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\chapter{Virtual large cardinals}
\label{chapter.virtual-large-cardinals}
\thispagestyle{fancy}

In this chapter we investigate the properties of virtual versions of well-known large cardinals, including measurables, strongs, supercompacts, Woodins and Vop\v enkas. This entails firstly analysing the relationships between them, and secondly looking at more general properties in terms of their behaviour in core models as well as their indestructibility. This virtual perspective also allows us to analyse virtualised versions of large cardinals that are otherwise inconsistent with $\zfc$, such as the Berkeley cardinals.

\section{Strongs \& supercompacts}

We start out with measurables, strongs and supercompacts. Their (non-virtual) definitions can be found in Section \ref{prelims.large-cardinals}.

\defi[][defi.strongsc]{
  For $\theta$ a regular uncountable cardinal, a cardinal $\kappa<\theta$ is...
  \begin{itemize}
    \item \textbf{faintly $\theta$-measurable} if, in a forcing extension of $V$, there is a transitive set $\N$ and an elementary embedding $\pi\colon H_\theta^V\to\N$ with $\crit\pi=\kappa$;
    \item \textbf{faintly $\theta$-strong} if, in a forcing extension of $V$, there is a transitive set $\N$ with $H_\theta^V\subset\N$ and an elementary embedding $\pi\colon H_\theta^V\to\N$ with $\crit\pi=\kappa$;
    \item \textbf{faintly $\theta$-supercompact} if, in a forcing extension of $V$, there is a transitive set $\N$ with ${^{<\theta}\N}\cap V\subset\N$ and an elementary embedding $\pi\colon H_\theta^V\to\N$ with $\crit\pi=\kappa$.
  \end{itemize}

  We further replace ``faintly'' by \textbf{virtually} when $\N\subset V$, we attach a \textbf{``pre''} if we do not assume that $\pi(\kappa)>\theta$, and we will leave out $\theta$ when it holds for all regular $\theta>\kappa$.
}

As a quick example of this terminology, a \textit{faintly prestrong cardinal} is a cardinal $\kappa$ such that for all regular $\theta>\kappa$, $\kappa$ is faintly $\theta$-measurable with $H_\theta^V\subset\N$.

\qquad Observe that whenever we have a virtual large cardinal that has its defining property for all regular $\theta$, we can assume that the target of the embedding is an \textit{element} of the ground model $V$ and not just a subset of $V$. Suppose, for instance, that $\kappa$ is virtually measurable and fix a regular $\theta>\kappa$ and set $\lambda:=(2^{<\theta})^+$. Take a generic elementary embedding $\pi:H_\lambda\to \M_\lambda$ witnessing that $\kappa$ is virtually $\lambda$-measurable.

\qquad Since $\abs{H_\theta}=2^{<\theta}$ it holds that $H_\theta\in H_\lambda$, so that the restriction $\pi\restr H_\theta\colon H_\theta\to\pi(H_\theta)$ witnesses that $\kappa$ is virtually $\theta$-measurable, and the target model $\M_\theta := \pi(H_\theta)$ is in $V$ because $\M_\lambda\subseteq V$ by assumption. Thus, the weaker assumption that the target model is a subset of the ground model only affects level-by-level virtual large cardinals. Indeed, as we will see in later sections, for virtually strong cardinals we may even further weaken the assumption that $\M_\theta\subseteq V$ to $H_\theta=H_\theta^{\M_\theta}$ (we do not know whether this holds level-by-level).

\qquad We note that even small cardinals can be faintly measurable: we may for instance have a precipitous ideal on $\omega_1$; see \cite[Theorem 22.33]{Jech}. The ``virtually'' adverb further implies that the cardinals are large cardinals in the usual sense, as Proposition \ref{prop.virtit} below shows.

\prop[Virtualised folklore][prop.virtit]{
  For any regular uncountable cardinal $\theta$, every virtually $\theta$-measurable cardinal is 1-iterable\footnote{See Section \ref{prelims.large-cardinals} for a definition of the 1-iterable cardinals.} (in particular, inaccessible).
}
\proof{
  Let $\kappa$ be virtually $\theta$-measurable, witnessed by a forcing $\mathbb P$, a transitive $\N\subset V$ and an elementary $\pi\colon H_\theta^V\to\N$ with $\pi\in V^{\mathbb P}$. If $\kappa$ is not a strong limit then we have a surjection $\pi(f)\colon\p(\alpha)\to\pi(\kappa)$ with $\ran\pi(f)=\ran f\subset\kappa$ for some $\alpha<\kappa$, $\contr$. Note that we used $\N\subset V$ to ensure that $\p(\alpha)^V=\p(\alpha)^{\N}$. The same argument shows that $\kappa$ is regular. By restricting the generic embedding and using that $\p(\kappa)^V = \p(\kappa)^N$ as $\N\subset V$ and $\p(\kappa)^V\subset\N$, we get that $\kappa$ is 1-iterable.
}

Along with the above definition of faint supercompactness we can also virtualise Magidor's characterisation of supercompact cardinals\footnote{See Section \ref{prelims.large-cardinals} for a definition of the non-virtual version of this characterisation.}, which was one of the original characterisations of the remarkable cardinals in \cite{Schindler}.

\defi{
  Let $\theta$ be a regular uncountable cardinal. Then $\kappa<\theta$ is \textbf{virtually $\theta$-Magidor-supercompact} if there are cardinals $\bar\theta<\kappa$ and $\bar\kappa<\bar\theta$, and a generic elementary $\pi\colon H_{\bar\theta}^V\to H_\theta^V$ such that $\crit\pi=\bar\kappa$ and $\pi(\bar\kappa)=\kappa$.
}

\cite{GitmanSchindler} observed that remarkable cardinals are precisely the virtually supercompacts. Surprisingly, they are also precisely the virtually strongs, which in turn makes virtually strongs and virtually supercompacts equivalent. The proofs of these equivalences were omitted in \cite{GitmanSchindler}, so we give proofs of these here. These are not the original proofs, however, but a slight improvement that allows us to get a more fine-grained level-wise corollary --- see Remark \ref{rema.levelwise-strong-supercompact}.

\theo[\cite{GitmanSchindler}][theo.rem]{
  For an uncountable cardinal $\kappa$, the following are equivalent:
  \begin{enumerate}
    \item $\kappa$ is virtually strong;
    \item $\kappa$ is virtually supercompact;
    \item $\kappa$ is virtually Magidor-supercompact.
  \end{enumerate}
}
\proof{
$(ii)\Rightarrow(i)$ is simply by definition.

\qquad $(i)\Rightarrow (iii)$: Fix $\theta>\kappa$. By $(i)$ there exists a generic elementary embedding $\pi\colon H_{(2^{<\theta})^+}^V\to\M$ with\footnote{The domain of $\pi$ is $H_{(2^{<\theta})^+}^V$ to ensure that $H_\theta^V\in\dom\pi$.} $\crit\pi=\kappa$, $\pi(\kappa)>\theta$, $H_{(2^{<\theta})^+}^V\subset\M$ and $\M\subset V$. Since $H_\theta^V, H_{\pi(\theta)}^{\M}\in\M$, Countable Embedding Absoluteness \ref{lemm.ctblabs} implies that $\M$ has a generic elementary embedding $\pi^*:H_\theta^V\to H_{\pi(\theta)}^{\M}$ with $\crit\pi^*=\kappa$ and $\pi^*(\kappa)=\pi(\kappa)>\theta$. Since $H_\theta^V=H_\theta^{\M}$ as $\M\subset V$ and $H_\theta^V\subset\M$, elementarity of $\pi$ now implies that $H_{(2^{<\theta})^+}^V$ has cardinals $\bar\theta<\kappa$ and $\bar\kappa<\bar\theta$, and a generic elementary $\sigma\colon H_{\bar\theta}^V\to H_\theta^V$ with $\crit\sigma=\bar\kappa$ and $\sigma(\bar\kappa)=\kappa$. This shows $(iii)$.

  \qquad $(iii)\Rightarrow (ii)$: Fix $\theta>\kappa$ and set $\delta:=(2^{<\theta})^+$. By $(iii)$ there exist cardinals $\bar\delta<\kappa$ and $\bar\kappa<\bar\delta$, and a generic elementary embedding $\pi\colon H_{\bar\delta}^V\to H_\delta^V$ with $\crit\pi=\bar\kappa$ and $\pi(\bar\kappa)=\kappa$. We will argue that $\bar\kappa$ is virtually $\bar\theta$-supercompact in $H_{\bar\delta}^V$, so that by elementarity $\kappa$ is virtually $\theta$-supercompact in $H_\delta^V$ and hence also in $V$ by the choice of $\delta$. Consider the restriction
  \eq{
    \sigma:=\pi\restr H_{\bar\theta}^V\colon H_{\bar\theta}^V\to H_\theta^V.
  }

  Note that $H_\theta^V$ is closed under ${<}\bar\theta$-sequences (and more) in $V$. Now define
  \eq{
    X := \bar\theta{+}1 \cup \{x\in H_\theta^V\mid\exists y\in H_{\bar\theta}^V\exists p\in\col(\omega, H_{\bar\theta}^V)\colon p\forces\dot\sigma(\check y)=\check x\}\in V.
  }

  Note that $\abs X = \abs{H_{\bar\theta}^V} = 2^{<\bar\theta}$ and that $\ran\sigma\subset X$. Now let $\overline\M\prec H_\theta^V$ be such that $X\subset\overline\M$ and $\overline\M$ is closed under ${<}\bar\theta$-sequences. Note that we can find such an $\overline\M$ of size $(2^{<\bar\theta})^{<\bar\theta} = 2^{<\bar\theta}$. Let $\M$ be the transitive collapse of $\overline\M$, so that $\M$ is still closed under ${<}\bar\theta$-sequences and we also still have that $\abs\M = 2^{<\bar\theta}<\bar\delta$, making $\M\in H_{\bar\delta}^V$.

  \qquad Countable Embedding Absoluteness \ref{lemm.ctblabs} then implies that $H_{\bar\delta}^V$ has a generic elementary embedding $\sigma^*\colon H_{\bar\theta}^V\to\M$ with $\crit\sigma^*=\bar\kappa$, showing that $\bar\kappa$ is virtually $\bar\theta$-supercompact in $H_{\bar\delta}^V$, which is what we wanted to show.
}

\rema[][rema.levelwise-strong-supercompact]{
  As mentioned above, the proof in fact shows something stronger: if $\kappa$ is virtually $(2^{<\theta})^+$-strong then it is virtually $\theta$-supercompact, and if it is virtually $(2^{<\theta})^+$-Magidor-supercompact then it is virtually $\theta$-supercompact. It is open whether these notions are equivalent level-by-level (see Question \ref{ques.remarkableequiv}).
}

As a corollary of the proof, we obtain the following weaker characterization of virtually strong cardinals:

\qprop[][prop.weakerstrong]{
    A cardinal $\kappa$ is virtually strong if and only if for every $\theta>\kappa$ there is a forcing extension, in which there is a transitive set $\N$ and an elementary embedding $\pi:H_\theta\to \N$ with $\crit \pi=\kappa$ and $H_\theta=H_\theta^{\N}$.
}

A key difference between the normal large cardinals and the virtual kinds is that we do not have a virtual version of the Kunen inconsistency\footnote{See Section \ref{prelims.large-cardinals} for a definition of the Kunen inconsistency.}: it is perfectly possible to have a generic elementary embedding $H_\theta^V\to H_\theta^V$ with $\theta$ much larger than the critical point, and thus in particular also the image of the critical point. Here is an example of such a virtualised Kunen inconsistency.

\prop[Folklore]{
  If $0^\sharp$ exists then there are inaccessible cardinals $\kappa<\theta$ such that, in a generic extension of $L$, there is an elementary embedding
  \eq{
    \pi\colon L_\theta\to L_\theta.
  }

  In other words, $\pi$ witnesses a strong failure of the virtualised Kunen inconsistency.
}
\proof{
  From $0^\sharp$ we get an elementary embedding $j\colon L\to L$. Let $C\subset\on$ be the proper class club of limit points of $j$ above $\crit j$, which then contains an $L$-inaccessible cardinal $\theta$ as there are stationarily many such. Restrict $j$ to $\pi:=j\restr L_\theta\colon L_\theta\to\N$ and note that $\N=L_\theta$ by condensation of $L$ and because $\theta$ is a limit point of $j$. Let $\kappa:=\crit\pi$. Now an application of Countable Embedding Absoluteness \ref{lemm.ctblabs} shows that a generic extension of $L$ contains an elementary embedding $\tilde\pi: L_\theta\to L_\theta$ with $\crit\tilde\pi = \kappa$.
}

This becomes important when dealing with the ``pre''-versions of the large cardinals. We next move to a virtualisation of the $\alpha$-superstrong cardinals.

\defi{
  Let $\theta$ be a regular uncountable cardinal and $\alpha$ an ordinal. Then a cardinal $\kappa<\theta$ is \textbf{faintly $(\theta,\alpha)$-superstrong} if it is faintly $\theta$-measurable, $H_\theta^V\subset\N$ and $\pi^\alpha(\kappa)\leq\theta$.\footnote{Here $\pi^1 = \pi$, $\pi^{\alpha+1}=\pi\circ\pi^{\alpha}$ and $\pi^\alpha(\kappa) = \sup_{\xi<\alpha}\pi^\xi(\kappa)$ when $\alpha$ is a limit ordinal.} We replace ``faintly'' by \textbf{virtually} when $\N\subset V$, we say that $\kappa$ is \textbf{faintly $\alpha$-superstrong} if it is faintly $(\theta,\alpha)$-superstrong for \textit{some} $\theta$, and $\kappa$ is simply \textbf{faintly superstrong} if it is faintly 1-superstrong.\footnote{Note that the conventions stated here are different from the ones in Definition \ref{defi.strongsc}.}
}

As in the non-virtual case, the virtually superstrongs surpass the virtually strongs in consistency strength. This then also implies that the superstrongs are stronger than the virtually supercompacts, which is \textit{not} the case outside the virtual world.

\prop[N.][prop.superstrong]{
  If $\kappa$ is faintly superstrong then $H_\kappa$ has a proper class of virtually strong cardinals, and thus also a proper class of virtually supercompact cardinals.
}
\proof{
  Fix a regular $\theta>\kappa$ and a generic embedding $\pi\colon H_\theta^V\to\N$ with $\crit\pi=\kappa$, $H_\theta^V\subset\N$ and $\pi(\kappa)\leq\theta$. Then $\pi(\kappa)$ is a $V$-cardinal, so that $H_{\pi(\kappa)}^V$ thinks that $\kappa$ is virtually strong. This implies that $H_\kappa^V$ thinks there is a proper class of virtually strong cardinals, using that $H_\kappa^V\prec H_{\pi(\kappa)}^V$.
}

The following theorem and its subsequent corollaries then show that the only thing stopping prestrongness from being equivalent to strongness is the existence of virtualised Kunen inconsistencies:

\theo[N.][theo.virtchar]{
  Let $\theta$ be an uncountable cardinal. Then a cardinal $\kappa<\theta$ is virtually $\theta$-prestrong iff either
  \begin{enumerate}
    \item $\kappa$ is virtually $\theta$-strong; or
    \item $\kappa$ is virtually $(\theta,\omega)$-superstrong.
  \end{enumerate}
}
\proof{
  $(\Leftarrow)$ is trivial, so we show $(\Rightarrow)$. Let $\kappa$ be virtually $\theta$-prestrong. Assume $(i)$ fails, meaning that there is a generic elementary embedding \hbox{$\pi\colon H_\theta\to\N$} for some transitive $\N\subseteq V$ with $H_\theta\subset\N$, $\crit\pi=\kappa$ and $\pi(\kappa)\leq\theta$.

  \qquad First, assume that there is some $n<\omega$ such that $\pi^n(\kappa)=\theta$. The proof of Proposition~\ref{prop.superstrong} shows that $\kappa$ is virtually strong in $H_{\pi(\kappa)}$. It follows that $\pi(\kappa)$ is virtually strong in $H_{\pi^2(\kappa)}$ by elementarity, and by applying elementarity repeatedly we get that $\pi^n(\kappa)=\theta$ is virtually strong in $\N$. Note that the condition $\pi^n(\kappa)=\theta$ implies that $\theta$ is inaccessible in $\N$, and hence a limit cardinal there.

