  We show that some of the most prominent large cardinal notions can be
characterized through the validity of certain combinatorial principles at
$\omega_2$ in forcing extensions by the pure side condition forcing introduced
by Neeman. The combinatorial properties that we make use of are natural
principles, and in particular for inaccessible cardinals, these principles are
equivalent to their corresponding large cardinal properties. Our
characterizations make use of the concepts of internal large cardinals
introduced in this paper, and of the classical concept of generic elementary
embeddings.
