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Preprint Number 1781

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1781. Mark Kamsma
Type space functors and interpretations in positive logic

Submission date: 7 May 2020


We construct a 2-equivalence CohTheory^{op} ≃ 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.

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Full text arXiv 2005.03376: pdf, ps.

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