The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories
Paper i proceeding, 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.

Författare

P. Clairambault

University of Bath

Peter Dybjer

Chalmers, Data- och informationsteknik, Datavetenskap

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)

Ämneskategorier

Data- och informationsvetenskap

DOI

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

ISBN

978-364221690-9

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2018-02-21