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Aspects of predicative algebraic set theory II:
 Realizability. Theoret. Comput. Sci.
, 2011
"... Abstract This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative category with small maps" which axiomatis ..."
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Abstract This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative category with small maps" which axiomatises the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are sets (in some axiomatic set theory). The main result of the present paper is that such predicative categories with small maps are stable under internal sheaves. We discuss the sheaf models of constructive set theory this leads to, as well as ideas for future work.
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 12 (5 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
Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets
, 2011
"... 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 s ..."
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Cited by 7 (1 self)
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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.
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
A unified approach to algebraic set theory
 the proceedings of the Logic Colloquium 2006, arXiv:0710.3066
, 2007
"... ..."
Three extensional models of type theory
, 2007
"... MartinLöf’s type theory 1 exists in two forms, differing in the formalisation of the identity types. In [15] Per MartinLöf formulated his type theory with ..."
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Cited by 2 (1 self)
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MartinLöf’s type theory 1 exists in two forms, differing in the formalisation of the identity types. In [15] Per MartinLöf formulated his type theory with
Abstract Nonwellfounded trees in categories
"... nonwellfounded sets, as well as nonterminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call Mtypes. We derive existence results for Mtypes in locally cartesian closed pretoposes with a natural numbers object, usi ..."
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nonwellfounded sets, as well as nonterminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call Mtypes. We derive existence results for Mtypes in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with Mtypes under various topostheoretic constructions; namely, slicing, formation of coalgebras (for a cartesian comonad), and sheaves for an internal site. 1
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.