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Geometric constructions preserving fibrations

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If you have a question about this talk, please contact Neel Krishnaswami.

Let π’ž be a representable 2-category, and 𝔗‒ a 2-endofunctor of the arrow 2-category π’žβ†“ such that (i) cod 𝔗‒ = cod: π’žβ†“ β†’ π’ž and (ii) 𝔗‒ preserves proneness (cartesianness) of morphisms in π’žβ†“. Then 𝔗‒ preserves fibrations and opfibrations in π’ž. The proof takes Street’s characterization of (e.g.) opfibrations as pseudoalgebras for 2-monads 𝔏B on slice categories π’ž/B and develops it by defining a 2-monad 𝔏‒ on π’žβ†“ that takes change of base into account, and uses known results on the lifting of 2-functors to pseudoalgebras.

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