elementary embeddings $j$ in set theory with $V$ and $M$ I'm confused by a variety of elementary non-trivial elementary embedings $j$ we might have.
There are 9 "syntactical" possiblities;Here $M$ is a transitive model. I'll name them with a wish that someone would tell me which are impossible, which are not useful and which are strong cardinals-like: $$j_1:L\to L,$$
$$j_2:L\to M,$$ $$j_3:L\to V,$$ $$j_4:M\to L,$$ $$j_5:M\to M,$$ $$j_6:M\to V,$$ $$j_7:V\to L,$$ $$j_8:V\to M,$$ $$j_9:V\to V.$$  What I know: $j_9$ is impossible by Kunen's barrier.$j_1$ implies $M\neq L$,$j_5$ implies $M\neq V$. $j_6$ and $j_8$ are a puzzle for me.
 A: For simplicity I work throughout in an appropriate class theory, say $\mathsf{NBG}$ or $\mathsf{MK}$. Also, I'll blackbox the fact that "There is a nonprincipal countably closed ultrafilter on some uncountable cardinal" is equivalent to "There is a measurable cardinal." This is a good exercise if you haven't seen it before! Also, Hamkins/Kirmayer/Perlmutter's Generalizations of the Kunen inconsistency is a good general source on this topic.

First, note that since $\mathsf{V=L}$ is a first-order sentence, if either $A$ or $B$ is $L$ and $A$ is elementarily embeddable into $B$ then $A=B=L$. So $(1)$, $(2)$, and $(4)$ are equivalent to each other while $(3)$ and $(7)$ are outright inconsistent. Also, $(1)$ is equivalent to $(5)$ since if $j:M\rightarrow M$ is nontrivial elementary then so is $\hat{j}: L^M=L\rightarrow L^M=L$.
Next up we have $(6)$. When I originally wrote this answer I didn't know anything about this possibility, but today I ran into a very valuable source (see below). As earlier, an embedding $M\rightarrow V$ restricts to an embedding $L\rightarrow L$, so $(6)$ is at least as strong as $0^\sharp$. The obvious question (especially in light of the next section of this answer!) is the relationship between $(6)$ and a measurable cardinal. As it turns out, $(6)$ is rather weak, at least in consistency strength, unless one adds additional demands on the embedding or the source model $M$. See the beginning of Section $2$ of Vickers/Welch, On elementary embeddings from an inner model to the universe (or the rest of the paper more generally).

Now we get to the fun stuff. The existence of an embedding of type $j_8$ is exactly equivalent to a measurable cardinal - and this is indeed the motivating observation which begins the whole study of elementary embeddings of inner models, so it is worth understanding well! Here's a sketch of the argument (due to Scott):

*

*One direction is easy: if $\mathcal{U}$ is a countably closed nonprincipal ultrafilter on an uncountable cardinal $\kappa$, then (utilizing Scott's trick to keep everything well-defined) we get an elementary embedding $\mathfrak{j}_\mathcal{U}$ from $V$ to the ultrapower $\prod V/\mathcal{U}$; by countable closure the latter is well-founded (and is easily seen to be set-like), and so it is definably-in-$V$ isomorphic to a unique inner model $M$. We usually conflate $\mathfrak{j}_\mathcal{U}$ with the induced elementary embedding of $V$ into this $M$.


*The other direction is a bit trickier: an arbitrary nontrivial elementary $j:V\rightarrow M$ might not "come from" an ultrafilter! However, it still implies the existence of an ultrafilter of the above type. Specifically, let $\kappa=crit(j)$ and consider $$\mathcal{U}_j:=\{A\subseteq\kappa: \kappa\in j(A)\}.$$ Then $\mathcal{U}_j$ is nonprincipal and countably closed.
One consequence of the above argument is that the a-priori-class-referring principle "There is a nontrivial elementary embedding of $V$ into some inner model $M$" is actually expressible in set theory alone. At the same time, it's important to note that we generally lose information in the $j\mapsto\mathcal{U}_j$ construction, in the sense that $\mathfrak{j}_{\mathcal{U}_j}\not=j$ in general. Figuring out how to "approximate" elementary embeddings by set-sized blobs of data is a general theme in inner model theory; for example, look at the notion of an extender, which is a generalization of an ultrafilter which lets us faithfully capture more embeddings.