  \qquad In particular, $\theta$ is virtually $\delta:=(\theta^+)^{\N}$-strong in $\N$, so that $\N$ has a generic elementary embedding $\sigma:H_\delta^{\N}\to \M$ with $\crit\sigma=\theta$ and $H_\theta^V\subseteq H_\delta^{\N}\subseteq\M$. Thus, $H_\theta^V\prec H_{\sigma(\theta)}^{\M}$, from which it follows that $\kappa$ is virtually strong in $H_{\sigma(\theta)}^{\M}$ and, in particular, virtually $\theta$-strong. But $H_{\sigma(\theta)}^{\M}$ must be correct about this since $H_\theta^{\M}=H_\theta^{\N}=H_\theta$, from which we can conclude that $\kappa$ is actually virtually $\theta$-strong, contradicting our assumption that $(i)$ fails.

  \qquad Next, assume that there is a least $n<\omega$ such that $\pi^{n+1}(\kappa)>\theta$. Since $\pi(\kappa)\leq\theta$, we have as before that  $\kappa$ is virtually strong in $H_{\pi(\kappa)}$. Since $H_{\pi^i(\kappa)}\prec H_{\pi^{i+1}(\kappa)}$ holds by elementarity, we have that $\kappa$ is virtually strong in $H_{\pi^n(\kappa)}$. Applying elementarity to the statement that $\kappa$ is virtually strong in $H_{\pi(\kappa)}$ we also get that  $\pi^n(\kappa)$ is virtually strong in $H_{\pi^{n+1}(\kappa)}^{\N}$.

  \qquad This means that there is some generic elementary embedding $\sigma\colon H_\theta\to\M$ with $H_\theta\subset\M$, $\M\subseteq H_{\pi^{n+1}(\kappa)}^{\N}$, $\crit\sigma=\pi^n(\kappa)$ and $\sigma(\pi^n(\kappa))>\theta$. Thus, again by elementarity, we get that $H_{\pi^n(\kappa)}  \prec H_{\sigma(\pi^n(\kappa))}^{\M}$. Since, as we already argued, $\kappa$ is virtually strong in $H_{\pi^n(\kappa)}$, this means that $\kappa$ is also virtually strong in $H_{\sigma(\pi^n(\kappa))}^{\M}$ and as $H_\theta^{\M} = H_\theta^{\N} = H_\theta$, this means that $\kappa$ is actually virtually $\theta$-strong, contradicting our assumption that $(i)$ fails.

 \qquad Finally, assume $\pi^n(\kappa)<\theta$ for all $n<\omega$ and let $\lambda = \sup_{n<\omega}\pi^n(\kappa)$. Since $\lambda\leq\theta$, we have that $\kappa$ is virtually $(\theta, \omega)$-superstrong by definition.
}

To get a better intuition for the virtual $\omega$-superstrongs, recall that a cardinal $\kappa$ is \textbf{virtually rank-into-rank} if there exists a cardinal $\theta>\kappa$ and a generic elementary embedding $\pi\colon H_\theta^V\to H_\theta^V$ with $\crit\pi=\kappa$. We then note that the virtually $\omega$-superstrongs coincide with the virtually rank-into-ranks.

\prop[N.][prop.superstrong-rank-into-rank]{
  A regular uncountable cardinal $\kappa$ is virtually $\omega$-superstrong iff it is virtually rank-into-rank.
}
\proof{
  If $\kappa$ is virtually $\omega$-superstrong, witnessed by a generic elementary embedding $\pi\colon H_\theta^V\to\N$, then $\lambda:=\sup_{n<\omega}\pi^n(\kappa)$ is well-defined. By restricting $\pi$ to $\pi\restr H_\lambda^V\colon H_\lambda^V\to H_\lambda^V$ we get a witness to $\kappa$ being virtually $\lambda$-rank-into-rank. Conversely, if $\kappa$ is $\theta$-rank-into-rank, witnessed by a generic embedding $\pi\colon H_\theta^V\to H_\theta^V$, then one readily checks that $\pi$ also witnesses that $\kappa$ is virtually $\omega$-superstrong.
}

Theorem \ref{theo.virtchar} also gives us the following surprising consistency result:

\coro[N.][coro.strong-meas-equiconsistent]{
  For any uncountable regular $\theta$, the existence of a virtually $\theta$-strong cardinal is equiconsistent with the existence of a faintly $\theta$-measurable cardinal.
}
\proof{
  The above Proposition \ref{prop.superstrong} and Theorem \ref{theo.virtchar} show that virtually $\theta$-prestrongs are equiconsistent with virtually $\theta$-strongs. Now note that Countable Embedding Absoluteness \ref{lemm.ctblabs} and condensation in $L$ imply that every faintly $\theta$-measurable cardinal is virtually $\theta$-prestrong in $L$.
}

\section{Woodins \& Vop\v enkas}

In this section we will analyse the virtualisations of the Woodin and Vop\v enka cardinals, which are defined using ``boldface'' variants of strongs and supercompacts.

\defi{
  Let $\theta$ be a regular uncountable cardinal. Then a cardinal $\kappa<\theta$ is \textbf{faintly $(\theta,A)$-strong} for a set $A\subset H_\theta^V$ if there exists a generic elementary embedding
\eq{
  \pi\colon (H_\theta^V,\in,A)\to(\M,\in,B)
}

with $\M$ transitive, such that $\crit\pi=\kappa$, $\pi(\kappa)>\theta$, $H_\theta^V\subset\M$ and $B\cap H_\theta^V = A$. $\kappa$ is \textbf{faintly $(\theta,A)$-supercompact} if we further have that ${^{<\theta}\M}\cap V\subset\M$ and say that $\kappa$ is \textbf{faintly $(\theta,A)$-extendible} if $\M=H_\mu^V$ for some $V$-cardinal $\mu$. We will leave out $\theta$ if it holds for all regular $\theta>\kappa$.
}

\defi{
  \label{defi.woodin}
  A cardinal $\delta$ is \textbf{faintly Woodin} if given any $A\subset H_\delta^V$ there exists a faintly $({<}\delta,A)$-strong cardinal $\kappa<\delta$.
}

As with the previous definitions, for both of the above two definitions we substitute ``faintly'' for \textbf{virtually} when $\M\subset V$, and substitute ``strong'', ``supercompact'' and ``Woodin'' for \textbf{prestrong}, \textbf{presupercompact} and \textbf{pre-Woodin} when we do not require that $\pi(\kappa)>\theta$.

\qquad We note in the following proposition that, in analogy with the real Woodin cardinals, virtually Woodin cardinals are Mahlo. This contrasts the virtually pre-Woodins since \cite{WilsonVopenka}, together with Theorem \ref{theo.vopwood} below, show that these can be singular.

\prop[Virtualised folklore][prop.woodmahlo]{
  Virtually Woodin cardinals are Mahlo.
}
\proof{
  Let $\delta$ be virtually Woodin. Note that $\delta$ is a limit of weakly compact cardinals by Proposition \ref{prop.virtit}, making $\delta$ a strong limit. As for regularity, assume that we have a cofinal increasing function $f\colon\alpha\to\delta$ with $f(0)>\alpha$ and $\alpha<\delta$, and note that $f$ cannot have any closure points since $f(0) > \alpha$ and $f$ is increasing. Fix a virtually $({<}\delta,f)$-strong cardinal $\kappa<\delta$; we claim that $\kappa$ is a closure point for $f$, which will yield our desired contradiction.

  \qquad Let $\gamma<\kappa$ and choose a regular $\theta \in (\max(f(\gamma), \kappa), \delta)$. We then have a generic embedding $\pi\colon (H_\theta^V,\in,f\cap H_\theta^V)\to(\N,\in,f^+)$ with $H_\theta^V\subset\N$, $\N\subset V$, $\crit\pi=~\kappa$, $\pi(\kappa)>\theta$ and $f^+$ is a function such that $f^+\cap H_\theta^V = f\cap H_\theta^V$. But then $f^+(\gamma) = f(\gamma) < \pi(\kappa)$ by our choice of $\theta$, so elementarity implies that $f(\gamma)<\kappa$, making $\kappa$ a closure point for $f$, $\contr$. This shows that $\delta$ is inaccessible.

  \qquad As for Mahloness, let $C\subset\delta$ be a club and $\kappa<\delta$ a virtually $({<}\delta,C)$-strong cardinal. Let $\theta \in (\min C,\delta)$ and let $\pi\colon H_\theta^V\to\N$ be the associated generic elementary embedding. Then for every $\gamma<\kappa$ there exists an element of $C$ below $\pi(\kappa)$, namely $\min C$, so by elementarity $\kappa$ is a limit of elements of $C$, making it an element of $C$. As $\kappa$ is regular, this shows that $\delta$ is Mahlo.
}

\qquad The well-known equivalence of the ``function definition'' and ``$A$-strong'' definition of Woodin cardinals\footnote{See Section \ref{prelims.large-cardinals} for this characterisation of (non-virtual) Woodin cardinals.} holds if we restrict ourselves to \textit{virtually} Woodins, and the analogue of the equivalence between virtually strongs and virtually supercompacts allows us to strengthen this:

\prop[Dimopoulos-Gitman-N.][prop.woodin]{
  For an uncountable cardinal $\delta$, the following are equivalent:
  \begin{enumerate}
    \item $\delta$ is virtually Woodin;
    \item for every $A\subset H_\delta^V$ there exists a virtually $({<}\delta,A)$-supercompact $\kappa<\delta$;
    \item for every $A\subset H_\delta^V$ there exists a virtually $({<}\delta,A)$-extendible $\kappa<\delta$;
    \item for every function $f\colon\delta\to\delta$ there are regular cardinals $\kappa<\theta<\delta$, where $\kappa$ is a closure point for $f$, and a generic elementary $\pi\colon H_\theta^V\to\M$ such that $\crit\pi=\kappa$, $H_\theta^V\subset\M$, $\M\subset V$ and $\theta = \pi(f\restr\kappa)(\kappa)$;
    \item for every function $f\colon\delta\to\delta$ there are regular cardinals $\kappa<\theta<\delta$, where $\kappa$ is a closure point for $f$, and a generic elementary $\pi\colon H_\theta^V\to\M$ such that $\crit\pi=\kappa$, ${^{<\pi(f)(\kappa)}\M}\subset\M$, $\M\subset V$ and $\theta = \pi(f\restr\kappa)(\kappa)$;
    \item for every function $f\colon\delta\to\delta$ there are regular cardinals $\bar\theta<\kappa<\theta<\delta$, where $\kappa$ is a closure point for $f$, and a generic elementary embedding $\pi\colon H_{\bar\theta}^V\to H_\theta^V$ with $\pi(\crit\pi)=\kappa$, $f(\crit\pi)=\bar\theta$ and $f\restr\kappa\in\ran\pi$.
  \end{enumerate}
}

\begin{figure}
  \begin{center}
    \includegraphics[scale=0.4]{\string~/gitsky/phd/gfx/woodin_equivalences.pdf}
    \caption{Proof strategy of Proposition \ref{prop.woodin}, dotted lines are trivial implications.}
  \end{center}
\end{figure}

\proof{
  Firstly note that $(iii)\Rightarrow(ii)\Rightarrow(i)$ and $(v)\Rightarrow(iv)$ are simply by definition.

  \qquad\framebox{$(i)\Rightarrow(iv)$} Assume $\delta$ is virtually Woodin, and fix a function $f\colon\delta\to\delta$. Let $\kappa<\delta$ be virtually $({<}\delta, f)$-strong and let $\theta<\delta$ be a regular cardinal such that $\sup_{\alpha\leq\kappa}f(\alpha)<\theta$. Then there is a generic elementary embedding
  \eq{
    \pi\colon (H_\theta, \in, f\cap H_\theta) \to (\M, \in, f^+)
  }

  such that $H_\theta\subseteq \M$, $f\cap H_\theta=f^+\cap H_\theta$, $\M\subset V$, and $\pi(\kappa)>\theta$. Note that, by our choice of $\theta$, $f\restr\kappa\in H_\theta^V$ and $\pi(f\restr\kappa)(\kappa)=f^+(\kappa)=f(\kappa)<\theta$.

 \qquad So it suffices to show that $\kappa$ is a closure point for $f$. Let $\alpha<\kappa$. Then
  \eq{
    f(\alpha)=f^+(\alpha)=\pi(f\restr\kappa)(\alpha)=\pi(f\restr\kappa)(\pi(\alpha))=\pi(f(\alpha)),
  }

  so $\pi$ fixes $f(\alpha)$ for every $\alpha<\kappa$. Now, if $\kappa$ was not a closure point of $f$ then, letting $\alpha<\kappa$ be the least such that $f(\alpha)\geq\kappa$, we have
  \eq{
    \theta > f(\alpha) = \pi(f(\alpha)) \geq \pi(\kappa) > \theta,
  }

  a contradiction. Note that we used that $\pi(\kappa)>\theta$ here, so this argument would not work if we had only assumed $\delta$ to be virtually pre-Woodin.

  \qquad \framebox{$(iv)\Rightarrow(vi)$} Assume $(iv)$ holds, let $f\colon\delta\to\delta$ be given and define $g\colon\delta\to\delta$ as $g(\alpha):=(2^{<\gamma_\alpha})^+$, where $\gamma_\alpha$ is the least regular cardinal above $|f(\alpha)|$. By $(iv)$ there is a $\kappa<\delta$ which is a closure point of $g$ (and so also a closure point of $f$), and there is a regular $\lambda\in(\kappa, \delta)$ for which there is a generic elementary embedding $\pi\colon H_\lambda\to\M$ with $\crit\pi=\kappa$, $H_\lambda\subset\M$, $\M\subset V$, and $\pi(g\restr\kappa)(\kappa)<\lambda$.

  \qquad Let $\theta$ be the least regular cardinal above $|\pi(f\restr\kappa)(\kappa)|$, and note that \hbox{$H_\theta\in H_\lambda$} by our definition of $g$. Thus, both $H_\theta$ and $H_{\pi(\theta)}^{\M}$ are elements of $\M$. An application of Countable Embedding Absoluteness \ref{lemm.ctblabs} then yields that $\M$ has a generic elementary embedding $\pi^*\colon H_\theta^{\M}\to H_{\pi(\theta)}^{\M}$ such that $\crit\pi^*=\kappa$, $\pi^*(\kappa)=\pi(\kappa)$, $\pi(f\restr\kappa)\in\ran\pi^*$, and $\pi(f\restr\kappa)(\kappa)<\theta$. By elementarity of $\pi$, $H_\theta$ has an ordinal $\bar\theta<\kappa$ and a generic elementary embedding $\sigma\colon H_{\bar\theta}\to H_\theta$ with $\sigma(\crit\sigma)=\kappa$, $f\restr\kappa\in\ran\sigma$ and $f(\crit\sigma)<\bar\theta$, which is what we wanted to show.

  \qquad \framebox{$(vi)\Rightarrow(v)$} Assume $(vi)$ holds and let $f\colon\delta\to\delta$ be given. Define $g\colon\delta\to\delta$ as $g(\alpha):=\langle(2^{<\gamma_\alpha})^+,f(\alpha)\rangle$, where $\gamma_\alpha$ is the least regular cardinal above $|f(\alpha)|$. In particular, $g$ codes $f$. By $(vi)$ there exist regular $\bar\kappa<\bar\lambda<\kappa<\lambda$ such that $\kappa$ is a closure point of $g$ (so also a closure point of $f$) and there exists a generic elementary embedding $\pi\colon H_{\bar\lambda}\to H_\lambda$ with $\crit\pi = \bar\kappa$, $\pi(\bar\kappa)=\kappa$, $g(\bar\kappa)<\bar\lambda$, and $g\restr\kappa\in\ran\pi$. Since $f$ is definable from $g$ and $g\restr\kappa\in\ran\pi$, it follows that $f\restr\kappa\in\ran \pi$. So let $\pi(\bar f)=f\restr\kappa$ with $\bar f\colon\bar\kappa\to\bar\kappa$. Now observe that $\bar f=f\restr\bar\kappa$ since for $\alpha<\bar\kappa$, we have $f(\alpha)=\pi(\bar f)(\alpha)=\pi(\bar f(\alpha))=\bar f(\alpha)$.

  \qquad Let $\bar\theta$ be the least regular cardinal above $|f(\bar\kappa)|$. By the definition of $g$, we have $H_{\bar\theta}\in H_{\bar\lambda}$. Now, following the $(iii)\Rightarrow(ii)$ direction in the proof of Theorem \ref{theo.rem} we get that $H_{\bar\lambda}$ has a generic elementary embedding \hbox{$\sigma\colon H_{\bar\theta}\to\M$} with $\M$ closed under ${<}\bar\theta$-sequences from $V$, $\crit\sigma=\bar\kappa$, $\sigma(\bar\kappa)> \bar\theta$, and $\sigma(\bar f)(\bar\kappa)<\bar\theta$. Let $\pi(\bar\theta)=\theta$ and $\pi(\M)=\N$. Now by elementarity of $\pi$, we get that there is a generic elementary embedding \hbox{$\sigma^*:H_\theta\to \N$} with $\crit\sigma^*=\kappa$, $\sigma^*(\kappa)>\theta$, and $\sigma^*(\pi(\bar f))(\kappa)=\sigma^*(f\restr\kappa)(\kappa)<\theta$.

  \qquad \framebox{$(vi)\Rightarrow(iii)$} Let $C$ be the club of all $\alpha$ such that $$(H_\alpha,\in, A\cap H_\alpha)\prec(H_\delta,\in,A).$$ Let $f\colon\delta\to\delta$ be given as $f(\alpha):=\bra{\gamma_0^\alpha,\gamma_1^\alpha}$, where $\gamma_0^\alpha$ is the first limit point of $C$ above $\alpha$ and the $\gamma_1^\alpha$ are chosen such that $\{\gamma_1^\alpha\mid\alpha<\beta\}$ encodes $A\cap\beta$ for cardinals $\beta$. This definition makes sense since $\delta$ is inaccessible by Proposition \ref{prop.virtit}.

  \qquad Let $\kappa<\delta$ be a closure point of $f$ such that there are regular cardinals $\bar\theta<\kappa<\theta$ and a generic elementary embedding $\pi\colon H_{\bar\theta}\to H_\theta$ such that $\pi(\crit\pi)=\kappa$, $f(\crit\pi)<\bar\theta$, and $f\restr\kappa\in\ran\pi$. Let $\bar\kappa=\crit \pi$. We claim that $\bar\kappa$ is virtually $({<}\delta, A)$-extendible. Since $\kappa\in C$ because it is a closure point of $f$, it suffices by the definition of $C$ to show that
  \eq{
    (H_\kappa,\in,A\cap H_\kappa)\models\godel{\text{$\bar\kappa$ is virtually $(A\cap H_\kappa)$-extendible}}. \tag*{$(1)$}
  }
 Let $\beta$ be the least element of $C$ above $\bar\kappa$ but below $\bar\theta$, and note that $\beta$ exists as $f(\bar\kappa)<\bar\theta$, and the definition of $f$ says that the first coordinate of $f(\bar\kappa)$ is a limit point of $C$ above $\bar\kappa$. It then holds that $$(H_{\bar\kappa},\in,A\cap H_{\bar\kappa}) \prec (H_\beta,\in,A\cap H_\beta)$$ as both $\bar\kappa$ and $\beta$ are elements of $C$. Since $f$ encodes $A$ in the manner previously described and $\pi(f\restr\bar\kappa)=f\restr\kappa$, we get that $\pi(A\cap H_{\bar\kappa}) = A\cap H_\kappa$, and thus
  \eq{
    (H_\kappa,\in,A\cap H_\kappa) \prec (H_{\pi(\beta)},\in,A^*) \tag*{$(2)$}
  }

  for $A^* := \pi(A\cap H_\beta)$. Now, as $(H_\gamma,\in, A\cap H_\gamma)$ and $(H_{\pi(\gamma)},\in,A^*\cap H_{\pi(\gamma)})$ are elements of $H_{\pi(\beta)}$ for every $\gamma<\kappa$, Countable Embedding Absoluteness \ref{lemm.ctblabs} implies that $H_{\pi(\beta)}$ sees that $\bar\kappa$ is virtually $({<}\kappa, A^*)$-extendible, which by $(2)$ then implies $(1)$, which is what we wanted to show.
}

As a corollary of the proof, we now have an analogue of Proposition~\ref{prop.weakerstrong} for virtually Woodin cardinals.

\qprop[Dimopoulos-Gitman-N.][prop.weakerwoodin]{
    A cardinal $\delta$ is virtually Woodin if and only if, for every $A\subseteq H_\delta$, there is a cardinal $\kappa$ satisfying the weakening of virtual $({<}\delta,A)$-strongness where $H_\theta=H_\theta^{\N}$ holds in place of $\N\subseteq V$, with $\N$ being the target of the generic embedding.
}

As a corollary we arrive at the surprising result that there is no distinction between the faint and virtual Woodin cardinals, in contrast with what we will see in Theorem \ref{theo.virtualsep}.

\coro[Gitman-N.][coro.faintvirtualWoodin]{
    Faintly Woodin cardinals are virtually Woodin.
}
\proof{
  Using Proposition~\ref{prop.weakerwoodin}, it suffices to observe that if $\theta$ is inaccessible and
  \eq{
    \pi:(H_\theta,\in,A)\to (\M,\in,B)
  }

  is a faintly $(\theta,A)$-strong embedding such that $A$ codes the sequence of $H_\lambda$ for $\lambda<\theta$, then $H_\theta=H_\theta^{\M}$.
}

We will now step away from the Woodins for a little bit, and introduce the Vop\v enkas. In anticipation of the next section we will work with the class-sized version here, but all the following results work equally well for inaccessible virtually Vop\v enka cardinals\footnote{Note however that we have to require inaccessibility here: see \cite{WilsonVopenka} for an analysis of the singular virtually Vop\v enka cardinals.}.

\defi[\gbc]{
  The \textbf{Generic Vop\v enka Principle} (\gvp) states that for any class $C$ consisting of structures in a common first-order language, there are distinct $\M,\N\in C$ and a generic elementary embedding $\pi\colon\M\to\N$.
}

We will be using a standard variation of \gvp\, involving the following \textit{natural sequences}:

\defi[\gbc]{
  Say that a class function $f\colon\on\to\on$ is an \textbf{indexing function} if it satisfies that $f(\alpha)>\alpha$ and $f(\alpha)\leq f(\beta)$ for all $\alpha<\beta$.
}

\defi[\gbc]{
  Say that an $\on$-sequence $\bra{\M_\alpha\mid\alpha<\on}$ is \textbf{natural} if there exists an indexing function $f\colon\on\to\on$ and unary relations $R_\alpha\subset V_{f(\alpha)}$ such that $\M_\alpha = (V_{f(\alpha)}, \in, \{\alpha\}, R_\alpha)$ for every $\alpha$. Denote this indexing function by $f^{\vec\M}$ and the unary relations as $R_\alpha^{\vec\M}$.
}

The following Theorem \ref{theo.vopwood} is then the main theorem of this section, showing that inaccessible cardinals are virtually Vop\v enka iff they are virtually pre-Woodin.

\theo[Dimopoulos-Gitman-N.; \gbc][theo.vopwood]{
  The following are equivalent:
  \begin{enumerate}
    \item \gvp\ holds;
    \item For any natural $\on$-sequence $\vec\M$ there exists a generic elementary embedding $\pi\colon\M_\alpha\to\M_\beta$ for some $\alpha<\beta$;
    \item $\on$ is virtually pre-Woodin;
    \item $\on$ is faintly pre-Woodin.
  \end{enumerate}
}
\proof{
  $(i)\Rightarrow(ii)$ and $(iii)\Rightarrow(iv)$ are trivial.

  \qquad $(iv)\Rightarrow(i)$: Assume $\on$ is faintly pre-Woodin and fix some $\on$-sequence \\$\vec\M:=\bra{\M_\alpha\mid\alpha<\on}$ of structures in a common language. Let $\kappa$ be $({<}\on,\vec\M)$-prestrong and fix some regular $\theta>\kappa$ satisfying that $\M_\alpha\in H_\theta^V$ for every $\alpha<\theta$, and fix a generic elementary embedding
  \eq{
    \pi\colon(H_\theta^V, \in, \vec\M) \to (\N, \in, \M^*)
  }

  with $H_\theta^V\subset\N$ and $\vec\M\cap H_\theta^V=\M^*\cap H_\theta^V$. Set $\kappa:=\crit\pi$.

  \qquad We have that $\pi\restr\M_\kappa\colon\M_\kappa\to\M^*_{\pi(\kappa)}$, but we need to reflect this embedding down below $\theta$ as we do not know whether $\M_{\pi(\kappa)}^*$ is on the $\vec\M$ sequence. Working in the generic extension, we have
  \eq{
    \N\models\exists\bar\kappa<\pi(\kappa)\exists\dot\sigma\in V^{\col(\omega, \M_{\bar\kappa}^*)}\colon \godel{\text{$\dot\sigma\colon \M^*_{\bar\kappa}\to\M^*_{\pi(\kappa)}$ is elementary}}.
  }

  Here $\kappa$ realises $\bar\kappa$ and $\pi\restr\M_\kappa$ realises $\sigma$. Note that $\M^*_\kappa = \M_\kappa$ since we ensured that $\M_\kappa\in H_\theta^V$ and we are assuming that $\vec\M\cap H_\theta^V = \M^*\cap H_\theta^V$, so the domain of $\sigma$ ($=\pi\restr\M_\kappa$) \textit{is} $\M_\kappa^*$ --- also note that $\sigma$ exists in a $\col(\omega,\M_\kappa)$ extension of $\N$ by an application of Countable Embedding Absoluteness \ref{lemm.ctblabs}. Now elementarity of $\pi$ implies that
  \eq{
    H_\theta^V\models\exists\bar\kappa<\kappa\exists\dot\sigma\in V^{\col(\omega, \M_{\bar\kappa})}\colon \godel{\text{$\dot\sigma\colon \M_{\bar\kappa}\to\M_{\kappa}$ is elementary}},
  }

  which is upwards absolute to $V$, from which we can conclude that $\sigma\colon\M_{\bar\kappa}\to\M_\kappa$ witnesses that \gvp\ holds.

  \qquad $(ii)\Rightarrow(iii)$: Assume $(ii)$ holds and assume that $\on$ is not virtually pre-Woodin, which means that there exists some class $A$ such that there are no virtually $A$-prestrong cardinals. This allows us to define a function $f\colon\on\to\on$ as $f(\alpha)$ being the least regular $\eta>\alpha$ such that $\alpha$ is not virtually $(\eta,A)$-prestrong.

  \qquad We also define $g\colon\on\to\on$ as taking $\alpha$ to the least strong limit cardinal above $\alpha$ which is a closure point for $f$. Note that $g$ is an indexing function, so we can let $\vec\M$ be the natural sequence induced by $g$ and $R_\alpha := A\cap H_{g(\alpha)}^V$. $(ii)$ supplies us with $\alpha<\beta$ and a generic elementary embedding\footnote{Note that $V_{g(\alpha)}=H_{g(\alpha)}^V$ since $g(\alpha)$ is a strong limit cardinal.}
  \eq{
    \pi\colon(H_{g(\alpha)}^V,\in,A\cap H_{g(\alpha)}^V)\to (H_{g(\beta)}^V, \in, A\cap H_{g(\beta)}^V).
  }

  Since $g(\alpha)$ is a closure point for $f$ it holds that $f(\crit\pi)<g(\alpha)$, so fixing a regular $\theta\in(f(\crit\pi), g(\alpha))$ we get that $\crit\pi$ is virtually $(\theta, A)$-prestrong, contradicting the definition of $f$. Hence $\on$ is virtually pre-Woodin.
}

\subsection{Weak Vop\v enka}

We now move to a \textit{weak} variant of \gvp, introduced in a category-theoretic context in \cite{AdamekRosicky}. It starts with the following equivalent characterisation of \gvp, which is the virtual analogue of the characterisation shown in \cite{AdamekRosicky}:

\lemm[Virtualised Ad\' amek-Rosick\' y; \gbc][lemm.adaros]{
  The following are equivalent\footnote{This is equivalent to saying that $\on$, viewed as a category, cannot be fully embedded into the category \textsf{Gra} of graphs, which is how it is stated in \cite{AdamekRosicky}.}:
  \begin{enumerate}
    \item \gvp\
    \item There is not a natural $\on$-sequence $\bra{\M_\alpha\mid\alpha<\on}$ satisfying that
      \begin{itemize}
        \item there is a generic homomorphism $\M_\alpha\to\M_\beta$ for every $\alpha\leq\beta$, which is unique in all generic extensions;
        \item there is \textit{no} generic homomorphism $\M_\beta\to\M_\alpha$ for any $\alpha<\beta$.
      \end{itemize}
    \item There is not a natural $\on$-sequence $\bra{\M_\alpha\mid\alpha<\on}$ satisfying that
      \begin{itemize}
        \item there is a homomorphism $\M_\alpha\to\M_\beta$ in $V$ for every $\alpha\leq\beta$, which is unique in all generic extensions;
        \item there is \textit{no} generic homomorphism $\M_\beta\to\M_\alpha$ for any $\alpha<\beta$.
      \end{itemize}
  \end{enumerate}
}
\proof{
  Note that the only difference between $(ii)$ and $(iii)$ is that the homomorphism exists in $V$, making $(ii)\Rightarrow(iii)$ trivial.

  \qquad $(iii)\Rightarrow(i)$: Assume that \gvp\ fails, meaning by Theorem \ref{theo.vopwood} that we have a natural $\on$-sequence $\vec\M_\alpha$ such that, in every generic extension, there is no homomorphism between any two distinct $\M_\alpha$'s. Define $\bra{\N_\kappa\mid\kappa\in\Card}$ as\footnote{$\coprod$ is the ``model-theoretic union'', also known as the coproduct.}
\eq{
  \N_\kappa:=\coprod_{\xi\leq\kappa}\M_\xi:= \{(x,\xi)\mid\xi\leq\kappa\land\xi\in\Card\land x\in\M_\xi\},
}

  with a unary relation $R^*$ given as $R^*(x, \xi)$ iff $\M_\xi\models R(x)$ and a binary relation $\sim^*$ given as $(x,\xi)\sim^*(x', \xi')$ iff $\xi=\xi'$. Whenever we have a homomorphism $f\colon\N_\kappa\to\N_\lambda$ we then get an induced homomorphism $\tilde f\colon\M_0\to\M_\xi$, given as $\tilde f(x) := f(x, 0)$, where $\xi\leq\kappa$ is given by preservation of $\sim^*$.

  \qquad For any two cardinals $\kappa<\lambda$ we have a homomorphism $j_{\kappa\lambda}\colon\N_\kappa\to\N_\lambda$ in $V$, given as $j_{\kappa\lambda}(x, \xi) := (x,\xi)$. This embedding must also be the \textit{unique} such, in all generic extensions, as otherwise we get a generic homomorphism between two distinct $\M_\alpha$'s. Furthermore, there cannot be any homomorphism $\N_\lambda\to\N_\kappa$ as that would also imply the existence of a generic homomorphism between two distinct $\M_\alpha$'s.

\qquad $(i)\Rightarrow(ii)$: Assume that we have an $\on$-sequence $\vec\M_\alpha$ as in the theorem, with generic homomorphisms $j_{\alpha\beta}\colon\M_\alpha\to\M_\beta$ that are unique in all generic extensions for every $\alpha\leq\beta$, with no generic homomorphisms going the other way.

