked references, and a Rust/Macroquad viewer. The viewer generates a plane wave, a monopole clutching sphere, an exterior Aharonov–Bohm loop with its phase, and the independent electric and magnetic periods of a periodic torus. It builds and renders all four scenes under Linux. The mathematical arguments use ordinary proofs and cited hyperbolic analysis.
Direct antecedents
The treatment of premetric fields and constitutive data follows Hehl and Obukhov. The differential-cohomology construction uses the Cheeger–Simons character description and Schreiber's cohesive framework. The Maxwell kernel on differential-cohomology classes is directly related to Becker, Schenkel, and Szabo. The Yang–Mills stack of Benini, Schenkel, and Schreiber is the direct precedent for parametrized Lorentzian connection solutions. Becker, Benini, Schenkel, and Szabo develop the two-character description of abelian duality. The full reference list is in the attachment.
- Hehl and Obukhov, Foundations of Classical Electrodynamics. https://link.springer.com/book/10.1007/978-1-4612-0051-2
- Cheeger and Simons, Differential Characters and Geometric Invariants. https://math.mit.edu/juvitop/pastseminars/notes_2019_Fall/cheeger-simons.pdf
- Schreiber, Differential Cohomology in a Cohesive Infinity-Topos. https://arxiv.org/abs/1310.7930
- Becker, Schenkel, and Szabo, Differential Cohomology and Locally Covariant Quantum Field Theory. https://arxiv.org/abs/1406.1514
- Benini, Schenkel, and Schreiber, The Stack of Yang–Mills Fields on Lorentzian Manifolds. https://arxiv.org/abs/1704.01378
- Becker, Benini, Schenkel, and Szabo, Abelian Duality on Globally Hyperbolic Spacetimes. https://arxiv.org/abs/1511.00316
Disclosure
AI assistance was used in research synthesis, mathematical exposition, and the companion software. This is an anonymous review copy.