Positively closed $Sh(B)$-valued models
Kristóf Kanalas
TL;DR
This work develops a topos-valued positive model theory for coherent theories, distinguishing globally defined positively closed models from locally defined strong positivity; it shows that in $Sh(B,τ_{coh})$ with a complete Boolean algebra $B$ these notions may diverge, yet a local criterion characterizes positive closure. The authors introduce type space functors $S_{\mathcal{C}}^{Sh(\mathcal{D})}$ and a minimal natural transformation $tp_M:M\Rightarrow S_{\mathcal{C}}$, then establish a factorization result: under the assumption that $Sh(B)$ can realize $\kappa$-types, any $M$ admits a factorization through a suitable $N$ with $\beta \le tp_N\circ\alpha$. Central to the analysis is the lattice $L^1(\Gamma M)$, the left Kan extension of subobject lattices, which encodes Lindenbaum–Tarski style information; strong positive closure is equivalent to $L^1(\Gamma M)$ being a Boolean algebra (and isomorphic to $B$ when complemented inside), while positive closure is captured by a retract-type condition and a unique $tp_M$ when certain realizability hypotheses hold. The paper provides explicit counterexamples (notably with $Open(Q)$ and $Closed(X)$) showing positive but not strong positive closure in $Sh(B)$-valued settings, and an internal example using an extremally disconnected Stone space. Overall, the results give both algebraic and geometric criteria for positivity properties in topos-valued models, and extend the theory to the infinitary fragment $L^g_{\kappa\kappa}$ with $\kappa$ weakly compact.
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 $\mathbf{Set}$-valued models of coherent theories they coincide. We prove that if $\mathcal{E}=Sh(B,τ_{coh})$ for a complete Boolean algebra, then positively closed but not strongly positively closed $\mathcal{E}$-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed $\mathcal{E}$-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.