\qquad We first note that we can for every $\alpha\leq\beta$ choose the $j_{\alpha\beta}$ in a $\col(\omega,\M_\alpha)$-extension, by a proof similar to the proof of Lemma \ref{lemm.ctblabs} and using the uniqueness of $j_{\alpha\beta}$. Next, fix a proper class $C\subset\on$ such that $\alpha\in C$ implies that
\eq{
  \sup_{\xi\in C\cap\alpha}\abs{\M_\xi}^V<\abs{\M_\alpha}^V.
}

and note that this implies that $V[g]\models\abs{\M_\xi}<\abs{\M_\alpha}$ for every $V$-generic \\$g\subset\col(\omega,\M_\xi)$. This means that for every $\alpha\in C$ we may choose some $\eta_\alpha\in\M_\alpha$ which is \textit{not} in the range of any $j_{\xi\alpha}$ for $\xi<\alpha$. But now define first-order structures $\bra{\N_\alpha\mid\alpha\in C}$ as $\N_\alpha := (\M_\alpha, \eta_\alpha)$. Then, by our assumption on the $\M_\alpha$'s and construction of the $\N_\alpha$'s, there can be no generic homomorphism between any two distinct $\N_\alpha$, showing that \gvp\ fails.
}

The \textit{weak} version of \gvp\ is then simply ``flipping the arrows around'' in the above characterisation of \gvp.

\xdefi[\gbc]{
  \textbf{Generic Weak Vop\v enka's Principle} (\gwvp) states that there does \textit{not} exist an $\on$-sequence of first-order structures $\bra{\M_\alpha\mid\alpha<\on}$ such that
  \begin{itemize}
    \item there is a homomorphism $\M_\beta\to\M_\alpha$ in $V$ for every $\alpha\leq\beta$, which is unique in all generic extensions;
    \item there is \textit{no} generic homomorphism $\M_\alpha\to\M_\beta$ for any $\alpha<\beta$.$\hfill\circ$
  \end{itemize}
}

We start by showing that \gwvp\ is indeed a weaker version of \gvp.

\prop{
  \gvp\ implies \gwvp.
}
\proof{
  Assume \gvp\ holds and \gwvp\ fails, and let $\bra{\M_\alpha\mid\alpha<\on}$ be an $\on$-sequence of first-order structures such that for every $\alpha\leq\beta$ there exists a homomorphism
  \eq{
    j_{\beta\alpha}\colon\M_\beta\to\M_\alpha
  }

  in $V$ which is unique in all generic extensions, with no generic homomorphisms going the other way. We can then find a proper class $C\subset\on$ such that $\abs{\M_\alpha}^V < \abs{\M_\beta}^V$ for every $\alpha<\beta$ in $C$. By \gvp\ there are then $\alpha<\beta$ in $C$ and a generic homomorphism
  \eq{
    \pi\colon\M_\alpha\to\M_\beta
  }

  in some $V[g]$. Here we may assume, as in the proof of Lemma \ref{lemm.adaros}, that $g\subset\col(\omega,\M_\beta)$. But then $\pi\circ j_{\beta\alpha} = \id$ by uniqueness of $j_{\beta\beta} = \id$, which means that $j_{\beta\alpha}$ is injective in $V[g]$ and hence also in $V$. But then $\abs{\M_\beta}^V\leq\abs{\M_\alpha}^V$, contradicting the definition of $C$.
}

Denoting the corresponding non-generic principle by \wvp\, Wilson showed the following surprising result:

\theo[\cite{WilsonWVP}]{
  \wvp\ is equivalent to $\on$ being a Woodin cardinal.
}

Given our Theorem \ref{theo.vopwood} we may then suspect that in the virtual world these two continue to be equivalent, which turns out to \textit{almost} be the case. In the proceeding argument we will be roughly following the structure of the argument in \cite{WilsonWVP}, but we have to diverge from it at several points in which the author is using the fact that they are working with class-sized elementary embeddings.

\qquad Indeed, in that paper they establish a correspondence between elementary embeddings and certain homomorphisms, a correspondence we will not achieve here. Proving that the elementary embeddings we \textit{do} get are non-trivial seems to furthermore require extra assumptions on our structures. In any case, let us begin.

\qquad Define for every strong limit cardinal $\lambda$ and $\Sigma_1$-formula $\varphi$ the relations
\eq{
  R^\varphi &:= \{x\in V\mid (V,\in)\models\varphi[x]\}\\
  R^\varphi_\lambda&:= \{x\subset H_\lambda^V\mid \exists y\in R^\varphi\colon y\cap H_\lambda^V = x\}
}

and given any class $A$ define the structure
\eq{
  \p_{\lambda,A} := (H_{\lambda^+}^V,R^\varphi_\lambda, \{\lambda\}, A\cap H_\lambda^V)_{\varphi\in\Sigma_1}.
}

Say that a homomorphism $h\colon\p_{\lambda,A}\to\p_{\eta,A}$ is \textbf{trivial} if
  \eq{
    h(x)\cap H_\eta^V = x\cap H_\eta^V
  }

  for every $x\in H_{\lambda^+}^V$. Note that $h$ can only be trivial if $\eta\leq\lambda$ since $h(\lambda)=\eta$.

\lemm[Gitman-N.; \gbc][lemm.hom]{
  Let $\lambda$ be a singular strong limit cardinal, $\eta$ a strong limit cardinal and $A\subset V$ a class. If there exists a non-trivial generic homomorphism $h\colon\p_{\lambda,A}\to\p_{\eta,A}$ then there is a non-trivial generic elementary embedding $\pi\colon(H_{\lambda^+}^V,\in,A\cap H_\lambda^V)\to(\M, \in, B)$ for some $\M$ such that, letting $\nu := \min\{\lambda,\eta\}$, it holds that $H_\nu^V=H_\nu^{\M}$, $A\cap H_\nu^V = B\cap H_\nu^V$ and $\crit\pi<\nu$.
}
\proof{
  Assume that we have a non-trivial homomorphism $h\colon\p_{\lambda,A}\to\p_{\eta,A}$ in a forcing extension $V[g]$, define in $V[g]$ the set
  \eq{
    \M^* := \{\bra{b,f} \mid b\in [H_\nu]^{<\omega}\land f\in H_{\lambda^+}^V\land f\colon H_\lambda^V\to H_\lambda^V\},
  }

  and define the standard relations $\in^*$ and $=^*$ on $\M^*$ as
  \eq{
    \bra{b_0,f_0} \in^* \bra{b_1, f_1} &\quad\text{iff}\quad b_0b_1\in h(\{xy\in[H_\lambda^V]^{<\omega}\mid f_0(x)\in f_1(y)\})\\
    \bra{b_0,f_0} =^* \bra{b_1, f_1} &\quad\text{iff}\quad b_0b_1\in h(\{xy\in[H_\lambda^V]^{<\omega}\mid f_0(x) = f_1(y)\})
  }

  Let $\M := \M^*/=^*$, and also call $\in^*$ the induced relation on $\M$, which is clearly well-defined. We then get a version of \L o\' s' Theorem, using that $h$ preserves all $\Sigma_1$-relations and that $H_\lambda^V\models\zfc^-$.

  \clai{
    For every formula $\varphi(v_1,\dots,v_n)$ and every $[b_1,f_1],\dots,[b_n,f_n]\in\M$ the following are equivalent:
    \begin{enumerate}
      \item $(\M,\in^*)\models\varphi[[b_1,f_1],\dots,[b_n,f_n]]$;
      \item $b_1\cdots b_n\in h(\{a_1\cdots a_n\mid\p_{\lambda,A}\models\varphi[f_1(a_1),\dots,f_n(a_n)]\})$.
    \end{enumerate}
  }

  \cproof{
    The proof is straightforward, using that $h$ preserves $\Sigma_1$-relations. We prove this by induction on $\varphi$. If $\varphi$ is $v_i\in v_j$ then we have that
    \eq{
      &(\M,\in^*)\models\varphi[[b_1,f_1],\dots,[b_n,f_n]]\\
      \Leftrightarrow&\bra{b_i,f_i}\in^*\bra{b_j,f_j}\\
      \Leftrightarrow&b_ib_j \in h(\{a_ia_j\in[H_\lambda^V]^{<\omega}\mid f_i(a_i)\in f_j(a_j)\})\\
      \Leftrightarrow&b_1\cdots b_n\in h(\{a_1\cdots a_n\mid f_i(a_i)\in f_j(a_j)\})\\
      \Leftrightarrow&b_1\cdots b_n\in h(\{a_1\cdots a_n\mid\p_{\lambda,A}\models\varphi[f_1(a_1),\dots,f_n(a_n)]\}).
    }

    The cases where $\varphi$ is $\psi\land\chi$ or $\lnot\psi$ is straightforward. If $\varphi$ is $\exists x\psi$ then
    \eq{
      &(\M,\in^*)\models\varphi[[b_1,f_1],\dots,[b_n,f_n]]\\
      \Leftrightarrow&\exists\bra{b,f}\in\M^*\colon(\M,\in)\models\psi[\bra{b,f}, \bra{b_1,f_1},\dots,\bra{b_n,f_n}]\\
      \Leftrightarrow&\exists\bra{b,f}\in\M^*\colon bb_1\cdots b_n\in h(\{a\vec a\mid\p_{\lambda,A}\models\psi[f(a), f_1(a_1),\dots,f_n(a_n)]\})\\
      \Leftrightarrow&b_1\cdots b_n\in h(\{\vec a\mid\p_{\lambda,A}\models\varphi[f_1(a_1),\dots,f_n(a_n)]\}),
    }

    finishing the proof.
  }

  Note that we have not shown that $(\M,\in^*)$ is well-founded, and indeed it might not be. However, the following claim will show that $(H_\nu^V,\in)$ is isomorphic to a rank-initial segment of $(\M^*, \in^*)$, giving well-foundedness up to that point at least. Define the function $\chi\colon(H_\nu^V,\in)\to(\M^*,\in^*)$ as $\chi(a):=[\bra{a},\text{pr}]$, where $\text{pr}(\bra{x}) := x$.

  \clai{
    \label{clai.pr}
    For every $[a,f]\in\M$ and $b\in H_\nu^V$,
    \eq{
      [a,f]\in^*\chi(b)\quad\Leftrightarrow\quad\exists c\in H_\nu^V\colon [a,f] = \chi(c).
    }
  }

  \cproof{
    We have that
    \eq{
      [a,f]\in^* \chi(b) = [\bra{b},\text{pr}] &\Leftrightarrow a\bra{b}\in h(\{x\bra{y}\mid f(x)\in y\})\\
      &\Leftrightarrow a\bra{b}\in h(\{x\bra{y}\mid\exists z\in y\colon f(x)=z\})\\
      &\Leftrightarrow \exists c\in b\colon a\bra{c}\in h(\{x\bra{z}\mid f(x)=z\})\\
      &\Leftrightarrow \exists c\in b\colon [a,f] = [\bra{c},\text{pr}] = \chi(c),
    }

    yielding the wanted.
  }

  This claim implies that by taking the transitive collapse of $\ran\chi\subset\M$ we may assume that $H_\nu^V = H_\nu^{\M}$. Now define
  \eq{
    B := \{[b,f]\in\M\mid b\in h(\{x\in H_\lambda^V\mid f(x)\in A\})\}.
  }

  and, in $V[g]$, let $\pi\colon (H_\lambda^V,\in,A\cap H_\lambda^V)\to(\M,\in, B)$ be given as $\pi(x) := [\bra{},c_x]$.

  \clai{
    $\pi$ is elementary.
  }

  \cproof{
     For $x_1,\dots,x_n\in H_\lambda^V$ it holds that
    \eq{
      (\M,\in^*, B)\models\varphi[\pi(x_1),\dots,\pi(x_n)]&\Leftrightarrow\ (\M,\in^*)\models\varphi[\pi(x_1),\dots,\pi(x_n)]\\
      &\Leftrightarrow\ \bra{}\in h(\{\bra{}\mid\p_{\lambda,A}\models\varphi[x_1,\dots,x_n]\})\\
      &\Leftrightarrow\ (H_{\lambda^+}^V,\in,A\cap H_\lambda^V)\models\varphi[x_1,\dots,x_n]
    }

    and we also get that, for every $x\in H_\lambda^V$,
    \eq{
      x\in A \Leftrightarrow \bra{}\in h(\{a\in H_\lambda^V\mid x\in A\}) \Leftrightarrow \pi(x)\in B,
    }

    which shows elementarity.
  }

  We next need to show that $B\cap H_\nu^V = A\cap H_\nu^V$, so let $x\in H_\nu^V$. Note that $x = [\bra{x}, \text{pr}]$ by Claim \ref{clai.pr} and the observation proceeding it, which means that
  \eq{
    x\in B \Leftrightarrow \bra{x}\in h(\{\bra{y}\in H_\lambda^V\mid y\in A\}) \Leftrightarrow x\in A.
  }

  The last thing we need to show is that $\crit\pi<\nu$. We start with an analogous result about $h$.

  \clai{
    \label{clai.move}
    There exists some $b\in H_\nu^V$ such that $h(b)\neq b$.
  }

  \cproof{
    Assume the claim fails. We now have two cases.\\

    \framebox{\textbf{Case 1:} $\lambda\geq\eta$}

    \qquad By non-triviality of $h$ there is an $x\in\p^V(H_\lambda^V)$ such that $h(x)\neq x\cap H_\eta^V$, which means that there exists an $a\in H_\eta^V$ such that $a\in h(x)\Leftrightarrow a\notin x$.

    \qquad  If $a\in x$ then $\{a\} = h(\{a\}) \subset h(x)$,\footnote{Note that as $h$ preserves $\Sigma_1$ formulas it also preserves singletons and boolean operations.} making $a\in h(x)$, $\contr$, so assume instead that $a\in h(x)$. Since $\eta$ is a strong limit cardinal we may fix a cardinal $\theta<\eta$ such that $a\in H_\theta^V$ and $H_\theta^V\in H_\eta^V$. We then have that\footnote{Note that we are using $\lambda\geq\eta$ here to ensure that $H_\theta^V\in\dom h$.}
    \eq{
      \{a\}\subset h(x)\cap H_\theta^V = h(x)\cap h(H_\theta^V) = h(x\cap H_\theta^V) = x\cap H_\theta^V,
    }

    so that $a\in x$, $\contr$.\\

    \framebox{\textbf{Case 2:} $\lambda<\eta$}

    \qquad In this case we are assuming that $h\restr H_\lambda^V = \id$, but $h(\lambda) = \eta > \lambda$. Since $\lambda$ is singular we can fix some $\gamma<\lambda$ and a cofinal function $f\colon\gamma\to\lambda$. Define the relation
    \eq{
      R := \{(\alpha,\beta,\bar\alpha,\bar\beta,g)\mid\godel{\text{$g$ is a cofinal function $g\colon\alpha\to\beta$}}\land g(\bar\alpha)=\bar\beta\}.
    }

    Then $R(\gamma,\lambda,\alpha,f(\alpha),f)$ holds by assumption for every $\alpha<\gamma$, so that $R$ holds for some $(\gamma^*,\lambda^*,\alpha^*,f(\alpha)^*,f^*)$ such that
    \eq{
      (\gamma^*,\lambda^*,\alpha^*,f(\alpha)^*,f^*)\cap H_\eta^V &= (h(\gamma),h(\lambda),h(\alpha),h(f(\alpha)),h(f)) \\
      &= (\gamma,\eta,\alpha,f(\alpha),h(f)),
    }

     using our assumption that $h$ fixes every $b\in H_\lambda^V$. Since $\gamma$, $\alpha$ and $f(\alpha)$ are transitive and bounded in $H_\lambda^V$, it holds that $h(\gamma)=\gamma^*$, $h(\alpha)=\alpha^*$ and that $h(f(\alpha))=f(\alpha)^*$. Also, since $\dom(f^*)=\gamma=\dom(f)$ we must in fact have that $f^*=h(f)$. But this means that $h(f)\colon\gamma\to\eta$ is cofinal and $\ran(h(f))\subset\lambda$, a contradiction!
  }

  To use the above Claim \ref{clai.move} to conclude anything about $\pi$ we will make use of the following standard lemma:

  \clai{
    \label{clai.pihagreement}
    For any $x\in H_\lambda^V$ it holds that $h(x) = \pi(x)\cap H_\eta^V$.
  }

  \cproof{
    For any $n<\omega$ and $\bra{a_1,\hdots,a_n}\in[H_\eta^V]^n$ we have that
    \eq{
      &\bra{a_1,\dots,a_n}\in\pi(x)\\
      \Leftrightarrow\ &(\M,\in)\models\bra{a_1,\dots,a_n}\in\pi(x)\\
      \Leftrightarrow\ &(\M,\in)\models\bra{[\bra{a_1},\text{pr}],\dots,[\bra{a_n},\text{pr}]}\in[\bra{},c_x]\\
      \Leftrightarrow\ &\bra{a_1,\hdots,a_n}\in h(\{\bra{x_1,\hdots,x_n}\mid\p_{\lambda,A}\models\bra{x_1,\dots,x_n}\in x\})\\
      \Leftrightarrow\ &\bra{a_1,\dots,a_n}\in h(x),
    }

    showing that $h(x) = \pi(x)\cap H_\eta^V$.
  }

  Now use Claim \ref{clai.move} to fix a $b\in H_\nu^V$ which is moved by $h$. Claim \ref{clai.pihagreement} then implies that
  \eq{
    \pi(b)\cap H_\eta^V = h(b)\cap H_\eta^V = h(b)\neq b = b\cap H_\eta^V,
  }

  showing that $\pi(b)\neq b$ and hence $\crit\pi<\nu$. This finishes the proof.
}

\defi{
  A cardinal $\kappa$ is \textbf{W-virtually pre-Woodin} if it is virtually pre-Woodin but without requiring that the target model is well-founded.
}

\theo[Gitman-N.; \gbc]{
  \gwvp\ holds iff $\on$ is W-virtually pre-Woodin.
}
\proof{
  $(\Leftarrow)$ is just observing that the virtualisation of the argument in \cite{WilsonWVP} that \wvp\ holds if \on\ is Woodin works in the W-virtually pre-Woodin case, so we only give a brief sketch.

