The biequivalence of locally cartesian closed categories and Martin-Lof type theories
Artikel i vetenskaplig tidskrift, 2014

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-Lof type theories with Pi, Sigma 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 Benabou-Hofmann interpretation of Martin-Lof 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-Lof type theories. As a second result, we prove that if we remove Pi-types, the resulting categories with families with only Sigma and extensional identity types are biequivalent to left exact categories.

Computer Science

Theory & Methods

Författare

P. Clairambault

Ecole Normale Superieure de Lyon

Peter Dybjer

Chalmers, Data- och informationsteknik, Datavetenskap

Mathematical Structures in Computer Science

0960-1295 (ISSN) 1469-8072 (eISSN)

Vol. 24 Art. no. e240606-

Ämneskategorier

Datavetenskap (datalogi)

DOI

10.1017/s0960129513000881