$ \newcommand{\Ab}{\mathbf{Ab}} \newcommand{\Sp}{\mathbf{Sp}} $In 0-groupoids (sets), the thick subcategories of the category of abelian groups $\Ab$ are given by the primes $p$ and $0$, which we can identify with $\mathbb F_p$ and $\mathbb Q$.
In $\infty$-groupoids, the correct notion of abelian group is "spectrum", and the thick subcategories of $\Sp$ are given by the Morava $K$-theories $K(n,p)$ for primes $p$ and $0\le n \le \infty$, where by convention $K(0,p) = H\mathbb Q$ and $K(\infty,p)=H\mathbb F_p$ for all $p$.
One frequently calls these thick subcategories "primes" or "characteristics", and thinks of them as the points of the "Balmer spectrum" of $\Ab$ or $\Sp$.
I am curious about the story for $n$-groupoids for $0<n<\infty$. I presume for each $n$ there is an appropriate notion of "$n$-abelian group"; for example, this should be a Picard groupoid for $n=1$. Write $\Sp_n$ for whatever this notion turns out to be. I think this is just spectra with homotopy groups concentrated in $[0,n]$, or maybe $[-n, n]$ for non-connective spectra.
What are the thick subcategories of $\Sp_n$?
