Sh(B)-Valued Models of $$(\kappa ,\kappa )$$-Coherent Categories
Year: 2025
Type: article
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 $$\textbf{Set}$$ 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 $$L_{\omega \omega }^g \subseteq L_{\omega \omega }$$ L ω ω g ⊆ L ω ω this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $$L_{\kappa \kappa }^g$$ L κ κ g when $$\kappa $$ κ is weakly compact. We present some further applications: first, a Sh(B)-valued completeness theorem for $$L_{\kappa \kappa }^g$$ L κ κ g ( $$\kappa $$ κ is weakly compact), second, that $$\mathcal {C}\rightarrow \textbf{Set} $$ C → Set regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.
Source: Applied Categorical Structures
Author: Kristóf Kanalas
Institution: Masaryk University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4408334199