Year: 2022
Type: article
Abstract: We construct a 2-equivalence $$\mathfrak {CohTheory}^{op }\simeq \mathfrak {TypeSpaceFunc}$$ CohTheory op ≃ TypeSpaceFunc . Here $$\mathfrak {CohTheory}$$ CohTheory is the 2-category of positive theories and $$\mathfrak {TypeSpaceFunc}$$ 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 $$\mathfrak {CohTheory}$$ 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.
Source: Archive for Mathematical Logic
Author: Mark Kamsma
Institution: University of East Anglia
Language: Latvian
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,780
Funder: University of East Anglia
OpenAlex ID: W3021736704