Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction, generalizing Hofstra's construction of Dialectica fibrations and Spivak's construction of generalized lenses. This construction turns a tower of Grothendieck fibrations into another tower of fibrations by iteratively twisting each of the components, using the opposite fibration construction.
Comment: v3: 18 pp. Project results from the American Mathematical Society's Math Research Community on Applied Category Theory 2022. v4: Final version for proceedings of MFPS 2024. Updated author affiliation. v5: Corrected author typo, added MPIM report no and DOI