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I’ve expanded Urs’s entry on interpretation: models of theories in are interpretations of theories in the internal logic of are (cartesian, regular, coherent, first-order, geometric) functors out of the syntactic categories of those theories. I also mention a concrete notion of interpretation (of models—of possibly different theories—in each other) that came into use among model theorists in the 80s and 90s, when people were trying to reconstruct -categorical theories from the automorphism groups of the countable model. I then mention some facts (I have proofs spelled out somewhere, but I’m sure they’re folklore) linking these concrete interpretations to the first notion of interpretation.
If I recall correctly, there isn’t a unified treatment in the various pages under the contexts of type theory and categorical logic treating the adjunction between taking syntactic categories and internal logics. I’ll try to get all that down in a single page in the future.
Perhaps we should have there something on Olivia Caramello’s bi-interpretation, e.g., as mentioned here. And the concept there of a ’bimodel’.
Wasn’t sure where to post this. Dimitris Tsementzis has a paper out on Homotopy Model Theory: Homotopy Model Theory I: Syntax and Semantics
”A model theory in the framework of Univalent Foundations requires a logic that allows us to define structures on homotopy (n-)types, similar to how first-order logic can define structures on sets. We define such an “n-level” logic for finite n. The syntax is based on a generalization of Makkai’s FOLDS, obtained by an operation that allows us to add equality sorts to FOLDS-signatures. We then give both a set-theoretic and a homotopy type-theoretic semantics for this logic and prove soundness for both with respect to an appropriate deductive system. As an application, we prove that univalent categories are axiomatizable in 1-logic.”
Brice Halimi is also working on a related approach. He has some slides on his website, but the paper isn’t available yet:
Halimi has also been doing some work connecting modal logic and differential geometry which David might be interested in:
Fibred Structures for Modal Logic
Finally, Halvorson and Tsementzis et al. have a series of papers on applications of category theory to question such as theoretical equivalence in philosophy of science:
Categories of Scientific Theories
Glymour and Quine on Theoretical Equivalence
Theoretical Equivalence in Classical Mechanics and its Relationship to Duality
Wasn’t sure where to post this.
In such cases I just try to put these References into aticles in a way that would help me myself to find them again later. Specifically regarding homotopy model theory, why not simply create this entry, cross-link both with homtopy typpe theory and with model theory and then record the references there.
Anyway, thanks for collecting all this!
I have created a web page for Brice Halimi as I know him slightly. One of his papers is in the Annales …. de Toulouse (Proceedings of a Conf. on homotopy and philosophy). I will put links on the new page.
Some of Tsementzis’ slides relating to the work cited above:
Perhaps we should have there something on Olivia Caramello’s bi-interpretation, e.g., as mentioned here. And the concept there of a ’bimodel’.
@David: the notion of interpretation and bi-interpretation defined by Olivia at the end of her 6.3 seems to be precisely the notion of interpretation in the entry.
If I’m reading the linked thread correctly, a ’bimodel’ describes a situation where, given an interpretation , we can take the reduct of any model of to obtain a model of (i.e. precomposing the logical functor by the logical functor .) Is that right?
Jesse, there are people here who know much more than I do, but I think there can be a bimodel without there being an interpretation between theories, just as the existence of a bimodule between two rings doesn’t indicate a morphism between them.
I added a brief remark (which was oddly missing) that interpretations of theories are the inverse image parts of geometric morphisms between their classifying toposes (up to Morleyization and taking pretopos completions of the syntactic sites.)
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