Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly equivalent to simplicial objects in spectra. For instance, these include nonnegatively-increasingly-filtered spectra and nonnegatively-homologically-graded chain complexes in spectra. See Walde. This is some sort of statement about the non-injectivity of $Psh_{Spt}(-)$ on $\Pi_0$ of the space of categories (or maybe pointed categories or something). Here is a question about $\Pi_1$:

Question: What is the space of auto-equivalences of the category of simplicial spectra?

To start, would be nice to enumerate the auto-equivalences up to homotopy.