The biequivalence of locally cartesian closed categories and martin-lof type theories
Clairambault, P. and Dybjer, P., 2011. The biequivalence of locally cartesian closed categories and martin-lof type theories. In: Typed Lambda Calculi and Applications - 10th International Conference, TLCA 2011, Proceedings. Vol. 6690 LNCS. Heidelberg: Springer, pp. 91-106. (Lecture Notes in Computer Science)
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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-Lof type theories with , , 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 -types the resulting categories with families are biequivalent to left exact categories.
|Item Type||Book Sections|
|Creators||Clairambault, P.and Dybjer, P.|
|Departments||Faculty of Science > Computer Science|
|Additional Information||10th International Conference on Typed Lambda Calculi and Applications, TLCA 2011. 1-3 June 2011. Novi Sad, Serbia.|
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