The biequivalence of locally cartesian closed categories and Martin-Löf type theories

@article{Clairambault2014TheBO,
  title={The biequivalence of locally cartesian closed categories and Martin-L{\"o}f type theories},
  author={Pierre Clairambault and Peter Dybjer},
  journal={Mathematical Structures in Computer Science},
  year={2014},
  volume={24}
}
Seely's paper Locally cartesian closed categories and type theory (Seely 1984) contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Π, Σ and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the Bénabou–Hofmann… 

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