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

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