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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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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
THE ASSOCIATED SHEAF FUNCTOR THEOREM IN ALGEBRAIC SET THEORY
"... Abstract. We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing develo ..."
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Abstract. We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites. 1.
Category Theory and Structuralism
, 2009
"... The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond ..."
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The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond Queneau) and in Mathematics (Nicolas Bourbaki). To the layman the structuralist movement in mathematics was perhaps most visible the form of New Math, which was strongly influenced by the Bourbaki school. It has been argued in (Aubin 1997) that there were cultural connections between these movements. (See also A. Aczel 2007.) Some of these interactions may be regarded as rather superficial. The epistemologist Piaget however was very much influenced by Bourbaki, and seems to have suggested that those patterns of thought used to explain cognitive development were closely related to the mathematical “mother structures ” found by Bourbaki. On a very general level, structuralism refers to a mode of thinking involving abstraction from specifics and systematic identification and naming of common patterns. It is the relation of objects under study to each other that is of importance rather than their specific appearance, or “nature”. In mathematics, Richard Dedekind may be said to be the first structuralist. He described the positive integers (1, 2, 3,...) as positions in an infinite progression of elements (a socalled simply infinite system) 1 � � 2