The natural functor from geometry-first generating objects to principal bundles with matter is faithful but neither essentially surjective nor full, so the two formulations of gauge theory are not categorically equivalent.
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The geometry-first formulation of gauge theory is not equivalent to the symmetry-first one
The natural functor from geometry-first generating objects to principal bundles with matter is faithful but neither essentially surjective nor full, so the two formulations of gauge theory are not categorically equivalent.