Positively closed Sh(B)-valued models

2024/09/17 by Kanalas, Kristóf
#Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2409.11231

Abstract

We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For Set-valued models of coherent theories they coincide. We prove that if E=Sh(X) for an extremally disconnected Stone space (or equivalently E=Sh(B,τ_coh) for a complete Boolean algebra) then i) E-valued types can be realized by E-valued models, and ii) positively closed but not strongly positively closed E-valued models (of coherent theories) exist, yet, there is an alternative local property that characterizes positively closed E-valued models. A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment L^g_κκ where κ is weakly compact.

Related

• Positive model theory of interpretations
• Sh(B)-valued models of (κ,κ)-coherent categories
• Positive model theory and infinitary logic
• Almost positively closed models
• Beautiful pairs