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We systematically extend the elementary differential and Riemannian geometryof classical $\mathrm{U}(1)$-gauge theory to the noncommutative setting bycombining recent advances in noncommutative Riemannian geometry with the theoryof coherent $2$-groups. We show that Hermitian line bimodules with Hermitianbimodule connection over a unital pre-$\mathrm{C}^\ast$-algebra with$\ast$-exterior algebra form a coherent $2$-group, and we prove that weakmonoidal functors between coherent $2$-groups canonically define bar orinvolutive monoidal functors in the sense of Beggs--Majid and Egger,respectively. Hence, we prove that a suitable Hermitian line bimodule withHermitian bimodule connection yields an essentially unique differentiablequantum principal $\mathrm{U}(1)$-bundle with principal connection and viceversa; here, $\mathrm{U}(1)$ is $q$-deformed for $q$ a numerical invariant ofthe bimodule connection. From there, we formulate and solve the interrelatedlifting problems for noncommutative Riemannian structure in terms of abstractHodge star operators and formal spectral triples, respectively; all the while,we account precisely for emergent modular phenomena of geometric nature. Inparticular, it follows that the spin Dirac spectral triple on quantum$\mathbf{C}\mathrm{P}^1$ does not lift to a twisted spectral triple on$3$-dimensional quantum $\mathrm{SU}(2)$ with the $3$-dimensional calculus butdoes recover Kaad--Kyed's compact quantum metric space on quantum$\mathrm{SU}(2)$ for a canonical choice of parameters.

Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds