A **commutative monoid with zero** is a commutative monoid $A$ together with an element $0_{A}$ such that $0_{A}a=a0_{A}=0_{A}$ for all $a\in A$. They are precisely the monoids (in the sense of monoidal categories) in the category of pointed sets, equipped with the smash product.
Passing to the derived world, commutative monoids get replaced by $\mathbb{E}_{\infty}$-monoids in spaces, while pointed sets get replaced by pointed spaces. So the natural analogue of a commutative monoid with zero in homotopy theory should be an $\mathbb{E}_{\infty}$-monoid in the symmetric monoidal $\infty$-category $(\mathcal{S}_*,\wedge,S^0)$ of pointed spaces.
**Questions:**
1. The May recognition principle states that a space is (weakly equivalent to) an infinite loop space iff it is a grouplike $\mathbb{E}_{\infty}$-monoid in spaces. Do we have such a recognition principle for $\mathbb{E}_{\infty}$-monoids in pointed spaces, or of some subclass of them?
2. Has the notion of an $\mathbb{E}_{\infty}$-monoid in pointed spaces been studied before, particularly in the classical homotopy theory literature, and certainly under another name?
3. The $\infty$-category of $\mathbb{E}_\infty$-spaces admits a tensor product $\otimes_\mathbb{F}$, having unit the geometric realisation
$$|\mathrm{N}_{\bullet}(\mathbb{F})|\cong\coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}$$
of the groupoid of finite sets and permutations $\mathbb{F}$. Is there such a tensor product in the $\infty$-category of $\mathbb{E}_{\infty}$-monoids in $\mathcal{S}_*$, and, if so, what is its monoidal unit?
4. A natural example of an $\mathbb{E}_\infty$-monoid in pointed spaces is $\Omega^\infty E$ for $E$ any $\mathbb{E}_{\infty}$-ring spectrum, as pointed by Maxime Ranzi in the comments. What are some other nice examples?
5. The $\infty$-category of $\mathbb{E}_{\infty}$-spaces admits a number of point-set models, such as special $\Gamma$-spaces, commutative monoids in $*$-modules, and commutative monoids in $\mathcal{I}$-spaces. Similarly, the $\infty$-category of spectra has $\mathbb{S}$-modules, symmetric spectra, and orthogonal spectra as point-set models (there's also very special $\Gamma$-spaces for connective spectra). Does the symmetric monoidal $\infty$-category $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S}_*)$ also admit point-set models, in the sense of a presentation by a monoidal model category?
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**Some Answers.** Since asking this question I found answers for some of the questions above. Maybe it's a good idea to write these below (in particular to avoid duplication of efforts).
4) Just as geometric realisations of nerves of symmetric monoidal categories provide examples of $\mathbb{E}_{\infty}$-spaces, geometric realisations of nerves of "symmetric monoidal categories with zero" give another class of examples of $\mathbb{E}_{\infty}$-monoids in pointed spaces. Indeed:
- Define a **monoidal category with zero** to be an $\mathbb{E}_{1}$-monoid in $(\mathrm{N}^\mathsf{D}_{\bullet}(\mathsf{Cats}_{\mathsf{2},*}),\wedge,\mathsf{pt}\coprod\mathsf{pt})$, where $\mathsf{Cats}_{\mathsf{2},*}$ is the $2$-category of small (weakly-)pointed categories;
- Concretely, such an object corresponds to a monoidal category $(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$ equipped with an object $\mathbf{0}_{\mathcal{C}}$ and natural families of isomorphisms
\begin{align*}
\delta^{\ell,\mathbf{0}}_{A} &\colon \mathbf{0}_{\mathcal{C}}\otimes A \longrightarrow \mathbf{0}_{\mathcal{C}},\\
\delta^{r,\mathbf{0}}_{A} &\colon A\otimes \mathbf{0}_{\mathcal{C}} \longrightarrow \mathbf{0}_{\mathcal{C}},
\end{align*}
called the **left and right annihilators** of $\mathcal{C}$, satisfying certain coherence conditions.
- Similarly, **braided** and **symmetric monoidal categories with zero** are $\mathbb{E}_{2}$- and $\mathbb{E}_{3}$($=\mathbb{E}_{4}=\cdots=\mathbb{E}_{\infty}$-)monoids in $(\mathrm{N}^\mathsf{D}_{\bullet}(\mathsf{Cats}_{\mathsf{2},*}),\wedge,\mathsf{pt}\coprod\mathsf{pt})$. They are described as above, but there are a few extra coherence conditions combining braidings and left/right annihilators.
- For example, any (braided, symmetric) bimonoidal category gives rise to such a (braided, symmetric) monoidal category with zero by forgetting the additive monoidal structure.
- By the symmetric monoidality of nerves and geometric realisations, it follows that, for any symmetric monoidal category with zero $\mathcal{C}$, the space $|\mathrm{N}_{\bullet}(\mathcal{C})|$ is an $\mathbb{E}_{\infty}$-monoid in $\mathcal{S}_*$.
- So for instance each of the examples [here](https://mathoverflow.net/questions/401555) give also examples of $\mathbb{E}_{\infty}$-monoids in pointed spaces, and there are of course many more.
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3) Yes, the $\infty$-category $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S}_*)$ has a tensor product, as shown by applying [Theorem 5.1, Gepner–Groth–Nikolaus](https://arxiv.org/abs/1305.4550) to $\mathcal{C}=\mathcal{S}_*$. By the same result, the unit for this tensor product is the free $\mathbb{E}_{\infty}$-monoid in $\mathcal{S}_*$. Indeed, just as the free $\mathbb{E}_{\infty}$-monoid in $\mathcal{S}$ is computed to be
\begin{align*}
\coprod_{n=0}^{\infty}*^{\times n}_{\mathsf{h}\Sigma_{n}} &\simeq \coprod_{n=0}^{\infty}\mathbf{E}\Sigma_{n}\times_{\Sigma_{n}}*
\\
&\simeq \coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}\\
&\cong |\mathrm{N}_{\bullet}(\mathbb{F})|,
\end{align*}
the geometric realisation of the groupoid of finite sets and permutations $\mathbb{F}$, the free $\mathbb{E}_{\infty}$-monoid in $\mathcal{S}_*$ is given by
\begin{align*}
\bigvee_{n=0}^{\infty}((S^{0})^{\wedge n})_{\mathsf{h}\Sigma_{n}} &\simeq \bigvee_{n=0}^{\infty}(S^{0})_{\mathsf{h}\Sigma_{n}}\\
&\simeq \bigvee_{n=0}^{\infty}\mathbf{E}\Sigma_{n,+}\times_{\Sigma_{n}}S^0,\\
&\simeq \bigvee_{n=0}^{\infty}(\mathbf{B}\Sigma_{n})_+\\
&\simeq \left(\coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}\right)_+\\
&\simeq |\mathrm{N}_{\bullet}(\mathbb{F})|_+\\
&\simeq |\mathrm{N}_{\bullet}(\mathbb{F}_+)|,
\end{align*}
where $\mathbb{F}_+$ is the symmetric monoidal category with zero obtained by freely adjoining an absorbing element $-\infty$ to $\mathbb{F}$ and extending the rest of the symmetric monoidal category structure of $\mathbb{F}$ to $\mathbb{F}_+$ suitably (in particular by defining $\langle n\rangle\oplus-\infty\overset{\mathrm{def}}{=}-\infty$ for all $\langle n\rangle\in\mathrm{Obj}(\mathbb{F}_+)$).