Characterizing large cardinals through Neeman's pure side condition forcing

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.