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A general construction of internal sheaves in algebraic set theory. Preliminary version available at [3
"... Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume ..."
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Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topostheoretic results.
CZF has the disjunction and numerical existence property. Available from the author’s web page www.amsta.leeds.ac.uk/Pure/staff/rathjen/preprints.html
, 2004
"... This paper proves that the disjunction property, the numerical existence property and Church’s rule hold true for Constructive ZermeloFraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom. As to the proof technique, it features a selfvalidating semantics fo ..."
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This paper proves that the disjunction property, the numerical existence property and Church’s rule hold true for Constructive ZermeloFraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom. As to the proof technique, it features a selfvalidating semantics for CZF that combines extensional Kleene realizability and truth. MSC:03F50, 03F35
LAWVERETIERNEY SHEAVES IN ALGEBRAIC SET THEORY
"... Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume ..."
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Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topostheoretic results.
CATEGORICAL LOGIC AND PROOF THEORY EPSRC INDIVIDUAL GRANT REPORT – GR/R95975/01
"... Abstract. I describe the main results obtained during the EPSRC postdoctoral fellowship that I held at the University of Cambridge. The fellowship focused on the interplay between category theory and mathematical logic. 1. Wellfounded trees Wtypes in categories. Types of wellfounded trees, or Wtyp ..."
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Abstract. I describe the main results obtained during the EPSRC postdoctoral fellowship that I held at the University of Cambridge. The fellowship focused on the interplay between category theory and mathematical logic. 1. Wellfounded trees Wtypes in categories. Types of wellfounded trees, or Wtypes, are one of the most important components of MartinLöf’s dependent type theories. They allow us to define a wide class of inductive types, play an essential role in the ‘setsastrees’ interpretation of constructive set theories, and contribute considerably to the prooftheoretic strength of dependent type theories. A categorical counterpart of Wtypes was introduced in [18] by defining Wtypes in a locally cartesian closed category to be initial algebras for endofunctors of a special kind, generally referred to as polynomial functors. In collaboration with Martin Hyland, I set out to investigate the consequences of the assumption that a locally cartesian closed category has Wtypes. To explore these consequences we introduced the notion of a dependent polynomial functor, a
for Constructive Set Theory
, 2008
"... This article presents a generalisation of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalisation of the Boolean models for classical set theory which are a kind of forcing, while realizability is a decidedly constructive method that has first ..."
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This article presents a generalisation of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalisation of the Boolean models for classical set theory which are a kind of forcing, while realizability is a decidedly constructive method that has first been develloped for number theory by Kleene and was later very fruitfully adapted to constructive set theory. In order to achieve the generalisation, a new kind of structure (applicative topologies) is introduced, which contains both elements of formal topology and applicative structures. The generalisation not only deepens the understanding of class models and leads to more efficiency in proofs about these kind of models, but also makes it possible to prove new results about the special cases which were not known before and to construct new models. Generalising Realizability and Heyting Models
researchshowcase@andrew.cmu.edu. LAWVERETIERNEY SHEAVES IN ALGEBRAIC SET THEORY
"... Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume ..."
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Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topostheoretic results.