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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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Cited by 10 (2 self)
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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 a categor ..."
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Cited by 8 (7 self)
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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
Oosten. Ordered partial combinatory algebras
 Mathematical Proceedings of the Cambridge Philosophical Society
, 1992
"... ..."
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 5 (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.
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.
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
Journal of Pure and Applied Algebra 210 (2007) 511–520 www.elsevier.com/locate/jpaa Cocomplete toposes whose exact completions are toposes
, 2006
"... Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere–Schanuel charact ..."
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Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere–Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos. We also show that for any topological space X, the exact completion of Sh(X) is a topos if and only if X is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.