Type space functors and interpretations in positive logic

Mark Kamsma

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Abstract

We construct a 2-equivalence CohTheoryop≃TypeSpaceFunc. Here CohTheory is the 2-category of positive theories and TypeSpaceFunc is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in CohTheory. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is ‘the same’ as the collection of its type spaces (i.e. its type space functor). In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.
Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalArchive for Mathematical Logic
Volume62
Issue number1-2
Early online date30 Mar 2022
DOIs
Publication statusPublished - Feb 2023

Keywords

  • Bi-intepretation
  • Coherent logic
  • Interpretation
  • Positive model theory
  • Stone duality
  • Type space

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