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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 ๐’ž be a representable 2-category, and ๐”—โ€ข a 2-endofunctor of the arrow 2-category ๐’žโ†“ such that (i) cod ๐”—โ€ข = cod: ๐’žโ†“ โ†’ ๐’ž and (ii) ๐”—โ€ข preserves proneness (cartesianness) of morphisms in ๐’žโ†“. Then ๐”—โ€ข preserves fibrations and opfibrations in ๐’ž. The proof takes Street’s characterization of (e.g.) opfibrations as pseudoalgebras for 2-monads ๐”B on slice categories ๐’ž/B and develops it by defining a 2-monad ๐”โ€ข on ๐’žโ†“ that takes change of base into account, and uses known results on the lifting of 2-functors to pseudoalgebras.

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