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Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories

There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces): Symmetric monoidal categories model all connective ...
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What is the group completion of the underlying multiplicative $\mathbb{E}_\infty$-monoid of the sphere spectrum?

I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$: We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and —...
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