I am reading section 26 of Kanamori's book The Higher Infinite. This section is about extenders, and I am struggling to understand what are the roles of these specific variables occurred in the definition of extenders. Before this my only knowledge about large cardinals are Jech's chapters on measurable cardinals and ultrapowers. Here is a Lemma I find difficult to understand:
26.1 Lemma.
(a) $M_\mathcal E = \{j_\mathcal E(f)(a) \mid a\in [\beta]^{<\omega} \wedge f\in {}^{[\zeta]^{|a|}}N\cap N\}$.
(b) For any $\gamma$ satisfying $|V_\gamma|^M\le \beta$: $V_\gamma^M\subseteq \operatorname{ran}(k_\mathcal E)$, $V^{M_\mathcal E}_\gamma=V^M_\gamma$, and $k_\mathcal E(x)=x$ for $x\in V^{M_{\mathcal E}}_\gamma$.
(c) $\operatorname{crit}(k_{\mathcal E})\ge \beta$, and so $\operatorname{crit}(j_{\mathcal E})=\kappa$ and $\beta\le j_{\mathcal E}(\zeta)$. If $\beta=j(\zeta)$, then $\operatorname{crit}(k_{\mathcal E})>\beta$, and so $\beta=j_{\mathcal E}(\zeta)$.
Here are my questions:
I suppose (a) says $M_\mathcal E$, the approximation for $M$ we constructed, consists of elements of the form $j_\mathcal E(f)(a)$. Intuitively, what does (b) and (c) say?
For the proof of (b), I can follow along the way to show $V_\gamma^M\subseteq \operatorname{ran}(k_\mathcal E)$. Henceforth, the only thing that matters is that $k_{\mathcal E}$ fixes elements in $x\in V_\gamma^{M_{\mathcal E}}$. But how does "the rest follows since the inverse of the inverse of $k_\mathcal E$ is just the collapsing isomorphism: $\operatorname{ran}(k_\mathcal E)\to M_\mathcal E$"? Specifically,
- what is the natural collapsing isomorphism from the range of $k_\mathcal E$ to $M_\mathcal E$?
- why is the inverse of $k_\mathcal E$ the collapsing isomorphism?
- how does this fact help prove the rest of (b)?