In Joel David Hamkins' "Forcing and Large Cardinals" a definition of $extender$ embeddings:


"An embedding $j:V \to M$ is an $extender$ embedding if every element of $M$ can be represented in the form $j(f)(\alpha)$ for some $f:\kappa \to V$ and $\alpha < j(\kappa)$, where $\kappa$ is the critical point of $j$."


Every ultra-power embedding by a measure $\mu$ on a measurable cardinal $\kappa$ is an extender embedding since $\kappa = [id]\_{\mu}$ can be the seed for such an embedding and therefore generate $M$ by representing every element as a $j(f)(\kappa)$ for an $f:\kappa \to V$. Such an infinite cardinal $\kappa$ is, of course, the critical point of $j$. 


However, not every extender embedding is an ultra-power embedding. For such embeddings j: M $\to$ N, how are the images of such embeddings different from ultra-power images? Or, how are the embeddings different? And, what mathematics can be done with extender embeddings which are not generated by it's critical point?



A question related to ultrapower embeddings.
I'm glad you're reading my book (in preparation).


There are a variety of large cardinal notions and large cardinal
embedding types that are witnessed by extender embeddings, but
which cannot be witnessed by ultrapower embeddings by an
ultrafilter on a measurable cardinal $\kappa$.


* Perhaps the easiest example arises from the iterations of a normal
measure $\mu$ on a measurable cardinal. If $j:V\to M$ is the
ultrapower by $\mu$, then in $M$ the cardinal $j(\kappa)$ is a
measurable cardinal with normal measure $j(\kappa)$, and one can
take the ultrapower of $M$ by $j(\mu)$. Eventually, one builds the
system of embeddings $$V\to M\_1\to M\_2\to M\_3\to\cdots,$$ where
$j\_{n,n+1}:M\_n\to M\_{n+1}$ is the ultrapower of $M\_n$ by
$j\_{0,n}(\mu)$, and where $j\_{i,j}$ is defined by composing the
maps at each step. Taking the direct limit of this system, one gets
a map $j\_{0,\omega}:V\to M\_\omega$, and one can show that this
limit is well-founded. This is an elementary embedding, but it is
not an ultrapower embedding by any ultrafilter on $\kappa$, because
$M\_\omega$ is well-founded, but is not closed under
$\kappa$-sequences, indeed, not even under $\omega$-sequences, as
the value $j\_{0,\omega}(\kappa)=\kappa\_\omega=\sup\_n \kappa\_n$,
where $\kappa\_n=j\_{0,n}(\kappa)$ is the critical sequence. That is,
$\kappa\_\omega$ has cofinality $\omega$ in $V$, but is a measurable
cardinal in $M\_\omega$. Meanwhile, however, one can prove that
every element of $M\_\omega$ has the form
$j(f)(\kappa\_0,\kappa\_1,\ldots,\kappa\_n)$ for some function
$f:[\kappa]^{\lt\omega}\to V$, and this observation can be used to
show that $j\_{0,\omega}$ has an extender representation. One can of
course continue the iterations through the ordinals, and these will
all be extender embeddings, and none of them is an ultrapower
embedding by an ultrafilter on $\kappa$.
* One can prove more generally that every extender embedding is the direct limit of the induced system of ordinary ultrapower embeddings. For any $\alpha\lt j(\kappa)$, one forms the hull $X=\{j(f)(\alpha)\mid f:\kappa\to V\}$, and this is an elementary substructure of $M$ containing the range of $j$. Following $j$ with the Mostowski collapse of that structure gives rise to a factor embeding $j\_0:V\to M\_0$, with $k:M\_0\to M$ the inverse collapse of $X$. These embeddings fit together into a directed system of embeddings, by means of Goedel pairing of the ordinals, and the extender embedding $j$ is the direct limit of the system.
* A cardinal $\kappa$ is [$\theta$-strong](http://cantorsattic.info/Strong) if there is an
embedding $j:V\to M$ with critical point $\kappa$ and
$V\_\theta\subset M$. Such kind of embeddings, for
$\kappa+2\leq\theta$, cannot arise from ultrapowers by a measure
on $\kappa$, since $\mu\notin M\_\mu$ for any such ultrapower
$j\_\mu:V\to M\_\mu$. But they can be represented by extender
embeddings, since one simply takes the Mostowski collapse of the
seed hull $\{j(f)(s)\mid f:V\_\kappa\to V, s\in V\_\theta\}\prec M$, and
then observes that an enumeration of $V\_\theta$ turns this into
the form you stated.
* A cardinal $\kappa$ is [$\theta$-tall](http://cantorsattic.info/Tall) if there is an
embedding $j:V\to M$ with critical point $\kappa$ and $\theta\lt j(\kappa)$ and
$M^\kappa\subset M$. Ultrapower embeddings by a measure on
$\kappa$ cannot achieve this when $\theta$ is larger than
$(2^\kappa)^+$, since one can count the number of functions to
get a bound on the size of $j(\kappa)$. But meanwhile, one can
realize tallness with extender embeddings.


