Axioms and (counter)examples in synthetic domain theory
type: Item
instance of: scholarly article
title: Axioms and (counter)examples in synthetic domain theory
zbMATH Open document ID: 0961.18006
DOI: 10.1016/S0168-0072(00)00014-2
author: Jaap van Oosten
author: Alex Simpson
published in: Annals of Pure and Applied Logic
publication date: 2001-05-16T00:00:00Z
review text: The goal of the subject which has become known as synthetic domain theory is to provide an axiomatics for categories of domains which should allow one to reason about domains as if they were simply sets, their `non-set-like' features being masked by the internal logic of the category within which one is working. In the present paper, the authors present a careful development of synthetic domain theory based upon the internal logic of an elementary topos: they also interpret their definitions in several particular toposes (both realizability toposes and Grothendieck toposes), which enables them to give counterexamples to various conjectures about synthetic domain theory which have been made in the past, for example that the initial lift-algebra is necessarily an internal `countable colimit'.
Mathematics Subject Classification ID: 18C50
Mathematics Subject Classification ID: 68Q55
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03D75
zbMATH DE Number: 1501508
zbMATH Keywords: synthetic domain theory
zbMATH Keywords: categories of domains
zbMATH Keywords: internal logic
zbMATH Keywords: elementary topos
Wikidata QID: Q114656181
reviewed by: Peter Johnstone
MaRDI profile type: Publication
cites work: New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic
cites work: Q4362914
cites work: Two models of synthetic domain theory
cites work: Q3978971
cites work: Q3671978
cites work: Q3978973
cites work: Q4281473
cites work: The Discrete Objects in the Effective Topos
cites work: A presentation of the initial lift-algebra
cites work: On a Topological Topos
cites work: Q4853985
cites work: Q5622402
cites work: The sequentially realizable functionals
cites work: A uniform approach to domain theory in realizability models
cites work: Sheaves in geometry and logic: a first introduction to topos theory
cites work: Q4261092
cites work: The modified realizability topos
cites work: Q4335371
cites work: Q4362975
cites work: Q3784071
cites work: Q4263806
cites work: Inductive construction of repletion
cites work: Metamathematical investigation of intuitionistic arithmetic and analysis. With contributions by C. A. Smorynski, J. I. Zucker and W. A. Howard
full work available at URL: s0168-0072(00)00014-2
OpenAlex ID: W2142922307
description: scientific article; zbMATH DE number 1501508