If you know what a supercompact cardinal is, there are many possibilities that come to mind for what the phrase “\(\Sigma_2\)-supercompact” could mean. Noting that \(\textrm{HOD}\) is \(\Sigma_2\)-definable, it is natural to throw \(\textrm{HOD}\)-supercompactness and supercompactness in \(\textrm{HOD}\) into this mix, particularly given the interest in large cardinals in \(\textrm{HOD}\) arising from Woodin’s \(\textrm{HOD}\) conjecture.
We report on joint work with Shoshana Friedman disentangling these various possibilities, in part building on work of Bea Adam-Day [1], with models separating examples of these variants that are inequivalent.

Bibliography

1. Beatrice Adam-Day,Indestructibility and \(C^{(n)}\)-Supercompact Cardinals,PhD Thesis, University of Leeds, 2024.