On the syntax of infinite objects: an extension of Martin-Lo¨f's theory of expressions
COLOG-88 Proceedings of the international conference on Computer logic
COLOG-88 Proceedings of the international conference on Computer logic
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Semantics of type theory: correctness, completeness, and independence results
Semantics of type theory: correctness, completeness, and independence results
Infinite objects in type theory
TYPES '93 Proceedings of the international workshop on Types for proofs and programs
Handbook of logic in computer science
On the Interpretation of Type Theory in Locally Cartesian Closed Categories
CSL '94 Selected Papers from the 8th International Workshop on Computer Science Logic
Extensional Equality in Intensional Type Theory
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Infinite trees and completely iterative theories: a coalgebraic view
Theoretical Computer Science
Mathematical Structures in Computer Science
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There are several approaches to the problem of giving a categorical semantics to Martin-Löf type theory with dependent sums and products and extensional equality types. The most established one relies on the notion of a type-category (or category with attributes) with ${\it \Sigma}$ and ${\it \Pi}$ types. We extend such a semantics by introducing coinductive types both on the syntactic level and in a type-category. Soundness of the semantics is preserved. As an example of such a category, we prove that the type-category built over a locally cartesian closed category ${\mathcal C}$ admits coinductive types whenever ${\mathcal C}$ has final coalgebras for all polynomial functors.