= Critical point of an elementary embedding
{title2=$\operatorname{crit}(j)$}
For an <elementary embedding> between suitable transitive models of <set theory>, its <critical point of an elementary embedding> is the least ordinal $\kappa$ with $j(\kappa)\ne\kappa$, provided such an ordinal exists. Since $j$ preserves ordinal order and fixes every $\alpha<\kappa$, one has $j(\kappa)>\kappa$.
In the large-cardinal setting $j:V_\lambda\to M$ with $\lambda$ inaccessible and $M$ transitive, the embedding fixes every element of the full ambient rank $V_\kappa$. One way to see the required size control is that $\kappa$ is inaccessible in the domain: each lower-rank set has size less than $\kappa$, so an enumeration of length below $\kappa$ is fixed pointwise once its values are fixed. Induction on rank then fixes all of $V_\kappa$.
The reference to the full rank domain matters. An arbitrary inner model may omit ambient sets of rank below $\kappa$, so a claim that its embedding fixes every ambient member of $V_\kappa$ would not even be defined on those omitted sets.
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