  \qquad Assume \on\ is W-virtually pre-Woodin and let $\vec\M$ be a counterexample to \gwvp, so that we in $V$ have homomorphisms $\M_\beta\to\M_\alpha$ for all $\alpha\leq\beta$. Work in some generic extension $V[g]$, fix a W-virtually $\vec\M$-prestrong cardinal $\kappa$ and let $\theta\gg\kappa$ be such that $\M_{\kappa+1}\in H_\theta^V$. Letting $\pi\colon (H_\theta^V,\in)\to(\M,\in^*)$ be the corresponding embedding we get that $\M_{\kappa+1}=\pi(\vec\M)_{\kappa+1}$, so that
  \eq{
    \pi\restr\M_\kappa\colon(\M_\kappa,\in)\to(\pi(\M_\kappa),\in^*)=(\pi(\vec\M)_{\pi(\kappa)},\in^*).
  }

  But then, by the choice of $\theta$ and elementarity of $\pi$, we get that $\M$ has a homomorphism
  \eq{
    h\colon(\pi(\vec\M)_{\pi(\kappa)}, \in^*)\to(\pi(\vec\M)_{\kappa+1},\in^*)=(\M_{\kappa+1}, \in),
  }

  making $h\circ(\pi\restr\M_\kappa)\colon(\M_\kappa,\in)\to(\M_{\kappa+1},\in)$ a counterexample to \gwvp.

  \qquad $(\Rightarrow)$: Assume that $\on$ is not W-virtually pre-Woodin. This means that there exists a class $A$ such that there are no W-virtually $A$-prestrong cardinals. We can therefore assign to any cardinal $\kappa$ the least cardinal $f(\kappa)>\kappa$ such that $\kappa$ is not W-virtually $(f(\kappa),A)$-prestrong.

  \qquad Also define a function $g\colon\on\to\Card$ as taking an ordinal $\alpha$ to the least singular strong limit cardinal above $\alpha$ closed under $f$. Then we are assuming that there is no non-trivial generic elementary embedding
  \eq{
    \pi\colon(H_{g(\alpha)}^V,\in,A\cap H_{g(\alpha)}^V)\to(\M,\in, B)
  }

  with $H_{g(\alpha)}^V\subset\M$ and $B\cap H_{g(\alpha)}^V = A\cap H_{g(\alpha)}^V$. Assume towards a contradiction that for some $\alpha,\beta$ there is a non-trivial generic homomorphism $h\colon\p_{g(\alpha),A}\to\p_{g(\beta), A}$. Lemma \ref{lemm.hom} then gives us a non-trivial generic elementary embedding
  \eq{
    \pi\colon(H_{g(\alpha)}^V,\in,A\cap H_{g(\alpha)}^V)\to(\M, \in, B)
  }

  for some transitive $\M$ such that $H_\nu^V\subset\M$ with $\nu := \min\{g(\alpha),g(\beta)\}$ and $A\cap H_\nu^V = B\cap H_\nu^V$, a contradiction! Therefore every generic homomorphism $h\colon\p_{g(\alpha),A}\to\p_{g(\beta),A}$ is trivial. Since there is a unique trivial homomorphism when $\alpha\geq\beta$ and no trivial homomorphism when $\alpha<\beta$ since $g(\alpha)$ is sent to $g(\beta)$, the sequence of structures
  \eq{
    \bra{\p_{g(\alpha),A}\mid \alpha\in\on}
  }

  is a counterexample to \gwvp, which is what we wanted to show.
}

\section{Berkeleys}

We next move to the higher realms of the virtual large cardinal hierarchy, and study cardinals whose non-virtual versions are inconsistent with \zfc.

\qquad In the virtual setting the virtually Berkeley cardinals, like all the other virtual large cardinals, are simply downwards absolute to $L$. It turns out that virtually Berkeley cardinals are natural objects, as the main theorem of this section, Theorem~\ref{theo.berkeleyequiv}, shows that these large cardinals are precisely what separates virtually pre-Woodins from the virtually Woodins, as well as separating virtually Vop\v enka cardinals from Mahlo cardinals, improving on a result in \cite{GitmanHamkins}.

\defi{
  Say that a cardinal $\delta$ is \textbf{virtually proto-Berkeley} if for every transitive set $\M$ such that $\delta\subset\M$ there exists a generic elementary embedding $\pi\colon\M\to\M$ with $\crit\pi<\delta$.

  \qquad If $\crit\pi$ can be chosen arbitrarily large below $\delta$ then $\delta$ is \textbf{virtually Berkeley}, and if $\crit\pi$ can be chosen as an element of any club $C\subset\delta$ we say $\delta$ is \textbf{virtually club Berkeley}.
}

Note that a quick application of Countable Embedding Absoluteness \ref{lemm.ctblabs} shows that virtually (proto-)Berkeley cardinals are downwards absolute to $L$.

\qquad We are not interested in the virtually proto-Berkeley cardinals for the same reason we are not interested in the proto-Berkeley cardinals, namely that if $\delta$ is virtually proto-Berkeley then every $\kappa>\delta$ is proto-Berkeley as well. The following theorem, which is a straight-forward virtualisation of the corresponding theorem in the non-virtual context, then shows that the least virtually proto-Berkeley cardinal is indeed virtually Berkeley.

\theo[Virtualised \cite{Cutolo} 2.1.14]{
  The least virtually proto-Berkeley cardinal is virtually Berkeley.
}
\proof{
  Let $\delta_0$ be the least virtually proto-Berkeley cardinal and assume that $\eta_0<\delta_0$ is least such that there exists a transitive set $\M_0$ with $\delta_0\in\M_0$ such that there are no generic elementary embeddings $\pi\colon\M\to\M$ with $\crit\pi\in(\eta_0,\delta_0)$.

  \qquad We will show that $\eta_0$ is virtually proto-Berkeley, which would contradict minimality of $\delta_0$, so let $\M$ be any transitive set with $\eta_0\in\M$. We can now fix some $\lambda>\delta_0$ such that $\M,\M_0\in V_\lambda$ and define
  \eq{
    \M' := V_\lambda\cup\{\{\bra{x, \eta_0,\M,\M_0}\mid x\in V_\lambda\}\}.
  }

  Note that $\eta_0$, $\M_0$ and $\M$ are all definable in $\M'$. Use the fact that $\delta_0$ is virtually proto-Berkeley to get a generic embedding $\pi\colon\M'\to\M'$, so that definability ensures that $\pi$ fixes $\eta_0$, $\M_0$ and $\M$.

  \qquad By choice of $\eta_0$ and because $\M_0$ is fixed by $\pi$ we get that $\crit\pi\leq\eta_0$, and since $\pi$ fixes $\eta_0$ we further have that $\crit\pi<\eta_0$. This implies that $\pi\restr\M\colon\M\to\M$ witnesses that $\eta_0$ is virtually proto-Berkeley, $\contr$.
}

Virtually (proto-)Berkeley cardinals turn out to be equivalent to their ``boldface'' versions, the proof of which is a straightforward virtualisation of Lemma 2.1.12 and Corollary 2.1.13 in \cite{Cutolo}.

\prop[Virtualised \cite{Cutolo} 2.1.12 and 2.1.13][prop.boldfaceberkeley]{
  If $\delta$ is virtually proto-Berkeley then for every transitive set $\M$ such that $\delta\subset\M$ and every subset $A\subset\M$ there exists a generic elementary embedding $\pi\colon(\M,\in,A)\to(\M,\in,A)$ with $\crit\pi<\delta$. If $\delta$ is virtually Berkeley then we can furthermore ensure that $\crit\pi$ is arbitrarily large below $\delta$.
}
\proof{
  Let $\M$ be transitive with $\delta\subset\M$ and $A\subset\M$. Let
  \eq{
    \N := \M\cup\{A, \{\{A, x\}\mid x\in\M\}\}\cup\{\{A,x\}\mid x\in\M\}
  }

  and note that $\N$ is transitive. Further, both $A$ and $\M$ are definable in $\N$ without parameters: $A$ is definable in $\N$ as the unique set such that there is a set $B$ ($=\{\{A,x\}\mid x\in\M\}$), all of whose elements are unordered pairs $\{A,x\}$ and for every $x\in\N$ such that $A\notin x$ and $x\notin\{A,B\}$ it holds that $\{A,x\}\in B$. $\M$ is then what remains if we remove all $x$ such that $A$ is in the transitive closure of $x$. But this means that a generic elementary embedding $\pi\colon\N\to\N$ fixes both $\M$ and $A$, giving us a generic elementary $\sigma\colon(\M,\in,A)\to(\M,\in,A)$ with $\crit\sigma=\crit\pi$, yielding the wanted conclusion.
}

The following is a straightforward virtualisation of the usual definition of the Vop\v enka filter (see e.g. \cite{Kanamori}):

\defi[\gbc]{
  Define the \textbf{virtually Vop\v enka filter} $F$ on $\on$ as $X\in F$ iff there is a natural $\on$-sequence $\vec\M$ such that $\crit\pi\in X$ for any $\alpha<\beta$ and any generic elementary $\pi\colon\M_\alpha\to\M_\beta$.
}

  Theorem \ref{theo.vopwood} shows that $\emptyset$ is in the virtually Vop\v enka filter iff \gvp\ fails, in analogy with the non-virtual case. Normality also holds in the virtual context, as the following proof shows:

\lemm[Virtualised folklore; \gbc]{
  The virtually Vop\v enka filter is a normal filter.
}
\proof{
  Let $F$ be the virtually Vop\v enka filter. We first show that $F$ is actually a filter. If $X\in F$ and $Y\supset X$ then $Y\in F$ simply by definition of $F$. If $X,Y\in F$, witnessed by natural sequences $\vec\M$ and $\vec\N$, then $X\cap Y\in F$ as well, witnessed by the natural sequence $\vec\P$ induced by the indexing function $f^{\vec\P} := \max(f^{\vec\M}, f^{\vec\N})$ and unary relations $R^{\vec\P}_\alpha := \text{Code}(\bra{R^{\vec\M}_\alpha, R^{\vec\N}_\alpha})$. Indeed, if $\pi\colon\P_\alpha\to\P_\beta$ is a generic elementary embedding with critical point $\mu$ then $\mu$ is also the critical point of both $\pi\restr\M_\alpha\colon\M_\alpha\to\M_\beta$ and $\pi\restr\N_\alpha\colon\N_\alpha\to\N_\beta$.

  \qquad For normality, let $X\in F^+$ be $F$-positive, where we recall that this means that $X\cap C\neq\emptyset$ for every $C\in F$, and let $f\colon X\to\on$ be regressive. We want to show that $f$ is constant on an $F$-positive set.

  \qquad Assume this fails, meaning that there are natural sequences $\vec\M^\gamma$ for $\gamma$ such that for any generic elementary $\pi\colon\M_\alpha^\gamma\to\M_\beta^\gamma$ satisfies that $f(\crit\pi)\neq\gamma$. Define a new natural sequence $\vec\N$ as induced by the indexing function $g\colon\on\to\on$ given as $g(\alpha):=\sup_{\gamma<\alpha}\rk\M_\alpha^\gamma + \omega$ and unary relations $R_\alpha^{\vec\N}$ given as
\eq{
  R_\alpha^{\vec\N} := \text{Code}(\bra{\bra{\M_\alpha^\gamma\mid\gamma<\alpha}, f\restr\alpha}).
}

Now since $X$ is $F$-positive there exists a generic elementary embedding
\eq{
  \pi\colon\N_\alpha\to\N_\beta
}

with $\crit\pi\in X$. As $f(\crit\pi)<\crit\pi$ we get that $\pi(f(\crit\pi))=f(\crit\pi)$, so that we have a generic elementary embedding
\eq{
  \pi\restr\M_\alpha^{f(\crit\pi)}\colon\M_\alpha^{f(\crit\pi)}\to\M_\beta^{f(\crit\pi)},
}

  but this contradicts the definition of $\vec\M^{f(\crit\pi)}$! Thus $F$ is normal.
}

  The reason why we are being careful in showing all these analogous properties for the virtual Vop\v enka filter is that not all of the properties carry over. Indeed, note that uniformity of filters is non-trivial as we are working with proper classes\footnote{This boils down to the fact that the class club filter is not provably normal in \gbc, see \cite{class-fodor}}, and we will see in Theorem \ref{theo.berkeleyequiv} that uniformity of this filter is equivalent to there being no virtually Berkeley cardinals --- the following lemma is the first implication:

\lemm[N.; \gbc][lemm.vopclub]{
  Assume $\gvp$ and that there are no virtually Berkeley cardinals. Then the virtually Vop\v enka filter $F$ on $\on$ contains every class club $C$.
}
\proof{
  The crucial extra property we get by assuming that there are not any virtually Berkeleys is that $F$ becomes uniform, i.e. contains every tail $(\delta,\on)\subset\on$. Indeed, assume that $\delta$ is the least cardinal such that $(\delta,\on)\notin F$. Let $M$ be a transitive set with $\delta\subset M$ and $\gamma<\delta$ a cardinal. As $(\gamma,\on)\in F$ by minimality of $\delta$, we may fix a natural sequence $\vec\N$ witnessing this. Let $\vec\M$ be the natural sequence induced by the indexing function $f\colon\on\to\on$ given by
  \eq{
    f(\alpha):=\max(\alpha+1, \delta+1)
  }

  and unary relations $R_\alpha := \{\bra{M, \N_\beta}\mid\beta\leq\alpha\}$. If $\pi\colon\M_\alpha\to\M_\beta$ is a generic elementary embedding with $\crit\pi\leq\delta$, which exists as $(\delta,\on)\notin F$, then $\pi(R_\alpha)=R_\beta$ implies that $\pi\restr\M\colon\M\to\M$ with $\crit\pi\leq\delta$. We also get that $\crit\pi>\gamma$, as
  \eq{
    \pi\restr\N_{\crit\pi}\colon\N_{\crit\pi}\to\N_{\pi(\crit\pi)}
  }

  is an embedding between two structures in $\vec\N$ and hence $\crit\pi>\gamma$ as $\vec\N$ witnesses that $(\gamma,\on)\in F$. This means that $\delta$ is virtually Berkeley, a contradiction. Thus $\crit\pi>\delta$, implying that $(\delta,\on)\in F$.

