Sh(B)-valued models of (κ,κ)-coherent categories

2023/06/08 by Kristóf Kanalas, Kanalas, Kristóf
Mathematics · #03C75 #03C90 #18C30 #18F10 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2306.05345

openalex publication_date 2023/06/08 · openalex created_date 2023/06/10 · openalex updated_date 2026/08/01

Abstract

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 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 ⊆ 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 C→ Set regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.

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