The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories

Abstract : Seely's paper "Locally cartesian closed categories and type theory" 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 Pi-types, Sigma-types 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 Pi-types the resulting categories with families are biequivalent to left exact categories.
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Contributor : Pierre Clairambault <>
Submitted on : Wednesday, December 14, 2011 - 6:35:14 PM
Last modification on : Wednesday, October 3, 2018 - 10:44:01 AM
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Pierre Clairambault, Peter Dybjer. The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories. TLCA 2011 - 10th Typed Lambda Calculi and Applications, Jun 2011, Novi Sad, Serbia. pp.91-106, ⟨10.1007/978-3-642-21691-6_10⟩. ⟨hal-00652087⟩



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