$Sh(B)$-valued models of $(κ,κ)$-coherent categories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915318608166912 |
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| author | Kanalas, Kristóf |
| author_facet | Kanalas, Kristóf |
| contents | A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/ sites.
As an application we identify $\mathbf{Set}$-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as "$Sh(B)$-valued models"). For the coherent fragment $L_{ωω}^g \subseteq L_{ωω}$ this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $L_{κκ}^g$ when $κ$ is weakly compact.
We present some further applications: first, a $Sh(B)$-valued completeness theorem for $L_{κκ}^g$ ($κ$ is weakly compact), second, that $\mathcal{C}\to \mathbf{Set} $ regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_05345 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $Sh(B)$-valued models of $(κ,κ)$-coherent categories Kanalas, Kristóf Category Theory 18F10, 18C30, 03C90, 03C75 A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/ sites. As an application we identify $\mathbf{Set}$-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as "$Sh(B)$-valued models"). For the coherent fragment $L_{ωω}^g \subseteq L_{ωω}$ this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $L_{κκ}^g$ when $κ$ is weakly compact. We present some further applications: first, a $Sh(B)$-valued completeness theorem for $L_{κκ}^g$ ($κ$ is weakly compact), second, that $\mathcal{C}\to \mathbf{Set} $ regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors. |
| title | $Sh(B)$-valued models of $(κ,κ)$-coherent categories |
| topic | Category Theory 18F10, 18C30, 03C90, 03C75 |
| url | https://arxiv.org/abs/2306.05345 |