The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories
Paper in proceedings, 2011

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 Π, ∑, 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 are biequivalent to left exact categories.

Author

P. Clairambault

University of Bath

Peter Dybjer

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

03029743 (ISSN) 16113349 (eISSN)

Vol. 6690 91-106
978-364221690-9 (ISBN)

Subject Categories

Computer and Information Science

DOI

10.1007/978-3-642-21691-6_10

ISBN

978-364221690-9

More information

Latest update

2/21/2018