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 on extender-non-measurable-ultra-power embeddings?

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?

In Joel David Hamkins' "Forcing and Large Cardinasls" 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 the embeddings are different? And, what mathematics can be done in an extender-non-measurable-ultra-power embeddings?

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 on extender-non-measurable-ultra-power embeddings?