Introduction to higher order categorical logic
Introduction to higher order categorical logic
Substitution up to isomorphism
Fundamenta Informaticae - Special issue: lambda calculus and type theory
TYPES '95 Selected papers from the International Workshop on Types for Proofs and Programs
On the Interpretation of Type Theory in Locally Cartesian Closed Categories
CSL '94 Selected Papers from the 8th International Workshop on Computer Science Logic
Normalization and the Yoneda embedding
Mathematical Structures in Computer Science
100 years of Zermelo's axiom of choice: what was the problem with it?
The Computer Journal
Towards Formalizing Categorical Models of Type Theory in Type Theory
Electronic Notes in Theoretical Computer Science (ENTCS)
On the algebraic foundation of proof assistants for intuitionistic type theory
FLOPS'08 Proceedings of the 9th international conference on Functional and logic programming
Categorial semantics of a solution to distributed dining philosophers problem
FAW'10 Proceedings of the 4th international conference on Frontiers in algorithmics
Proof-relevance of families of setoids and identity in type theory
Archive for Mathematical Logic
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We give an intuitionistic view of Seely's interpretation of Martin-Lof's intuitionistic type theory in locally cartesian closed categories. The idea is to use Martin-Lof type theory itself as metalanguage, and E-categories, the appropriate notion of categories when working in this metalanguage. As an E-categorical substitute for the formal system of Martin-Lof type theory we use E-categories with families (E-cwfs). These come in two flavours: groupoid-style E-cwfs and proof-irrelevant E-cwfs. We then analyze Seely's interpretation as consisting of three parts. The first part is purely categorical: the interpretation of groupoid-style E-cwfs in E-locally cartesian closed categories. (The key part of this interpretation has been type-checked in the Coq system.) The second is a coherence problem which relates groupoid-style E-cwfs with proof-irrelevant ones. The third is a purely syntactic problem: that proof-irrelevant E-cwfs are equivalent to traditional lambda calculus based formulations of Martin-Lof type theory.