Topoi for Photons:A Categorical Formulation of Electromagnetic Duality

We propose a categorical formulation of electromagnetic (Maxwell) duality by identifying a class of Grothendieck topoi (and suitable stacky sheaf models) whose internal cohomological data encode electric and magnetic sectors and admit a canonical duality isomorphism. Understanding this duality should ultimately improve our success-rate studying unobserved photonic laws. Building on the program of topospotentials and G-Theory–Maxwell correspondences, we formulate the Toposic Maxwell Duality Conjecture which asserts a natural equivalence between internal hypercohomology functors associated to dual gauge stacks. We provide precise definitions of the topos model E over a smooth spacetime manifold M, the gauge stack G encoding U(1)-connections (and higher analogues), and the pair of functors F,G selecting electric/magnetic sectors. We prove a partial result: on compact orientable surfaces (notably M=T2) the conjectured duality reduces to Poincaré/Čech–de Rham duality and can be established up to the expected torsion and orientation twists. A worked toy computation on the two-torus exhibits the isomorphism of the relevant cohomological invariants. We conclude with numerical/heuristic checks (lattice discretizations and spectral invariants), physical implications for photonic systems, and explicit open problems, though we affirm the development of experiments, a variety of checks and the debate related to potential implications needs to broaden.