What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess?


Ok, I already posted this question, but a couple of notational errors and assumptions were made in the previous version. Hopefully, this new equivalent version will be better understood.


Let $U$ be a nonprinciple $\lambda$-complete ultrafilter over $\lambda$.
Let $\pi\_U(f)$ for a function $f$ with domain $\lambda$ be defined as follows:
$$\pi\_U(f)=\{\pi\_U(g):\{\alpha<\lambda:g(\alpha)\in f(\alpha)\}\in U\}$$


Let $M$ be $\{\pi\_U(f):\mathrm{Dom}(f)=\lambda\}$. Finally, let $\lambda\_0=\lambda$ and:
$$\lambda\_{n+1}=|\{\pi\_U(f):\{\alpha<\lambda:g(\alpha)\in \lambda\_n\}\in U\}|$$


**Then, which of the following are always true:**


* If $\lambda$ is $\theta$-strong, then $V\_\theta\subset M$.
* If $\lambda$ is $\theta$-supercompact, then $M^\theta\subset M$.
* If $\lambda$ is $n$-superstrong, then $V\_{\lambda\_n}\subset M$.
* If $\lambda$ is $n$-huge, then $M^{\lambda\_n}\subset M$.



What strengthenings of measurability do the Mostowski collapses of ultrapowers possess?
Let's adopt Miha's interpretation of your question: which kinds of large cardinal properties can be witnessed by ultrapower embeddings? 


Here, there are a variety of things one can say. 


If one allows $\kappa$-complete measures on arbitrary sets, then basically all the usual large cardinal properties can be witnessed with suitably chosen ultrapower embeddings. For example, supercompactness embeddings $j:V\to M$ are ultrapowers by normal fine measures on $P\_\kappa\theta$, and these kinds of embeddings are also $\alpha$-strong, when $\beth\_\alpha\leq\theta$. 


Hugeness and almost hugeness are also witnessed by ultrapower embeddings on a suitable set, so this means that (assuming appropriate consistency strength), we can realize all the properties on your list with ultrapower embeddings. 


But meanwhile, one cannot expect to *define* the various extender-based large cardinal notions in ZFC using only ultrapowers. For example, it is relatively consistent that $\kappa$ is strong, but for large enough $\alpha$, there is no ultrapower embedding $j:V\to M$ by a measure on any set for which $V\_\alpha\subset M$. 


If you mean to allow only ultrapowers by measures on $\kappa$ itself, then much of those embeddings are not possible. 


* No ultrapower embedding by a measure on $\kappa$ can realize $\kappa+2$ strongness, since $M\_{j(\kappa)}$ will have size $2^\kappa$ in $V$ and therefore cannot contain all of $V\_{\kappa+2}$.
* In particular, no ultrapower embedding by a measure on $\kappa$ can realize $2^\kappa$-supercompactness, for then it would realize $\kappa+2$-strongness.
* Meanwhile, there is a kind of border right at that level. There is an interesting theorem of Woodin showing how to make the least measurable cardinal also a little supercompact, and in that model, we get that the $\kappa^+$-supercompactness of $\kappa$ is witnessed by a normal ultrapower on $\kappa$. Necessarily, by the previous observations, $\kappa^+<2^\kappa$ in this model.
(Arthur Apter also has some work on this.)


That last example would be perhaps one of the most interesting situations fitting under the umbrella of this interpretation of your question. 


