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 spectra;
- In Modelling Connective Spectra via Multicategories, Fuentes-Keuthan proves the statement in the title, and also refines in Theorem 5.3 Thomason's result, showing that symmetric monoidal groupoids suffice to model all connective spectra;
- In Modeling Stable One-Types, Johnson–Osorno showed that Picard categories model $1$-truncated spectra;
- In Stable homotopy hypothesis in the Tamsamani model, Moser–Ozornova–Paoli–Sarazola–Verdugo proved that symmetric monoidal weak $n$-groupoids model $n$-truncated spectra.
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).
