Laver function

If κ is a supercompact cardinal, a Laver function is a function ƒ:κ → V_κ such that for every set x and every cardinal λ ≥ |TC(x)| + κ there is a supercompact measure U on [λ]^<κ such that if j _U is the associated elementary embedding then j _U(ƒ)(κ) = x. (Here V_κ denotes the κ-th level of the cumulative hierarchy, TC(x) is the transitive closure of x)

The original application of Laver functions was the following theorem of Laver. If κ is supercompact, there is a κ-c.c. forcing notion (P, ≤) such after forcing with (P, ≤) the following holds: κ is supercompact and remains supercompact after forcing with any κ-directed closed forcing.

There are many other applications, for example the proof of the consistency of the proper forcing axiom.