Virtual large cardinals

We introduce the concept of virtual large cardinals and apply it to obtain a hierarchy of new large cardinal notions between ineffable cardinals and 0#. Given a large cardinal notion A characterized by the existence of elementary embeddings j:Vα→Vβ satisfying some list of properties, we say that a cardinal is virtuallyA if the embeddings j:VαV→VβV exist in the generic multiverse of V. Unlike their ideological cousins generic large cardinals, virtual large cardinals are actual large cardinals that are compatible with V=L. We study virtual versions of extendible, n-huge, and rank-into-rank cardinals and determine where they fit into the large cardinal hierarchy.