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.

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.

Example: For one example, note that the usual Dold Kan correspondence really comes in two dual flavors: from a simplicial spectrum, you can extract the spectral chain complex of "left normalized chains" or the chain complex of "right normalized chains". Composing one of these functors with its inverse gives an automorphism of simplicial spectra. Is this automorphism trivial?