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There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces):

Are the following three variants on these results true?

  • Do symmetric monoidal categories with zero model all $\mathbb{E}_\infty$-spaces with zero?
  • Do (braided, symmetric) semiring categories model all connective ($\mathbb{E}_{2}$-, $\mathbb{E}_{\infty}$-)ring spaces?
  • Do (braided, symmetric) ring categories model all connective $1$-truncated ($\mathbb{E}_{2}$-, $\mathbb{E}_{\infty}$-)ring spectra? (Related: see Modeling Stable One-Types, Remark 3.2).
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