  \qquad Note that the class $C_0\subset\on$ of limit ordinals is in $F$, since it is the diagonal intersection of the tails $(\alpha+1,\on)$. Now let $C\subset\on$ be a class club, and let
  \eq{
    C := \{a_\alpha\mid\alpha<\on\}
  }

  be its increasing enumeration. Then $C \supset C_0\cap\triangle_{\alpha<\on}(a_\alpha,\on)$, implying that $C\in F$.
}

\theo[N.; \gbc][theo.prewoodwood]{
  If there are no virtually Berkeley cardinals then $\on$ is virtually pre-Woodin iff $\on$ is virtually Woodin.
}
\proof{
  Assume $\on$ is virtually pre-Woodin, so \gvp\ holds by Theorem \ref{theo.vopwood} and we can let $F$ be the virtually Vop\v enka filter. The assumption that there are not any virtually Berkeley cardinals implies that for any class $A$ we not only get a virtually $A$-prestrong cardinal, but we get stationarily many such. Indeed, assume this fails --- we will follow the proof of Theorem \ref{theo.vopwood}.

  \qquad Failure means that there is some class $A$ and some class club $C$ such that there are no virtually $A$-prestrong cardinals in $C$. Since there are no virtually Berkeley cardinals, Lemma \ref{lemm.vopclub} imples that $C\in F$, so there exists some natural sequence $\vec\N$ such that whenever $\pi\colon\N_\alpha\to\N_\beta$ is an elementary embedding between two distinct structures of $\vec\N$ it holds that $\crit\pi\in C$. Define $f\colon\on\to\on$ as sending $\alpha$ to the least cardinal $\eta>\alpha$ such that $\alpha$ is not virtually $(\eta,A)$-prestrong if $\alpha\in C$, and set $f(\alpha):=\alpha$ if $\alpha\notin C$. Also define $g\colon\on\to\on$ as $g(\alpha)$ being the least strong limit cardinal in $C$ above $\alpha$ which is a closure point for $f$.

  \qquad Now let $\vec\M$ be the natural sequence induced by $g$ and
  \eq{
    R_\alpha := \code(\bra{A\cap H_{g(\alpha)}^V, \N_\alpha})
  }

  and apply \gvp\ to get $\alpha<\beta$ and a generic elementary embedding $\pi\colon\M_\alpha\to\M_\beta$, which restricts to
  \eq{
    \pi\restr(H_{g(\alpha)}^V,\in,A\cap H_{g(\alpha)}^V)\colon(H_{g(\alpha)}^V,\in,A\cap H_{g(\alpha)}^V)\to(H_{g(\beta)}^V,\in,A\cap H_{g(\beta)}^V),
  }

  making $\crit\pi$ virtually $(g(\alpha),A)$-prestrong and thus $\crit\pi\notin C$. But as we also get the embedding $\pi\restr\N_\alpha\colon\N_\alpha\to\N_\beta$, we have that $\crit\pi\in C$ by definition of $\vec\N$, $\contr$.

  \qquad Now fix any class $A$ and some large $n<\omega$ and define the class
  \eq{
    C := \{\kappa\in\Card\mid (H_\kappa^V,\in,A\cap H_\kappa^V)\prec_{\Sigma_n}(V,\in,A)\}.
  }

  This is a club and we can therefore find a virtually $A$-prestrong cardinal $\kappa\in C$. Assume that $\kappa$ is not virtually $A$-strong and let $\theta$ be least such that it is not virtually $(\theta,A)$-strong. Fix a generic elementary embedding
  \eq{
    \pi\colon(H_\theta^V,\in,A\cap H_\theta^V)\to (M,\in,B)
  }

  with $\crit\pi=\kappa$, $H_\theta^V\subset M$, $M\subset V$, $A\cap H_\theta^V = B\cap H_\theta^V$ and $\pi(\kappa)<\theta$.

  \qquad Now $\pi(\kappa)$ is inaccessible, and $(H_{\pi(\kappa)}^V, \in, A\cap H_{\pi(\kappa)}^V) = (H_{\pi(\kappa)}^M,\in,B\cap H_{\pi(\kappa)}^M)$ believes that $\kappa$ is virtually $(A\cap H_{\pi(\kappa)}^V)$-strong as in the proof of Theorem \ref{theo.virtchar}, meaning that $(H_\kappa^V,\in,A\cap H_\kappa^V)$ believes that there is a proper class of virtually $(A\cap H_\kappa^V)$-strong cardinals. But $\kappa\in C$, which means that
  \eq{
    (V,\in,A)\models\godel{\text{There exists a proper class of virtually $A$-strong cardinals}},
  }

  implying that \on\ is virtually Woodin.
}

Next, the following result is an improvement of a theorem in \cite{GitmanHamkins}, reducing the assumption of the existence of $0^\sharp$ to the existence of a virtually Berkeley cardinal, which we will later see is optimal:

\theo[N.; \gbc][theo.berkgvpmahlo]{
  If there exists a virtually Berkeley cardinal $\delta$ then $\gvp$ holds and $\on$ is not Mahlo.
}
\proof{
  If $\on$ was Mahlo then there would in particular exist an inaccessible cardinal $\kappa>\delta$, but then $H_\kappa^V\models\godel{\text{there exists a virtually Berkeley cardinal}}$, contradicting the incompleteness theorem, as we would have shown that
  \eq{
    \gbc+\Phi\proves\textsf{Con}(\gbc+\Phi),
  }

  witnessed by the model $(H_\kappa^V, \p^V(H_\kappa^V))$, with $\Phi$ being ``There exists a virtually Berkeley cardinal''.

  \qquad To show $\gvp$ we show that $\on$ is virtually pre-Woodin, which is equivalent by Theorem \ref{theo.vopwood}. Fix therefore a class $A$ --- we have to show that there exists a virtually $A$-prestrong cardinal. For every cardinal $\theta\geq\delta$ there exists a generic elementary embedding
  \eq{
    \pi_\theta\colon(H_\theta^V,\in,A\cap H_\theta^V)\to(H_\theta^V, \in, A\cap H_\theta^V)
  }

  with $\crit\pi<\delta$. By the pigeonhole principle we thus get some $\kappa<\delta$ which is the critical point of proper class many $\pi_\theta$, showing that $\kappa$ is virtually $A$-prestrong, making $\on$ virtually pre-Woodin.
}

%The proof of the following theorem is similar to the proof of Theorem 11 in \cite{GitmanHamkins}. Here we improve that result by reducing the assumption from the existence of $0^\sharp$ to the existence of a virtually Berkeley cardinal, which we will later see is optimal.
%
%\theo[N.][theo.clubforcing]{
%  If there exists a virtually Berkeley cardinal $\delta$ then there is a definable $\gbc$-preserving class forcing notion $\mathbb P$ such that whenever $C\subset\mathbb P$ is class generic over $V$ then
%  \eq{
%    V[C]\models\godel{\text{\gvp\ holds but \on\ is not Mahlo}}.
%  }
%}
%\proof{
%  Define $\mathbb P$ as the class forcing adding a class club which avoids the regular cardinals; conditions are closed bounded sets containing no regular cardinals, ordered by end-extension. This forcing is ${\leq}\gamma$-distributive for every ordinal $\gamma$, as it even contains a dense ${\leq}\gamma$-closed subclass: the class of conditions with elements above $\gamma$. This implies that the forcing adds no sets and preserves $\gbc$.\footnote{See e.g. Example 2.2.7 and Corollary 4.3.7 in \cite{Krapf}.}
%
%  \qquad Letting $G\subset\mathbb P$ be $V$-generic, $C:=\bigcup G$ is a class club containing no regular cardinals and $C$ and $G$ are interdefinable, so that $V[C]=V[G]\models\godel{\text{$\on$ is not Mahlo}}$. We need to show that $\gvp$ holds in $V[C]$. As $\mathbb P$ is definable we can let $\mathbb P^\theta$ be $H_\theta^V$'s version of $\mathbb P$, for any ordinal $\theta$.
%
%  \clai{
%    For any set $a$ and any natural number $n$ in the meta-theory, let $D_{a,n}\subset\mathbb P$ be the class of $\mathbb P$-conditions $c$ for which there exists an $\omega$-club of cardinals $\theta$ such that
%    \begin{enumerate}
%      \item $H_\theta^V\prec_{\Sigma_n} V$;
%      \item $c\cap\theta\subset\mathbb P^{H_\theta^V}$ is $H_\theta^V$-generic;
%      \item there exists a generic elementary $\pi\colon(H_\theta^V,\in,a,c\cap\theta)\to(H_\theta^V,\in,a,c\cap\theta)$ with critical point below $\delta$.
%    \end{enumerate}
%
%    Then $D_{a,n}$ is a definable dense subclass of $\mathbb P$.
%  }
%
%  \cproof{
%    Let $d$ be a $\mathbb P$-condition --- we are looking for a $\bar c\in D_{a,n}$ extending $d$. Pick a cardinal $\theta>\delta$ above the rank of $a$ of countable cofinality which we may also assume satisfies $H_\theta^V\prec_{\Sigma_n} V$ since this is the case for club many $\theta$. Fix a sequence $\bra{\theta_k\mid k<\omega}$ cofinal in $\theta$. We will now construct $c\subset\theta$ end-extending $d$ which is $H_\theta^V$-generic for $\mathbb P^{H_\theta^V}$. For each $k<\omega$ define the class
%    \eq{
%      D_k := \bigcap\{D\subset\mathbb P^{H_\theta^V}\mid\text{$D$ is open dense and $\Sigma_k^{H_\theta^V}(H_{\theta_k}^V)$}\}.
%    }
%
%    As $\mathbb P^{H_\theta^V}$ is ${\leq}\theta_k$-distributive for all $k<\omega$ we get that all the $D_k$'s are open dense subclasses of $\mathbb P^{H_\theta^V}$. Note that the $D_k$'s also suffice for genericity, as any subclass of $\mathbb P^{H_\theta^V}$ which is definable in $H_\theta^V$ contains some $D_k$. We can therefore build an $H_\theta^V$-generic $g\subset\mathbb P^{H_\theta^V}$ in $V$ such that $c:=\bigcup g$ end-extends $d$, by the usual diagonalisation argument. Since $\delta$ is virtually Berkeley, Proposition \ref{prop.boldfaceBerkeley} implies that we get a generic elementary embedding
%    \eq{
%      \pi\colon(H_\theta^V,\in,c,a)\to(H_\theta^V,\in,c,a)
%    }
%
%    with $\crit\pi<\delta$. Defining $\bar c:=c\cup\{\theta\}$ we get that $\bar c\in\mathbb P$ because $\theta$ is singular, so we have verified that $\bar c\in D_{a,n}$ end-extends $d$, implying that $D_{a,n}$ is dense in $\mathbb P$.
%  }
%
%  Now we use the claim to show that $\gvp$ holds in $V[C]$, so let us now work in $V[C]$. Fix some $\on$-sequence $\vec\M = \bra{\M_\alpha\mid\alpha<\on}$ of structures in a common language $\mathcal L$. As we are working with definable classes there is some $m<\omega$ and some $\Sigma_m$-formula $\varphi(v,a,C)$ with a set parameter $a\in V$ and a class parameter $C$ defining $\vec\M$.
%
%  \qquad Let $n<\omega$ be much larger than $m$ --- we will get back to exactly how large later. By the claim we get an $\omega$-club $C_{\text{claim}}$ of cardinals $\theta$ such that the corresponding initial segment of $C$ is in $D_{\bra{a,\mathcal L},n}$. This means that for every $\theta\in C_{\text{claim}}$ there exists a generic elementary embedding
%  \eq{
%    \pi\colon(H_\theta^V,\in,C\cap\theta,a,\mathcal L)\to(H_\theta^V,\in,C\cap\theta,a,\mathcal L),
%  }
%
%  with $\crit\pi<\delta$.
%
%  \qquad We now claim that $(H_\theta^V, \in, C\cap\theta)\prec_{\Sigma_m}(V, \in, C)$. To see this, suppose that $(H_\theta^V,\in, C\cap\theta)\models\psi(b)$ for some $\Sigma_m$-formula $\psi(v)$ and $b\in H_\theta^V$. This is then forced over $H_\theta^V$ by some $\mathbb P^{H_\theta^V}$-condition $c$, an initial segment of $C\cap\theta$. By choosing $n$ very large, what we meant was that $n$ is large enough to express this forcing relation\footnote{Note that this $n$ works for all the $\theta\in C_{\text{claim}}$ since $H_{\theta_0}^V\prec_{\Sigma_n}H_{\theta_1}^V$ for all $\theta_0,\theta_1\in C_{\text{claim}}$.}, so since $H_\theta^V\prec_{\Sigma_n}V$ we get that $c$ also forces $\psi(b)$ over $V$. So $(V,\in,C)\models\psi(b)$.
%
%  \qquad It follows that the definition of the class $\vec\M$ is absolute to $(H_\theta^V, \in, C\cap\theta)$ since $\vec\M$ was defined by the $\Sigma_m$-formula $\varphi(v,a,C)$, which also means that $\pi$ fixes $\vec\M$ as $\pi$ fixes both $\mathcal L$, $a$ and $C$ by the choice of $\pi$. We then have that $\pi(\M_\kappa)=\pi(\vec\M)_{\pi(\kappa)}=\M_{\pi(\kappa)}$, so that $\pi\restr\M_\kappa\colon\M_\kappa\to\M_{\pi(\kappa)}$ witnesses $\gvp$.
%}

\theo[N.; \gbc][theo.berkeleyequiv]{
  The following are equivalent:
  \begin{enumerate}
    \item if \gvp\ holds then $\on$ is Mahlo;
    \item $\on$ is virtually pre-Woodin iff $\on$ is virtually Woodin;
    \item There are no virtually Berkeley cardinals.
  \end{enumerate}
}
\proof{
  $(iii)\Rightarrow(ii)$ is Theorem \ref{theo.prewoodwood}, and the contraposed version of $(i)\Rightarrow(iii)$ is Theorem \ref{theo.berkgvpmahlo}. For $(ii)\Rightarrow(i)$ note that \gvp\ implies that \on\ is virtually pre-Woodin by Theorem \ref{theo.vopwood}, which by $(ii)$ means that it is virtually Woodin and the usual proof shows that virtually Woodins are Mahlo\footnote{See e.g. Exercise 26.10 in \cite{Kanamori}.}, showing $(i)$.
}

This also immediately implies the following equiconsistency, as virtually Berkeley cardinals have strictly larger consistency strength than virtually Woodin cardinals:

\qcoro[N.]{
  The existence of an inaccessible virtually pre-Woodin cardinal is equiconsistent with the existence of an inaccessible virtually Woodin cardinal.
}

\section{Behaviour in core models}
\label{section.core-models}

Most of the cardinals turn out to be downwards absolute to most inner models, including $L$.

\prop[][prop.absemb]{
  For any regular uncountable cardinal $\theta$, faintly $\theta$-measurable cardinals are downwards absolute to any transitive class $\U\subset V$ satisfying \\$\zf^-+\dc$.
}
\proof{
  Let $\kappa$ be faintly $\theta$-measurable, witnessed by a forcing poset $\mathbb P$ and a $V$-generic $g\subset\mathbb P$ such that, in $V[g]$, there is a transitive $\M$ and an elementary embedding $\pi\colon H_\theta^V\to\M$ with $\crit\pi=\kappa$. Fix a transitive class $\U\subset V$ which satisfies $\zf^-+\dc$. Restricting the embedding to $\pi\restr H_\theta^{\U}\colon H_\theta^{\U}\to\N$ we can now apply the Countable Absoluteness Lemma \ref{lemm.ctblabs} to $\pi\restr H_\theta^{\U}$ to get that there exists an embedding $\pi^*\colon H_\theta^{\U}\to\N^*$ in a generic extension of $U$, making $\kappa$ faintly $\theta$-measurable in $\U$.
}

\theo[N.][theo.core-model-behaviour]{
  Let $\theta$ be a regular uncountable cardinal.
  \begin{enumerate}
    \item $L\models\godel{\text{faintly $\theta$-measurables are equivalent to virtually $\theta$-prestrongs}}$.
    \item Assume that $L[\mu]$ exists. It then holds that\\
      $L[\mu]\models\godel{\text{faintly $\theta$-measurables are equivalent to virtually $\theta$-measurables}}$.
    \item Assume there is no inner model with a Woodin. It then holds that\\
      $K\models\godel{\text{faintly $\theta$-measurables are equivalent to virtually $\theta$-measurables}}$.
  \end{enumerate}
}
\proof{
  For $(i)$ simply note that if $\pi\colon L_\theta\to\N$ is a generic elementary embedding with $\N$ transitive, then by condensation we have that $\N=L_\gamma$ for some $\gamma\geq\theta$, so that $\pi$ also witnesses the virtual $\theta$-prestrongness of $\crit\pi$.

