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19
Oosten. Ordered partial combinatory algebras
 Mathematical Proceedings of the Cambridge Philosophical Society
, 1992
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Aspects of predicative algebraic set theory I: Exact Completion
 Ann. Pure Appl. Logic
"... This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on ..."
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This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on
Inductive Types and Exact Completion
 Ann. Pure Appl. Logic
, 2002
"... Using the theory of exact completions, we show that a specific class of pretopoi, consisting of what we might call "realizability pretopoi", can act as categorical models of certain predicative type theories, including MartinLof type theory. Our main theoretical instrument for doing so is ..."
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Using the theory of exact completions, we show that a specific class of pretopoi, consisting of what we might call "realizability pretopoi", can act as categorical models of certain predicative type theories, including MartinLof type theory. Our main theoretical instrument for doing so is a categorical notion, the notion of weak Wtypes, an "intensional" analogue of the "extensional " notion of Wtypes introduced in an article by Moerdijk and Palmgren ([6]). 1
A Characterization Of The Left Exact Categories Whose Exact Completions Are Toposes
, 1999
"... We characterize the categories with finite limits whose exact completions are toposes. We review the examples in the literature and also find new examples and counterexamples. ..."
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Cited by 11 (2 self)
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We characterize the categories with finite limits whose exact completions are toposes. We review the examples in the literature and also find new examples and counterexamples.
Realizability Categories
"... This thesis contains a collection of results of my Ph.D. research in the area of realizability and category theory. My research was an exploration of the intersection of these areas focused on gaining a deeper understanding rather than on answering a specific question. This gave us some theorems tha ..."
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This thesis contains a collection of results of my Ph.D. research in the area of realizability and category theory. My research was an exploration of the intersection of these areas focused on gaining a deeper understanding rather than on answering a specific question. This gave us some theorems that help to define what realizability
Closure Operators in Exact Completions
, 2001
"... In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions. The study of separated objects in exact completions ..."
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In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions. The study of separated objects in exact completions will then allow us to give conceptual proofs of local cartesian closure of di#erent categories of pseudo equivalence relations. Finally, we characterize when certain categories of sheaves are toposes. 1.
A categorical version of the BrouwerHeytingKolmogorov interpretation
, 2002
"... In this paper we interpret (fragments of) intuitionistic logic in categories with weak closure properties, such as quasi left exact categories and locally cartesian closed categories (LCCC) with sums. We also interpret the full choice scheme in an LCCC. The interpretation can be seen as a categorica ..."
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In this paper we interpret (fragments of) intuitionistic logic in categories with weak closure properties, such as quasi left exact categories and locally cartesian closed categories (LCCC) with sums. We also interpret the full choice scheme in an LCCC. The interpretation can be seen as a categorical form of the usual BrouwerHeytingKolmogorov (BHK) interpretation. The standard interpretation of geometric logic in a pretopos is obtained by applying the image functor to the BHKinterpretation The standard interpretation of...
Relative Completions
, 2002
"... We introduce a relativised version of the regular and exact completion. ..."
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We introduce a relativised version of the regular and exact completion.
Diaconescu’s Theorem and the Principle of Propositional Extensionality
, 2007
"... A wellknown result in topos theory is Diaconescu’s Theorem, which says that toposes that satisfy IAC (the Internal Axiom of Choice) are boolean. It has been suggested that PPE, the Principle of Propositional Extensionality, which says that equivalent propositions are the same (p ↔ q ∈ Ω ⇒ p = q) is ..."
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A wellknown result in topos theory is Diaconescu’s Theorem, which says that toposes that satisfy IAC (the Internal Axiom of Choice) are boolean. It has been suggested that PPE, the Principle of Propositional Extensionality, which says that equivalent propositions are the same (p ↔ q ∈ Ω ⇒ p = q) is essential for the proof. This note shows that this is correct: there are models of HigherOrder Intuitionistic Logic (triposes) that validate choice at all types, but not classical logic. The proof is essentially an application of the work of Matías Menni on exact completions and toposes [2].