What are examples of non-equivalent virtualizations of a large cardinal?

Question

This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization process.

If $A$ is a large cardinal property characterized by the existence of suitable embeddings. A cardinal is virtually $A$ if the embeddings characterizing $A$ exist in some set-forcing extensions. Such a reformulation of a large cardinal axiom is called a virtualization of $A$.

Remarkable cardinal is an instance of a virtual large cardinal axiom. By definition, it is the virtualization of a certain characterization of supercompact cardinals presented by Magidor (See ZBL0263.02034). Later Gitman and Schindler proved that the virtualization of a particular characterization of strong cardinals gives rise to an equivalent version of remarkable cardinals as well:

Theorem. The followings are equivalent:

(1) $\kappa$ is remarkable.

(2) (Virtualization of Magidor's characterization of supercompacts) For every $\lambda>\kappa$, there is $\overline{\lambda}<\kappa$ such that in a set-forcing extension there is an elementary embedding $j:V_{\overline{\lambda}}^{V}\rightarrow V_{\lambda}^{V}$ with $j(crit(j))=\kappa$.

(3) (Virtualization of a characterization of strong cardinals) For every $\lambda>\kappa$ there is $\alpha>\lambda$ and a transitive $M$ with $V_{\lambda}\subseteq M$ such that in a set forcing extension there is $j:V_{\alpha}^V\rightarrow M$ with $crit(j)=\kappa$ and $j(\kappa)>\lambda$.

Thus, virtualization doesn't preserve the consistency strength order of large cardinal axioms strictly because there are characterizations of totally different large cardinals whose virtualizations are equivalent.

Conversely, one may ask whether different characterizations of the same large cardinal axiom are always equivalent. According to what Joel mentioned in his comment here, the answer is negative! So:

Question 1. What are examples of large cardinal axioms which virtualization of their different characterizations gives rise to essentially different virtual large cardinals? Can Vopěnka's principle (which has many characterizations) be such a large cardinal axiom with non-equivalent virtual forms?

As the consistency strength of the virtualization of a certain large cardinal axiom highly depends on one's choice of its characterization, it is reasonable to ask about the possible criteria which guides us to choose the right characterization to virtualize.

Question 2. What are the criteria for choosing a certain characterization of a large cardinal to virtualize among many others?

Answer 1

An important feature which separates the notion of virtual large cardinals from the related notion of generic large cardinals is that we only consider embeddings on set-sized structures. Since most large cardinals are characterized by embeddings of the entire universe, there will always be some arbitrary choices made in the characterization we pick because we need to reduce to a set-sized embedding.

It is true that Schindler originally thought of remarkable cardinals as virtual supercompacts using the Magidor characterization. But in fact, the most natural characterization of supercompact cardinals in terms of set-sized embedding: $\kappa$ is supercompact if for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive model $N$ closed under $\lambda$-sequences with an elementary embedding $j:V_\alpha\to N$ with critical point $\kappa$, when virtualized gives a remarkable cardinal! (Note that $N$ is assumed to be closed under $\lambda$-sequences in $V$.) So remarkable cardinals is not an example you are looking for. Vopenka's Principle comes closer.

Bagaria showed that Vopenka's Principle is equivalent to the existence of a $C^{(n)}$-extendible cardinal for every $1\leq n<\omega$. $C^{(n)}$ is the collection of all cardinal $\alpha$ such that $V_\alpha\prec_{\Sigma_n} V$. A cardinal $\kappa$ is $C^{(n)}$-extendible if for every $\alpha\in C^{(n)}$ above $\kappa$, there is an elementary $j:V_\alpha\to V_\beta$ with critical point $\kappa$ and $\beta\in C^{(n)}$. It is not difficult to see that we can always obtain a $C^{(n)}$-extendibility embedding with the additional property that $j(\kappa)>\alpha$. The proof makes use of the Kunen inconsistency and it turns out that this is fundamental.

Kunen's inconsistency does not hold for virtual embeddings in the sense that in a forcing extension we can have embeddings $j:V_\lambda\to V_\lambda$ with $\lambda$ much greater than the supreumum of the critical sequence of $j$. This has the effect of messing up some properties of virtual $C^{(n)}$-extendible cardinals and virtual Vopenka's Principle. We showed with Joel Hamkins that virtual $C^{(n)}$-extendible cardinals with the assumption that $j(\kappa)>\alpha$ are not equivalent to virtual $C^{(n)}$-extendible cardinals, call them virtual weakly $C^{(n)}$-extendible cardinals, without this assumption (see here). However the two notions are equiconsistent, so the consistency strength is not affected by the different characterizations.

The consequence for virtual Vopenka's Principle is that it is no longer equivalent to the existence of a virtual $C^{(n)}$-extendible cardinal for every $1\leq n<\omega$, but to the existence of the virtual weak $C^{(n)}$-extendibles.

It seems that the only instance of equivalent characterizations producing different virtual notions all revolve around the consequences of the failure of the Kunen Inconsistency.

In practice the most robust embedding characterizations for virtualizing have the form $j:V_\alpha\to V_\beta$, namely the target model has the form $V_\beta$. In my opinion, the reason that it appears that strong cardinals and supercompact cardinals have equivalent virtual versions is that strong cardinals simply don't have a virtual characterization because they don't have a "robust" embedding characterization for virtualizing. A discussion of this can be found in our joint paper with Ralf Schindler (see here).