  \qquad $(ii)$: Assume that $V=L[\mu]$ for notational simplicity and let $\kappa$ be faintly $\theta$-measurable, witnessed by a generic elementary embedding $\pi\colon L_\theta[\mu]\to\N$ existing in some generic extension $V[g]$. By condensation we get that $\N=L_\gamma[\overline\mu]$ for some $\gamma\geq\theta$ and $\overline\mu\in V[g]$, but we are not guaranteed that $\overline\mu\in V$ here. Let $\lambda$ be the unique measurable cardinal of $V=L[\mu]$.

  \qquad Note that $\bar\mu$ is a measure on $\pi(\lambda)\geq\lambda$. If $\pi(\lambda)=\lambda$ then $L[\mu]=L[\bar\mu]$ by \cite[Kunen's Theorem 20.10]{Kanamori} and we trivially get that $\N\subset V$. Assume thus that $\pi(\lambda)>\lambda$, which implies that $L[\bar\mu]$ is an internal iterate of $L[\mu]$ by \cite[Kunen's Theorem 20.12]{Kanamori}. In particular it then holds that $L[\bar\mu]\subset~L[\mu]$, so again we get that $\N\subset V$.

  \qquad $(iii)$: Assume that $V=K=L[\mathcal E]$ and fix a faintly $\theta$-measurable cardinal $\kappa$, witnessed by a generic embedding $\pi\colon L_\theta[\mathcal E]\to\N=L_\gamma[\overline{\mathcal E}]$ in some generic extension $V[g]$. Now coiterate\footnote{See Section \ref{prelims.core-model-theory} for more information about the core model $K$ and coiterations.} $L[\mathcal E]$ with $L[\overline{\mathcal E}]$, and denote the last models by $\P$ and $\Q$. Since $K=K^{V[g]}$ and as $K$ is universal we get that $\Q\init\P$. Then the $L[\overline{\mathcal E}]$-to-$\Q$ branch did not drop, giving us an iteration embedding $i\colon L[\overline{\mathcal E}]\to\Q$.

  \qquad Note that $\crit i\geq\kappa$ as $\overline{\mathcal E}$ is simply the pointwise image of $\mathcal E$ under $\pi$, so nothing below $\kappa$ is touched and is therefore not used in the comparison either. This means that $\crit(i\circ\pi)=\kappa$, so that $(i\circ\pi)\colon L_\theta[\mathcal E]\to\Q$ witnesses that $\kappa$ is virtually $\theta$-measurable, since $\Q\init\P$ implies that $\Q\subset K$.
}

Note that the proofs of $(ii)$ and $(iii)$ above do not show that $\kappa$ is virtually $\theta$-prestrong, as it might still be the case that $\bar\mu\neq\mu$ or $\bar{\mathcal E}\neq\mathcal E$, so we cannot conclude that $L_\theta[\mu]\subset L_\theta[\bar\mu]$ or $L_\theta[\mathcal E]\subset L_\theta[\overline{\mathcal E}]$. It might still hold however; see Question \ref{ques.prestrong-equivalence}.

\section{Separation results}

Having proven many positive results about the relations between the virtual large cardinals in the previous sections, this section is dedicated to the negatives. More precisely, we will aim to \textit{separate} many of the defined notions (potentially under suitable large cardinal assumptions). See Figure \ref{fig.virtual-separations} for an overview of the relations between the various level-wise faint- and virtual large cardinals.

\qquad Our first separation result is that the virtuals form a level-by-level hierarchy.

\theo[N.]{
  Let $\alpha<\kappa$ and assume that $\kappa$ is faintly $\kappa^{+\alpha+2}$-measurable. Then
  \eq{
    L_\kappa\models\godel{\text{There is a proper class of $\lambda$ which are virtually $\lambda^{+\alpha+1}$-strong}}.
  }
}
\proof{
  Write $\theta:=\kappa^{+\alpha+1}$. Then by Theorem \ref{theo.virtchar} we get that either $\kappa$ is faintly $\theta^+$-strong in $L$ or otherwise, in particular, $L_\kappa$ thinks that there is a proper class of remarkables. In the second case we also get that $L_\kappa$ thinks that there is a proper class of $\lambda$ such that $\lambda$ is virtually $\lambda^{+\alpha+1}$-strong and we would be done, so assume the first case. Then $L_\kappa\prec_2 L_{\theta^+}$, so define for each $\xi<\kappa$ the sentence $\psi_\xi$ as
  \eq{
    \psi_\xi:\equiv\exists\lambda>\xi\colon\godel{\text{$\lambda$ is virtually $\lambda^{+\alpha+1}$-strong}}.
  }

  Then $\psi_\xi$ is $\Sigma_2(\{\alpha,\xi\})$ since being virtually $\beta$-strong is a $\Delta_2(\{\beta\})$-statement. As $L_{\theta^+}\models\psi_\xi$ for all $\xi<\kappa$ we also get that $L_\kappa\models\psi_\xi$ for all $\xi<\kappa$, which is what we wanted to show.
}

As we are only assuming $\kappa$ to be \textit{faintly} measurable in the above, this also shows that the faintly $\kappa^{+\alpha+1}$-measurable cardinals $\kappa$ form a strict hierarchy whenever $\alpha<\kappa$.

\qquad A separation result in a similar vein is the following, showing that it is consistent to have an inaccessible faintly measurable cardinal which is not weakly compact:

\prop[N.]{
  \label{prop.faintly-not-weakly-compact}
  Assuming $\kappa$ is measurable, there is a generic extension of $V$ in which $\kappa$ is inaccessible and faintly measurable, but not weakly compact.
}
\proof{
  Let $\mathbb P$ be the forcing notion that adds a $\kappa$-Suslin tree $\T$. By \cite{Kunen} it then holds that $\mathbb P * \T\cong\add(\kappa^+, 1)$, a ${<}\kappa^+$-closed forcing, which preserves the measurability of $\kappa$. Further, the $\mathbb P$ forcing is shown to preserve the inaccesibility of $\kappa$, making $\kappa$ inaccesible and faintly measurable in $V[g]$. Lastly, it cannot be weakly compact in $V[g]$ because $\T$ is a $\kappa$-tree without a branch, by definition.
}

Next, we show that the virtuals are in fact different from the faints. This is trivial in general as successor cardinals can be faintly measurable and are never virtually measurable, but the separation still holds true if we rule out this successor case.

For a slightly more fine-grained distinction let us define an intermediate large cardinal between the faintly and virtual.

\defi{
  Let $\kappa<\theta$ be infinite regular cardinals. Say that $\kappa$ is \textbf{faintly $\theta$-power-$\Phi$} for $\Phi\in\{\text{measurable}, \text{prestrong}, \text{strong}\}$ if it is faintly $\theta$-$\Phi$, witnessed by an embedding $\pi\colon H_\theta^V\to\N$, and $\p^V(\kappa) = \p^{\N}(\kappa)$.
}

Note that the proof of Lemma \ref{prop.virtit} shows that faintly power-measurables are also 1-iterable and so in particular weakly compact. Our separation result is then the following:

\theo[Gitman-N.][theo.virtualsep]{
  For $\Phi\in\{\text{measurable}, \text{prestrong}, \text{strong}\}$, if $\kappa$ is virtually $\Phi$, then there exist forcing extensions $V[G]$ and $V[G^*]$ such that
  \begin{enumerate}
    \item in $V[G]$, $\kappa$ is inaccessible and faintly $\Phi$, but not faintly power-$\Phi$, and
    \item in $V[G^*]$, $\kappa$ is faintly power-$\Phi$, but not virtually $\kappa^{++}$-prestrong.
  \end{enumerate}
}
\proof{
  We start with $(i)$. Let us assume that $\kappa$ is virtually measurable. This implies, in particular, that for every regular $\theta>\kappa$, we have generic embeddings $\pi:H_\theta\to \M$ with $\crit \pi=\kappa$ such that $\M\in V$. Thus, by Proposition~\ref{prop.coll}, we can assume that the generic embedding $\pi$ exists in a $\col(\omega,H_\theta)$-extension.

\qquad Let $\mathbb P_\kappa$ be the Easton support iteration that adds a Cohen subset to every regular $\alpha<\kappa$, and let $G\subset\mathbb P_\kappa$ be $V$-generic. Standard computations show that $\mathbb P_\kappa$ preserves all inaccessible cardinals.

\qquad Fix a regular $\theta\gg\kappa$ and let $h\subseteq \col(\omega,H_\theta)$ be $V[G]$-generic. In $V[h]$, we must have an elementary embedding $\pi:H_\theta\to \M$ with $\crit \pi=\kappa$ and $\M\in V$, and we can assume without loss that $\M$ is countable. Obviously, $\pi\in V[G][h]$. Working in $V[G][h]$, we will now lift $\pi$ to an elementary embedding on $H_\theta[G]$. To ensure that such a lift exists, it suffices to find in $V[G][h]$ an $\M$-generic filter for $\pi(\mathbb P_\kappa)$ containing $\pi''G$\footnote{This standard lemma is referred to in the literature as the \textbf{lifting criterion}.}. Observe first that $\pi''G=G$ since the critical point of $\pi$ is $\kappa$ and we can assume that $\mathbb P_\kappa\subseteq V_\kappa$. Next, observe that $\pi(\mathbb P_\kappa)\cong \mathbb P_\kappa*\mathbb P_{\text{tail}}$, where $\mathbb P_{\text{tail}}$ is the forcing beyond $\kappa$. Since $\M[G]$ is countable, we can build an $\M[G]$-generic filter $G_{\text{tail}}$ for $\mathbb P_\text{tail}$ in $V[G][h]$. Thus, $G*G_{\text{tail}}$ is $\M$-generic for $\pi(\mathbb P_\kappa)$, and so we can lift $\pi$ to $\pi:H_\theta[G]\to \M[G][G_{\text{tail}}]$. Since $\theta$ was chosen arbitrarily, we have just shown that $\kappa$ is faintly measurable in $V[G]$.

\qquad Now suppose that $\kappa$ is faintly power-measurable in $V[G]$. Fix regular $\theta<\bar\theta$ and a generic elementary embedding $\sigma:H_{\bar\theta}[G]\to \N$ with $\crit\sigma=\kappa$ and $\p(\kappa)^{V[G]}=\p(\kappa)^{\N}$. By elementarity, $H_{\sigma(\theta)}^{\N}=\sigma(H_\theta)[\sigma(G)]$ is a forcing extension of $K=\sigma(H_\theta)$ by $\sigma(G)=G*\bar G_{\text{tail}}\subseteq \sigma(\mathbb P_\kappa)\cong \mathbb P_\kappa*\bar{\mathbb P}_{\text{tail}}$. Thus, we have the restrictions $\sigma:H_\theta\to K$ and $\sigma:H_\theta[G]\to K[G][\bar G_{\text{tail}}]$. Let us argue that $\p^{V[G]}(\kappa)\subseteq \p^{K[G]}(\kappa)$, and hence we have equality. Suppose $A\subseteq\kappa$ in $V[G]$ and let $\dot A$ be a nice $\mathbb P_\kappa$-name for $A$, which can be coded by a subset of $\kappa$. Since $\crit\sigma=\kappa$, we have that $\dot A\in K$, and hence $A=\dot A_G\in K[G]$. But now it follows that the $K[G]$-generic for ${\rm Add}(\kappa,1)$, the forcing at stage $\kappa$ in $\sigma(\mathbb P_\kappa)$, cannot be in $V[G]$. Thus, we have reached a contradiction, showing that $\kappa$ cannot be faintly power-measurable in $V[G]$.

\qquad If $\Phi = \text{measurable}$, then we are done at this point. For $\Phi = \text{prestrong}$ we simply note that $G\in\M[G*G_{\text{tail}}]$ so that $H_\theta^{V[G]}\subseteq\N[G*G_{\text{tail}}]$ as well, and since we lifted $\pi$, we still have $\pi(\kappa) > \theta$ in the $\Phi = \text{strong}$ case.

  \qquad For $(ii)$, we change $\mathbb P_\kappa$ to only add Cohen subsets to \textit{successor} cardinals $\lambda<\kappa$ and call the resulting forcing $\mathbb P^*_\kappa$. Let $G^*\subseteq \mathbb P^*_\kappa$ be $V$-generic. We verify that $\kappa$ is faintly-$\Phi$ as above by lifting an embedding $\pi:H_\theta\to \M$, with $\M\subseteq V$, to
  \eq{
    \pi:H_\theta[G^*]\to \M[G^*][G_{\text{tail}}]
  }

  in a collapse extension $V[G^*][h]$. The lifted embedding is $\kappa$-powerset preserving because a subset of $\kappa$ from $\M[G^*][G_{\text{tail}}]$ has to already be in $\M[G^*]$ as $\mathbb P_{\text{tail}}$ is ${\leq}\kappa$-closed, and $\M[G^*]\subseteq V[G^*]$. So it remains to show that $\kappa$ is not virtually $\kappa^{++}$-prestrong. Suppose it is and fix a generic embedding $\sigma:H_{\kappa^{++}}[G^*]\to K[G^*][G_{\text{tail}}]$ with $\crit\sigma=\kappa$ and $H_{\kappa^{++}}[G^*]\subseteq K[G^*][G_{\text{tail}}]$. It follows that the generic subset of $\kappa^+$ added at stage $\kappa^+$ by the tail forcing must be $V[G^*]$-generic, which is contradictory.
}

Starting from a much stronger hypothesis, it can be shown that a power-measurable cardinal need not even be virtually $\kappa^+$-measurable.

\qquad To see this, we first observe that virtually $\theta$-measurable cardinals $\kappa$ are $\Pi^2_1$-indescribable for all $\theta>\kappa$. The proof is identical to the standard Hanf-Scott proof that measurable cardinals are $\Pi^2_1$-indescribable; see e.g. Hanf-Scott's Proposition 6.5 in \cite{Kanamori}. It should be noted that we crucially need the ``virtual'' property for the proof to go through.

\qquad Using this indescribability fact, the proof of the following theorem is precisely the same as Hamkins' Proposition 8.2 in \cite{HolySchlicht}:

\theo[Virtualised Hamkins][theo.Hamkins]{
Assuming $\kappa$ is a $\kappa^{++}$-tall cardinal,\footnote{Recall that $\kappa$ is \textbf{$\kappa^{++}$-tall} if there is an elementary embedding $j\colon V\to M$ with $\crit j=\kappa$, $^\kappa M\subset M$ and $j(\kappa)>\kappa^{++}$.} there is a forcing extension in which $\kappa$ is not virtually $\kappa^+$-measurable, but becomes measurable in a further $\text{Add}(\kappa^+,1)$-generic extension.
}

This then immediately gives the separation result.

\coro[][coro.virtualsep]{
    Assuming $\kappa$ is a $\kappa^{++}$-tall cardinal, it is consistent that $\kappa$ is faintly power-measurable, but not virtually $\kappa^+$-measurable.
}
\proof{
	By the above Theorem \ref{theo.Hamkins} we may assume that $\kappa$ is not virtually $\kappa^+$-measurable but that it is measurable in $V^{\mathbb P}$ for $\mathbb P:=\text{Add}(\kappa^+,1)$, so that $\kappa$ is faintly power-measurable.
}

In contrast to the above separation result, note that we showed in Corollary~\ref{coro.faintvirtualWoodin} and Theorem~\ref{theo.vopwood} that the faint-virtual distinction vanishes when we are dealing with virtual pre-Woodin or Woodin cardinals.

