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We provide a pedagogical introduction to the theory of principal 2-bundleswith adjusted connections and show how they enter the description of geometricand non-geometric T-dualities as proposed in arXiv:2204.01783. This descriptioncombines the torus fibrations as well as the gerbe containing the Kalb-Ramond$B$-field into a single geometric object, a particular case of a non-abeliangerbe. The $B$-field and the metric are encoded in the connection of thiscategorified principal bundle, and a T-duality is described as a particularspan or correspondence of such bundles. The formalism is manifestly covariantunder the full T-duality group, and it readily reproduces key examples from theliterature.

T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections