Cummings and Shelah developed a generalised notion of the dominating number and used a non-linear iteration of Hechler forcing to fix the dominating number for \(\lambda\) and \(2^\lambda\) for all regular \(\lambda\) with minimal restrictions. We would like to find an inner model for this global property, but the techniques available for finding inner models assuming only \(0^\sharp\) cannot be used with this forcing.

Therefore, in joint work with Sy-David Friedman, we restrict ourselves first to finding an inner model of Global Domination, a global property where the dominating number is less than \(2^\lambda\) for all regular \(\lambda\). Using perfect tree forcing Friedman and I get Global Domination in an inner model for inaccessible cardinals. We would like to extend this to all regular cardinals by sneaking in some Hechler forcing at successors, but run into problems with the mix of forcings at the successors of inaccessibles. The solution has a lot in common with making chocolate mousse.