Appears in collection : XVII International Luminy Workshop in Set Theory / XVII Atelier International de Théorie des Ensembles

Universally Baire sets play an important role in studying canonical models with large cardinals. But to reach higher large cardinals more complicated objects, for example, canonical subsets of universally Baire sets come into play. Inspired by core model induction, we introduce the definable powerset $A^{\infty }$ of the universally Baire sets $\Gamma ^{\infty }$ and show that, after collapsing a large cardinal, $L(A^{\infty })$ is a model of determinacy and its theory cannot be changed by forcing. Moreover, we show a similar result for adding a club filter to the model constructed over universally Baire sets.