In the book "handbook of set theory", in the chapter "the covering lemma", definition 3.24, Mitchell defines what is mouse.

However he did not give any definition of $\mathcal{U}_\gamma$ and $\mathcal{U}\restriction \gamma$, before he uses them. What is his definition regarding these notations?

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    $\begingroup$ @AlecRhea In my experience, $\upharpoonright$ is most often used for domain restriction, not codomain restriction. But more generally, it is often used for restriction of any kind. $\endgroup$ Nov 29, 2023 at 11:06
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    $\begingroup$ I guess it is implicit that $\mathcal U$ is a sequence of objects indexed by ordinals. $\endgroup$ Nov 29, 2023 at 13:34
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    $\begingroup$ (However, I think there is a condition missing. There would usually be the requirement that if $\mathcal{U}_\gamma\neq\emptyset$ then $\gamma=\kappa^{++\mathrm{Ult}(J_\gamma[\mathcal{U}],\mathcal{U}_\gamma)}$, whereas in 3.24 it just seems to say that $\gamma=\kappa^{++J_\gamma[\mathcal{U}]}$. Of course the latter literally means that $J_\gamma[\mathcal{U}]\models$"$\kappa^+$ is the largest cardinal".) $\endgroup$
    – Farmer S
    Nov 29, 2023 at 14:43
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    $\begingroup$ Have you checked the Chapter "Beginning Inner Model Theory" also written by Mitchell? $\endgroup$
    – Hanul Jeon
    Nov 29, 2023 at 17:27
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    $\begingroup$ That makes sense, but I haven't seen that distinction before, whereas I do often see $f\upharpoonright A$ for the domain restriction case $A\subseteq X$. $\endgroup$ Nov 29, 2023 at 18:56


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