A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(μ+), B) → (H(κ+), A) (where H(κ+) is the set of all sets of cardinality hereditarily less than κ+) with critical point μ and j(μ) = κ. Analogously, κ is a quasicompact cardinal if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(κ+), A) → (H(μ+), B) with critical point κ and j(κ) = μ. H(λ) consists of all sets whose transitive closure has cardinality less than λ.
Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards.
(Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.)
Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders.
For current inner models, the elementary embeddings included are determined by their effect on P(κ) (as computed at the stage the embedding is included), where κ is the critical point.