What determines when an inner model is "canonical"? I've read several places that L, the constructible universe, is the least canonical inner model. Grigor Sargsyan explains in his slides that L is canonical due to $\mathbb{R}^L$ being $\Sigma_2^1$ and for every real number $x$, $x\in L$ iff $x$ is $\Delta_2^1$ in a countable ordinal. Why does this amount to being canonical? Or is the concept of being canonical just not defined rigorously?
Thanks in advance.
 A: Canonical inner models $M$ are definable without parameters and, in fact, "locally" definable, that is, there is a formula $\psi_M$ such that $M=\{x\mid \psi_M(x)\}$. (Of course, in $\mathsf{ZF}$ and its extensions, classes are definable, but we allow set parameters in general.)
For $L$ this "locality" is true in the strongest possible sense, as running the definition of $L$ inside any transitive set model of enough set theory $N$ gives us as output the set $L_\alpha$, where $\alpha=\mathsf{ORD}\cap N$. 
(That said, the definability may depend on additional assumptions on the universe, such as there not being inner models with certain large cardinals. But this is fine; for example, there is a canonical inner model $K$ definable in the absence of Woodin cardinals. If the universe is rich enough to see Woodin cardinals, the definition of $K$ will not result in a proper class. Even in this case, there are still inner models where the construction of $K$ can be carried out, that is, models where there are no inner models with Woodin cardinals, and so we can define $K$ inside them.) 
Canonical inner models are generically invariant, so that passing to forcing extensions of the universe does not change the interpretation of the definition of $M$. The local nature of the definition can then be used as a basis for an "inductive" construction of the model: For example, there is a version of the model $K$ defined inside, say, $J_{\omega_1}(\mathbb R)$, and this version is just an initial segment of $K$, of height $\omega_1$. Pass to a forcing extension where $\omega_1$ is collapsed, and $\omega_2$ becomes the new $\omega_1$. In the $J_{\omega_1}(\mathbb R)$ of this extension, we get a version of $K$, of height $\omega_1$. This is the initial segment of the $K$ of the ground model, of height $\omega_2$ in the ground model.
In the absence of certain large cardinals, these inner models are "close" to the true universe in several ways. This is usually formalized in terms of covering. Jensen's covering theorem, for example, tells us that either $0^\sharp$ exists, or else for any set $X$ of ordinals there is a set $Y$ of ordinals, $Y\in L$, such that $X\subseteq Y$, and $|Y|\le|X|+\aleph_1$. There are appropriate versions of covering (more technical to state) as we look at inner models that allow the existence of stronger large cardinals than $L$ does. 
These inner models come with additional structure, that allows them to reconstruct themselves, meaning in particular that $K$ satisfies the statement $V=K$, but also that there is a well-ordering of the whole inner model, definable within the inner model. When restricted to the reals, this well-ordering is a (light-face) well-ordering of appropriate complexity within the natural hierarchy of definability for reals. For a long while this just means projective; so $\mathbb R^L$ and the well-ordering of $\mathbb R^L$ are $\Sigma^1_2$, similarly, $\mathbb R^{L[\mu]}$ and the well-ordering of 
$\mathbb R^{L[\mu]}$ are $\Sigma^1_3$ for $L[\mu]$ the canonical inner model for a measurable cardinal, etc. "Appropriate" means here that $\Sigma^1_2$ is only possible for $\mathbb R^L$ (this is a theorem of Mansfield); once our model $M$ sees Woodin cardinals, $\mathbb R^M$ can no longer be $\Sigma^1_3$ (this is related to results on projective absoluteness), and the more Woodin cardinals we see, the higher in the projective hierarchy the set of reals. In the presence of infinitely many Woodin cardinals, the set of reals is no longer projective but, at least for all the canonical inner models currently understood, they are universally Baire. 
These additional structure of the canonical inner models provides them with a fine structure theory, similar to what we have in $L$. For example, in $L$, if $L_\alpha$ satisfies enough set theory, and $X$ is an elementary substructure of $L_\alpha$, then its transitive collapse is an $L_\beta$. This is no longer true in general for other canonical inner models $M$, but we have that if we look at an initial segment $P$ of $M$, take an elementary substructure $X$, and collapse it, the resulting transitive structure $Q$ can be compared with $P$ in a way that says that $Q$ "comes" before $P$ in a natural order. More precisely, this translates into there being iterations of $P$ and $Q$, $P\to S$ and $Q\to R$, that end in models $S$ and $R$ with $R$ being a genuine initial segment of $S$. In $L$, the iterations are trivial, in that $S=P$ and $R=Q$. In larger canonical inner models, we still have $S=P$, but $R$ has genuinely iterated. In even larger canonical inner models, we may have to iterate $P$ as well, but only to an ultrapower. 
This possibility of comparison of models via iterations is key. The iterations come from the large cardinals in the models: We can form ultrapowers in the presence of measurable cardinals, for example, and iterate this process transfinitely many times, taking appropriate direct limits at limit stages. The more complex the model we consider, the more complicated the shapes of the iterations. But we require that there is a definable "strategy" telling us how the iterations ought to proceed. 
Now, since the fine structure of these models is so important, we can consider fine structural models for their own sake. The models I have described above are usually called core models; more precisely, $L$ is the core model unless $0^\sharp$ exists, $L[\mu]$ is the core model if there are inner models with measurable cardinals, but $0^\dagger$ does not exist, and so on: Each of these models $M$ is the core model, unless we see (a real coding) a nontrivial elementary embedding of $M$ to itself. This is the "official" version of the notion of canonical inner model. Below, I briefly mention some variants.  
More generally, some authors use the term "canonical" to refer to fine structural models in general. There is some leeway here, on the details of the fine structure hierarchy that is adopted, and studying these hierarchies on their own is itself interesting. The theory at the level of Woodin cardinals was developed using what we call Mitchell-Steel models. Currently, the preferred approach is using a hierarchy due to Jensen. 
Finally, much of the above admits appropriate relativizations, so we may start with a small set structure "at the bottom", and build $K$ as before "on top" of it. (This allows us to bypass some obstacles; for example, if the structure already had Woodin cardinals, then our $K$ sees those Woodin cardinals, but no more.) These structures are typically reals, or countable and transitive. We can go beyond that, and study objects such as $K(\mathbb R)$, but significant complications arise then since, for example, we lose the self-well-orderings. Note that if the structure we add at the bottom is "canonical" in some sense, then one could make a claim of calling these more general models canonical as well.
