Comprehension categories and the semantics of type dependency
Theoretical Computer Science
Computation and reasoning: a type theory for computer science
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Mathematical theory of domains
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A Categorical Approach to Realizability and Polymorphic Types
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On the Interpretation of Type Theory in Locally Cartesian Closed Categories
CSL '94 Selected Papers from the 8th International Workshop on Computer Science Logic
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LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
A General Notion of Realizability
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Developing theories of types and computability via realizability
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The realizability approach to computable analysis and topology
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Foundation of a computable solid modelling
Theoretical Computer Science
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Comparing Functional Paradigms for Exact Real-Number Computation
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
What do types mean?: from intrinsic to extrinsic semantics
Programming methodology
A General Notion of Realizability
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
A non-topological view of dcpos as convergence spaces
Theoretical Computer Science - Topology in computer science
A Cartesian closed extension of the category of locales
Mathematical Structures in Computer Science
Compactly generated domain theory
Mathematical Structures in Computer Science
A Convenient Category of Domains
Electronic Notes in Theoretical Computer Science (ENTCS)
On the Relationship between Filter Spaces and Weak Limit Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
RZ: A Tool for Bringing Constructive and Computable Mathematics Closer to Programming Practice
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Reducibility of domain representations and cantor–weihrauch domain representations
Mathematical Structures in Computer Science
Electronic Notes in Theoretical Computer Science (ENTCS)
Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
Computability in computational geometry
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
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It is well known that one can build models of full higher-order dependent-type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra. But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily define the category of ERs and equivalence-preserving continuous mappings over the standard category Top0 of topological T0-spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category--in contradistinction to Top0--is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein), a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of finite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models.