Some contributions to presheaf model theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917254721961984 |
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| author | Brunner, Andreas Morgan, Charles Pinto, Darllan Conceição |
| author_facet | Brunner, Andreas Morgan, Charles Pinto, Darllan Conceição |
| contents | This paper makes contributions to ``pure'' sheaf model theory, the part of model theory in which the models are sheaves over a complete Heyting algebra. We start by outlining the theory in a way we hope is readable for the non-specialist. We then give a careful treatment of the interpretation of terms and formulae. This allows us to prove various preservation results, including strengthenings of the results of \cite{BM14}. We give refinements of Miraglia's work on directed colimits, \cite{M88}, and an analogue of Tarski's theorem on the preservation of $\forall_2$-sentences under unions of chains. We next show various categories whose objects are (pairs of) presheaves and sheaves with various notions of morphism are accessible in the category theoretic sense. Together these ingredients allow us ultimately to prove that these categories are encompassed in the AECats framework for independence relations developed by Kamsma in \cite{K22}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some contributions to presheaf model theory Brunner, Andreas Morgan, Charles Pinto, Darllan Conceição Logic 03C90, 03C40, 03C45, 03C68 This paper makes contributions to ``pure'' sheaf model theory, the part of model theory in which the models are sheaves over a complete Heyting algebra. We start by outlining the theory in a way we hope is readable for the non-specialist. We then give a careful treatment of the interpretation of terms and formulae. This allows us to prove various preservation results, including strengthenings of the results of \cite{BM14}. We give refinements of Miraglia's work on directed colimits, \cite{M88}, and an analogue of Tarski's theorem on the preservation of $\forall_2$-sentences under unions of chains. We next show various categories whose objects are (pairs of) presheaves and sheaves with various notions of morphism are accessible in the category theoretic sense. Together these ingredients allow us ultimately to prove that these categories are encompassed in the AECats framework for independence relations developed by Kamsma in \cite{K22}. |
| title | Some contributions to presheaf model theory |
| topic | Logic 03C90, 03C40, 03C45, 03C68 |
| url | https://arxiv.org/abs/2409.18089 |