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Definable Categorical Equivalence

Published online by Cambridge University Press:  01 January 2022


This article proposes to explicate theoretical equivalence by supplementing formal equivalence criteria with preservation conditions concerning interpretation. I argue that both the internal structure of models and choices of morphisms are aspects of formalisms that are relevant when it comes to their interpretation. Hence, a formal criterion suitable for being supplemented with preservation conditions concerning interpretation should take these two aspects into account. The two currently most important criteria—generalized definitional equivalence (Morita equivalence) and categorical equivalence—are not optimal in this respect. I put forward a criterion that takes both aspects into account: the criterion of definable categorical equivalence.

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I am indebted to Charlotte Werndl for her invaluable support and advice. I also thank Hajnal Andréka, Thomas Barrett, Alexander Bors, Jeremy Butterfield, Adam Caulton, Erik Curiel, Marton Gömöri, Sam Fletcher, Hannes Leitgeb, Sebastian Lutz, István Németi, Karl-Georg Niebergall, Miklos Redei, Bryan Roberts, and Jim Weatherall as well as two anonymous referees for very helpful feedback and discussions.


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