Categoricity and closure of amalgamation bases under unions

Let K\operatorname{\mathcal{K}} be an abstract elementary class, let Hanf(K)\operatorname{Hanf}(\operatorname{\mathcal{K}}) be its Hanf number, and let LS(K)LS(\operatorname{\mathcal{K}}) be its Löwenheim–Skolem number. For a cardinal μ\mu, let Kμam\operatorname{\mathcal{K}}^{am}_\mu be the class of amalgamation bases in K\operatorname{\mathcal{K}} of cardinality μ\mu. Closure conjecture for amalgamation bases. If there exists a λHanf(K)\lambda\geq\operatorname{Hanf}(\operatorname{\mathcal{K}}) such that K\operatorname{\mathcal{K}} is categorical in λ\lambda, then Kμam\operatorname{\mathcal{K}}^{am}_\mu is closed under unions of length <μ+<\mu^+ for all μ\mu with LS(K)μ<λLS(\operatorname{\mathcal{K}})\leq\mu<\lambda. This is presented as a weakening of Grossberg's Intermediate Categoricity Conjecture; the source does not give a resolution.

Sources & referencesView supporting material

Primary source

Monica VanDieren, “Categoricity in Abstract Elementary Classes with No Maximal Models”, arXiv:math/0510579 (2005).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.