If $\kappa$ is an inaccessible cardinal then $V_\kappa$ is a model of $\sf ZFC_2$ ($\sf ZFC$ with a second-order replacement axiom).

If there are many inaccessible cardinals then there are many models of $\sf ZFC_2$, but one can add all sort of $\varphi$ which describe $V_\kappa$ completely. For example if $\varphi$ is "there is no inaccessible cardinal" then $\sf ZFC_2+\varphi$ is only satisfied by $V_\kappa$ if $\kappa$ is the least inaccessible cardinal. We can continue and state that there is exactly one, or two, or $\alpha$ inaccessible cardinals for every "small enough" $\alpha$. We can continue by adding more and more information (e.g. there is no inaccessible which is the limit of inaccessibles; there is only one inaccessible which has a stationary set of inaccessibles; and so on).

Let us say that $\varphi$ in $n$-order logic is a categorizer for $\kappa$ if $V_\kappa$ is the unique model of $\sf ZFC_2+\varphi$. We say that $\kappa$ is **$\Pi^n_m$-categorical** (or $\Sigma^n_m$-categorical) if there is a $\Pi^n_m$ ($\Sigma^n_m$) sentence $\varphi$ which is a categorizer for $\kappa$.

On the other hand, we say that $\kappa$ is $\Pi^n_m$-indescribable if for every $R\subseteq V_\kappa$ and a $\Pi^n_m$ sentence $\psi$ such that $\langle V_\kappa,\in,R\rangle\models\psi$ there is some $\alpha<\kappa$ such that $\langle V_\alpha,\in,R\cap V_\alpha\rangle\models\psi$. (Similarly, of course, we define $\Sigma^n_m$-indescribable cardinals.)

Note that inaccessible cardinals are $\Pi^1_0$-indescribable, but the least inaccessible is $\Pi^0_n$-categorical for some $n$ (because the statement "there are no inaccessible cardinals is a first-order statement).

> **Question:** Is there a [deep?] connection between the two notions?