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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
Aspects of predicative algebraic set theory II: Realizability. Accepted for publication in Theoretical Computer Science
 In Logic Colloquim 2006, Lecture Notes in Logic
, 2009
"... This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how ..."
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Cited by 7 (1 self)
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This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how these predicative categories
A unified approach to algebraic set theory
 the proceedings of the Logic Colloquium
, 2006
"... This short paper provides a summary of the tutorial on categorical logic given by the second named author at the Logic Colloquium in Nijmegen. Before we ..."
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This short paper provides a summary of the tutorial on categorical logic given by the second named author at the Logic Colloquium in Nijmegen. Before we
Sheaves for predicative toposes
 ArXiv:math.LO/0507480v1
"... Abstract: In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal systems. Among our technical results, we prove that ..."
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Abstract: In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal systems. Among our technical results, we prove that all the notions of a “predicative topos ” that we consider, are stable under presheaves, while most are stable under sheaves. 1
Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets
, 2009
"... Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of MartinLöf. The theory, CETCS, provides a str ..."
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Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of MartinLöf. The theory, CETCS, provides a structuralist foundation for constructive mathematics in the style of Bishop.
Wtypes in sheaves
, 2008
"... In this small note we give a concrete description of Wtypes in categories of sheaves. It can be shown that any topos with a natural numbers object has all Wtypes. Although there is this general result, it can be useful to have a concrete description of Wtypes in various toposes. For example, a co ..."
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In this small note we give a concrete description of Wtypes in categories of sheaves. It can be shown that any topos with a natural numbers object has all Wtypes. Although there is this general result, it can be useful to have a concrete description of Wtypes in various toposes. For example, a concrete description of Wtypes in the effective topos can be found in [2, 3], and a concrete description of Wtypes in categories of presheaves was given in [5]. It was claimed in [5] that Wtypes in categories of sheaves are computed as in presheaves (Proposition 5.7 in loc.cit.) and can therefore be described in the same way. Unfortunately, this claim is incorrect, as the following (easy) counterexample shows. Let f: 1 → 1 be the identity map on the terminal object. The Wtype associated to f is the initial object, which, in general, is different in categories of presheaves and sheaves. This means that we still lack a concrete description of Wtypes in categories of sheaves. This note aims to fill this gap.