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Logic"],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We construct a 2-equivalence <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {CohTheory}^{op }\\simeq \\mathfrak {TypeSpaceFunc}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>CohTheory<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mi>op<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>\u2243<\/mml:mo>\n                    <mml:mi>TypeSpaceFunc<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Here <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {CohTheory}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>CohTheory<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the 2-category of positive theories and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {TypeSpaceFunc}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>TypeSpaceFunc<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {CohTheory}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>CohTheory<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is \u2018the same\u2019 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.<\/jats:p>","DOI":"10.1007\/s00153-022-00825-7","type":"journal-article","created":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T17:04:23Z","timestamp":1648659863000},"page":"1-28","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Type space functors and interpretations in positive logic"],"prefix":"10.1007","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0419-7061","authenticated-orcid":false,"given":"Mark","family":"Kamsma","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,30]]},"reference":[{"issue":"3","key":"825_CR1","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1017\/S1755020316000186","volume":"9","author":"TW Barrett","year":"2016","unstructured":"Barrett, T.W., Halvorson, H.: Morita equivalence. 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