An ultrafilter (or measure)U on P_δ(λ) is fine iff for all \alphaamp;amp;amp;amp;lt;λ we have \{a\inP_δ(λ)\mid \alpha\in a\}\inU.
The ultrafilter U is normal iff it is δ-plete and for all F:P_δ(λ)oλ， if F is regressive U-ae (i.e.， if \{a\mid F(a)\in a\}\inU) then F is constant U-ae， i.e.， there is an \alphaamp;amp;amp;amp;lt;λ such that \{a\mid F(a)=\alpha\}\inU.
δ is superpact iff for all λ there is a normal fine measure U on P_δ(λ).
It is a standard result that δ is superpact iff for all λ there is an elementary embedding j:Vo M with {\rm cp}(j)=δ， j(δ)amp;amp;amp;amp;gt;λ， and j''λ\in M (or， equivalently，{}^λ M\subseteq M).
In fact， given such an embedding j， we can define a normal fine U on P_δ(λ) by
A\inU iff j''λ\in j(A).
Conversely， given a normal fine ultrafilter U on P_δ(λ)， the ultrapower embedding generated by U is an example of such an embedding j. Moreover， if U_j is the ultrafilter on P_δ(λ) derived from j as explained above， then U_j=U.
Another characterization of superpactness was found by Magidor， and it will play a key role in these lectures; in this reformulation， rather than the critical point，δ appears as the image of the critical points of the embeddings under consideration. This version seems ideally designed to be used as a guide in the construction of extender models for superpactness， although recent results suggest that this is， in fact， a red herring.
The key notion we will be studying is the following:
Definition. N\subseteq V is a weak extender model for `δ is superpact’ iff for all λamp;amp;amp;amp;gt;δ there is a normal fine U on P_δ(λ) such that:
P_δ(λ)\cap N\in U， and
U\cap N\in N.
This definition couples the superpactness of δ in N directly with its superpactness in V. In the manuscript， that N is a weak extender model for `δ is superpact’ is denoted by o^N_{\rm long}(δ)=\infty. Note that this is a weak notion indeed， in that we are not requiring that N=L[\vec E] for some (long) sequence \vec E of extenders. The idea is to study basic properties of N that follow from this notion， in the hopes of better understanding how such an L[\vec E] model can actually be constructed.
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