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

Published online by Cambridge University Press:  29 April 2014

PIERRE CLAIRAMBAULT  and
PETER DYBJER
PIERRE CLAIRAMBAULT
Affiliation:
CNRS, ENS de Lyon, Inria, UCBL, Université de Lyon, Laboratoire LIP, 46 Allée d'Italie, 69364 Lyon, France Email: pierre.clairambault@ens-lyon.fr
PETER DYBJER
Affiliation:
Department of Computer Science and Engineering, Chalmers University of Technology, S-412 96 Göteborg, Sweden Email: peterd@chalmers.se
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Abstract

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 interpretation of Martin-Löf type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development, we employ categories with families as a substitute for syntactic Martin-Löf type theories. As a second result, we prove that if we remove Π-types, the resulting categories with families with only Σ and extensional identity types are biequivalent to left exact categories.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 24 , Issue 6 , December 2014 , e240606
Copyright
Copyright © Cambridge University Press 2014 

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