\qquad Our next separation result is concerning the virtually prestrong and virtually strong cardinals.

\coro[N.][coro.rank-into-rank-prestrong-strong]{
  There exists a virtually rank-into-rank cardinal iff there is an uncountable cardinal $\theta$ and a virtually $\theta$-prestrong cardinal which is not virtually $\theta$-strong.
}
\proof{
  $(\Leftarrow)$ is directly from the above Proposition \ref{prop.superstrong-rank-into-rank} and Theorem \ref{theo.virtchar}.

  \qquad $(\Rightarrow)$: Here we have to show that if there exists a virtually rank-into-rank cardinal then there exists a $\theta>\kappa$ and a virtually $\theta$-prestrong cardinal which is not virtually $\theta$-strong. Let $(\kappa,\theta)$ be the lexicographically least pair such that $\kappa$ is virtually $\theta$-rank-into-rank, which trivially makes $\kappa$ virtually $\theta$-prestrong. If $\kappa$ was also virtually $\theta$-strong then it would be $\Sigma_2$-reflecting\footnote{See Exercise 20.18 in \cite{Jech}, and Section \ref{prelims.large-cardinals} for a definition of $\Sigma_n$-reflecting cardinals.}, so that the statement that there exists a virtually rank-into-rank cardinal would reflect down to $H_\kappa^V$, contradicting the minimality of $\kappa$.
}

\qquad Figure \ref{fig.virtual-separations} summarises the separation results along with the results from Section \ref{section.core-models}. Note that it \textit{might} be the case that virtually $\theta$-measurables are always virtually $\theta$-prestrong (and hence also equivalent in $L[\mu]$ and $K$ below a Woodin cardinal); see Question \ref{ques.prestrong-equivalence}.

\figx[fig.virtual-separations][200]{virtual-separations.pdf}{Direct implications between virtuals, where the red lines with crosses indicate that $\zfc$ does not prove the reverse implication.}

\section{Indestructibility}

It is well-known that supercompact cardinals $\kappa$ can be made indestructible by all ${<}\kappa$-directed closed forcings by a suitable \textit{Laver preparatory forcing}, which is the main theorem in the seminal paper \cite{laver-indestructibility}. A natural question, then, is whether similar results hold for the faintly and virtual versions. We noted in Proposition \ref{prop.virtit} that the virtuals are weakly compact, so the following theorem from \cite{mousestack} shows that the consistency strength of indestructible virtual supercompacts is very large, potentially even in the realm of supercompacts:

\theo[Schindler]{
  The consistency strength of a weakly compact cardinal $\kappa$ which is indestructible by ${<}\kappa$-directed closed forcing is larger than the consistency strength of a proper class of strong cardinals and a proper class of Woodin cardinals.
}

This gets close to resolving the question about the indestructible virtuals, so what about the faintly supercompact cardinals? To make things a bit easier for ourselves, let us make the notion a bit stronger.

\defi{
  Fix uncountable cardinals $\kappa<\theta$. Then $\kappa$ is \textbf{generically setwise $\theta$-supercompact} if there exists a generic extension $V[g]$, a transitive $\N\in V[g]$ and a generic elementary embedding $\pi\colon H_\theta^V\to\N$, $\pi\in V[g]$, with $\crit\pi=\kappa$, $\pi(\kappa)>\theta$ and $V[g]\models{^{<\theta}}\N\subset\N$. If it holds for all $\theta>\kappa$ then we say that $\kappa$ is \textbf{generically setwise supercompact}.
}

Note that the only difference between a generically setwise $\theta$-supercompact cardinal and a virtually $\theta$-supercompact cardinal is that the former is closed under sequences \textit{in the generic extension}, where the latter is only closed under sequences in $V$; i.e., that $V\cap{^{<\theta}}\N\subset\N$.

\qquad Ostensibly this seems to be an incredibly strong notion, as the target model now inherits a lot of structure from the generic extension. A first stab at an upper consistency bound could be to note that if there exists a proper class of Woodin cardinals then $\omega_1$ is generically setwise supercompact. This can be shown using the countable stationary tower, see 2.7.7 and 2.7.8 in \cite{stationary-tower}.

\qquad But, surprisingly, the following result from Usuba shows that they can exist in $L$:

\qtheo[\cite{usuba}][theo.usuba]{
  If $\kappa$ is virtually extendible then $\col(\omega,{<}\kappa)$ forces that $\omega_1$ is generically setwise supercompact.
}

It turns out that this slightly stronger notion \textit{does} have indestructiblity properties. We warm up by firstly showing that they are indestructible by small forcing.

\prop[N.-Schlicht]{
  Generic setwise supercompactness of $\kappa$ is indestructible for forcing notions of size $<\kappa$.
}
\proof{
  Fix a forcing $\mathbb P$ of size ${<}\kappa$ and assume without loss of generality that $\mathbb P\in H_\kappa^V$, and fix also a cardinal $\theta>\kappa$. Using the setwise supercompactness of $\kappa$ we may fix a forcing $\mathbb Q$ and a $V$-generic $h\subseteq\mathbb Q$ such that in $V[h]$ we have an elementary $\pi\colon M:=H_\theta^V\rightarrow\N$ in $V[h]$ with $\N$ ${<}\theta$-closed.

  \qquad Let $g\subset\mathbb P$ be $V[h]$-generic and work in $V[g\times h]$. By the Lifting Criterion \ref{prop.lifting-criterion} we get a lift $\tilde{\pi}\colon M[g]\rightarrow \N[g]$ of $\pi$. If $\kappa$ is a limit cardinal, then we may choose a cardinal $\lambda<\kappa$ such that $\mathbb P\in H_\lambda^V$. Since $\mathbb P$ has the $\lambda^+$-cc in $V$ we get that $\pi(\mathbb P)=\mathbb P$ has the $\lambda^+$-cc in $\N$ and hence in $V[h]$ as well, making $\N[g]$ ${<}\theta$-closed by Lemma \ref{lemm.lucke-schlicht} and we are done.

  \qquad If $\kappa=\nu^+$, then there are no cardinals between $\nu$ and $\pi(\kappa)$ in $\N$ and hence $|\theta|\leq \nu$. Thus it suffices to show that $\N[g]$ is $\nu$-closed. Since $\pi(\mathbb P)=\mathbb P$ has size ${\leq}\nu$ in $V$, it has the $\nu^{+V[h]}$-cc in $V[h]$. Therefore $\N[g]$ is again ${<}\theta$-closed by Lemma \ref{lemm.lucke-schlicht}.
}

Next, we show that these generic setwise supercompact cardinals $\kappa$ \textit{are} in fact indestructible for ${<}\kappa$-directed closed forcings, without having to do any preparation forcing at all.

\theo[N.-Schlicht]{
  Generic setwise supercompactness of $\kappa$ is indestructible for ${<}\kappa$-directed closed forcings.
}
\proof{
  Suppose that $\kappa$ is generically setwise supercompact, $\mathbb P$ is a ${<}\kappa$-directed closed forcing and $g$ is $\mathbb P$-generic over $V$. We will show that $\kappa$ is generically setwise supercompact in $V[g]$.

  \qquad In $V$ fix a regular $\theta>\kappa$ such that $\mathbb P\in H_\theta^V$, and let $\mathbb Q$ be the forcing given by the definition of setwise supercompactness. Let $h$ be $\mathbb Q$-generic over $V[g]$. Let $\pi\colon H_\theta^V\rightarrow \N$ be as in the definition of generically setwise supercompactness, so that $\pi\in V[h]\subset V[g\times h]$. Work in $V[g\times h]$.

  \qquad We may assume that $\theta=\theta^{<\theta}$ holds, as otherwise we just replace $\mathbb Q$ with $\mathbb{Q}*\col(\theta,\theta^{<\theta})$ --- we retain the ${<}\theta$-closure of $\N$ because $\col(\theta,\theta^{<\theta})$ is ${<}\theta$-closed. We can further assume that $\abs{\N}=\theta^{<\theta}=\theta$, as otherwise we can take a hull of $\N$ containing $\ran(\pi)$ and recursively close under ${<}\theta$-sequences, ending up with a ${<}\theta$-closed elementary substructure $\h\prec\N$ containing $\ran(\pi)$ -- now replace $\N$ by the transitive collapse of $\h$.

  \clai{
    There is a $\pi(\mathbb P)$-generic filter $\tilde{g}$ over $\N$ that extends $\pi[g]$.
  }

  \cproof{
    Since $\N$ is (in particular) $\abs{\mathbb P}$-closed in $V[h]$ and $\mathbb P$ is trivially $\abs{\mathbb P}^+$-cc, Lemma \ref{lemm.lucke-schlicht} implies that $\N$ is still $\abs{\mathbb P}$-closed in $V[g\times h]$. As below, we thus still get that $\pi[g]\in\N$. Now work in $V[h]$, where we have full ${<}\theta$-closure of $\N$.

    \qquad Since $\pi(\mathbb P)$ is directed, there is a condition $q\leq \pi[g]$ in $\pi(\mathbb P)$. Using the fact that $\abs{\N}=\theta$ and $\pi(\mathbb P)$ is ${<}\theta$-closed, we can construct a $\pi(\mathbb P)$-generic filter $\tilde{g}$ over $\N$ with $q\in g$. Namely, enumerate the dense subsets of $\pi(\mathbb P)$ that are elements of $\N$ in order type $\theta$ and use the fact that the initial segments of the sequence, and of the corresponding sequence of conditions that we construct, are in $\N$. Then $\tilde{g}$ is as required.
  }

    Since we now have that $\pi[g]\subseteq \tilde{g}$ by the claim, the lifting criterion implies that we can lift $\pi$ to $\tilde{\pi}\colon H_\theta^V[g]\rightarrow\N[\tilde{g}]$.

    \qquad It thus remains to see that $\N[\tilde{g}]$ is ${<}\theta$-closed. To see this, take a sequence $\vec{x}=\langle x_i\mid i< \gamma\rangle $ with $\gamma<\theta$ and $x_i\in\N[\tilde{g}]$ and find names $\sigma_i\in\N$ with $\sigma_i^{\tilde{g}}=x_i$ for all $i<\gamma$. Since $^{<\theta}\N\subseteq\N$ we have that $\vec{\sigma}=\langle \sigma_i\mid i<\gamma\rangle\in\N$, and from $\vec{\sigma}$ we obtain a canonical name $\vec{\sigma}^\bullet\in\N$ with $\vec{\sigma}^{\bullet {\tilde{g}}}=\vec{x}\in\N[\tilde{g}]$.
}

Investigating further, we also show indestructibility for some forcings that do not fall into the above-mentioned categories.

\prop[N.-Schlicht]{
  Generic setwise supercompactness of a regular cardinal $\kappa$ is indestructible for $\add(\omega, \kappa)$. If $\kappa$ is a successor cardinal then it is also indestructible for $\col(\omega, {<}\kappa)$.
}
\proof{
  Let $g$ be $\add(\omega,\kappa)$-generic over $V$. In $V$ fix a regular $\theta>\kappa$ and let $\mathbb Q$ be the forcing given by the definition of generic setwise supercompactness. Let $h$ be $\mathbb Q$-generic over $V[g]$ and work in $V[g\times h]$.

  \qquad Let $\pi\colon H_\theta^V\rightarrow\N$ be as in the definition of generically setwise supercompactness. Moreover, let $\tilde{g}$ be $\add(\omega,\pi(\kappa))$-generic over $V[g\times h]$. Since $\pi[g]=g$, the lifting criterion allows us to extend $\pi$ to some $\tilde{\pi}\colon H_\theta^V[g]\rightarrow\N[g\times \tilde{g}]$. To show that $\N[g\times \tilde{g}]$ is ${<}\theta$-closed in $V[g\times h\times \tilde{g}]$, it suffices that $\add(\omega,\pi(\kappa))$ has the ccc by Lemma \ref{lemm.lucke-schlicht}.

  \qquad For $\col(\omega,{<}\kappa)$, we proceed similarly. Assume that $\kappa=\nu^+$. Take $\col(\omega,{<}\kappa)$-, $\mathbb Q$- and $\col(\omega,{<}\pi(\kappa))$-generic filters $g$, $h$ and $\tilde{g}$. $\pi$ and $\N$ are as above. Since $\nu<\kappa<\theta<\pi(\kappa)$ and there are no cardinals between $\nu$ and $\pi(\kappa)$ (in $\N$ and thus also in $V[h]$), ${<}\theta$-closure means $\nu$-closure (in any model containing $V[h]$). By Lemma \ref{lemm.lucke-schlicht}, it is thus sufficient to know that $\col(\omega,{<}\pi(\kappa))$ has the $\nu^+$-cc in $V[g\times h]$. This is because $\pi(\kappa)=\nu^{+\N}\leq\nu^{+V[g\times h]}$.
}

Usuba's Theorem \ref{theo.usuba} shows that the \textit{consistency strength} of these generically setwise supercompact cardinals is small, but do they appear naturally anywhere? The following result shows that we cannot find any in either $L$, $L[\mu]$ nor $K$ below a measurable cardinal:

\prop[N.-Schlicht]{
  No cardinal $\kappa$ is generically setwise supercompact in either $L$, $K$ below a measurable cardinal, or $L[\mu]$ with $\mu$ being a normal ultrafilter.
}
\proof{
  Assume first that $V=L$ and that $\kappa$ is generically setwise supercompact. Let $g$ be a generic filter and $\pi\colon L_\theta\rightarrow\N$ an embedding in $V[g]$ with $\pi\restr L_{\kappa^{+L}} \in\N$. Then $\N=L_\alpha$ for some $\alpha$ by condensation and thus $\pi\restr H_{\kappa^{+L}}\in L$. But this would induce a ${<}\kappa$-complete ultrafilter on $\kappa$, contradicting $V=L$.

  \qquad Assume now that there exists no inner model with a measurable cardinal and let $K$ be the core model. Let $\kappa$ be generically setwise supercompact in $K$ and $\pi\colon K|\theta\to\N$ be the generic elementary embedding. As above we get that
  \eq{
    \pi\restr H_{\kappa^{+K}}\in\N,
  }

  so that again $\N$ has a measure on $\kappa$. But that means that $K|\kappa$ has a measurable cardinal, which is a contradiction since $\kappa$ is inaccessible (in both $V$ and $K$) and therefore $K|\kappa\models\zfc$, contradicting our assumption that no inner model with a measurable cardinal exists.

  \qquad Assume now that $V=L[\mu]$ and that $\kappa$ is generically setwise supercompact, witnessed by a generic embedding $\pi\colon L_\theta[\mu]\to L_\alpha[\bar\mu]$. In particular this means that $\pi\restr L_{\kappa^{+L[\mu]}}[\mu]\in L_\alpha[\bar\mu]$. If $\crit\mu<\kappa$ then $\mu=\bar\mu$ and
  \eq{
    \p^{L[\mu]}(\kappa)=\p^{L[\bar\mu]}(\kappa),
  }

  so that both $\pi(\kappa)$ and $\kappa$ are now measurable cardinals in $L[\bar\mu]$, contradicting Solovay's Lemma \cite[20.2]{Kanamori}. So $\crit\mu\geq\kappa$.

  \qquad If $\pi(\crit\mu)>\crit\mu$ then by Kunen's Theorem \cite[20.12]{Kanamori} we get that $L[\bar\mu]$ is an iterate of $L[\mu]$. But iteration embeddings preserve the subsets of their critical point, so again we have that $\p^{L[\mu]}(\kappa)=\p^{L[\bar\mu]}$ and we get the same contradiction as before.

  \qquad Lastly, if $\crit\mu>\kappa$ and $\pi(\crit\mu)=\crit\mu$ then $\mu=\bar\mu$ by Kunen's Theorem \cite[20.10]{Kanamori}, so we get a contradiction as in the $\crit\mu<\kappa$ case.
}

\end{document}