DMCA
On the Interpretation of Type Theory in Locally Cartesian Closed Categories (1994)
| Venue: | Proceedings of Computer Science Logic, Lecture Notes in Computer Science |
| Citations: | 55 - 1 self |
BibTeX
@INPROCEEDINGS{Hofmann94onthe,
author = {Martin Hofmann},
title = {On the Interpretation of Type Theory in Locally Cartesian Closed Categories},
booktitle = {Proceedings of Computer Science Logic, Lecture Notes in Computer Science},
year = {1994},
pages = {427--441},
publisher = {Springer}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
. We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory. 1 Introduction and Motivation Interpreting dependent type theory in locally cartesian closed categories (lcccs) and more generally in (non split) fibrational models like the ones described in [7] is an intricate problem. The reason is that in order to interpret terms associated with substitution like pairing for \Sigma -types or application for \Pi-types one needs a semantical equivalent to syntactic substitution. To clarify the issue let us have a look at the "naive" approach described in Seely's seminal paper [14] which contains a subtle inaccuracy. Assume some dependently typed calculus like the one defined in [10] and an lccc C (a category ...
Keyphrases
type theory locally cartesian intensional type theory naive approach syntactic substitution sigma type motivation interpreting dependent type theory dependent type theory pi-types one fibrational model subtle inaccuracy seminal paper typed calculus intricate problem extensional type theory semantical equivalent non split semantic function
