Semirings, also called rigs, are rings without negatives: their underlying additive monoids are not groups (in other words, while rings are monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, semirings are monoids in $(\mathsf{CMon},\otimes_{\mathbb{N}},\mathbb{N})$). Examples include all rings, but also objects like

- The
**natural numbers semiring**$(\mathbb{N},+,\cdot,0,1)$, the monoidal unit of $(\mathsf{Semirings},\otimes_{\mathbb{N}},\mathbb{N})$; - The
**Boolean semiring**$\mathbb{B}=\{0,1\}$, with semiring structure characterised by $1+1=1$; - The
**tropical semiring**$\mathbb{T}=(\mathbb{R}\cup\{\infty\},\min,+)$, and its sibling $\mathbb{A}=(\mathbb{R}\cup\{-\infty\},\max,+)$, sometimes called the**Arctic semiring**; - The semiring $(\mathrm{Idl}(R),+,\cdot,(0),R)$ of ideals of a ring $R$.

When we pass to the $\infty$-world, we replace the symmetric monoidal category $(\mathsf{Sets},\times,\mathrm{pt})$ of sets and maps between them by the symmetric monoidal $\infty$-category $(\mathcal{S},\times,\mathrm{pt})$ of spaces, and also commutative monoids and groups by $\mathbb{E}_{\infty}$-monoids and $\mathbb{E}_\infty$-groups.

As a result, the immediate analogues of $\mathsf{CMon}$ and $\mathsf{Ab}$ become the $\infty$-categories $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S})$ and $\mathsf{Grp}_{\mathbb{E}_{\infty}}(\mathcal{S})$ of $\mathbb{E}_\infty$-spaces and grouplike $\mathbb{E}_{\infty}$-spaces. The latter of these is equivalent to the $\infty$-category of connective spectra $\mathsf{Sp}_{\geq0}$, which embeds into the $\infty$-category of all spectra $\mathsf{Sp}$, the “true” analogue of $\mathsf{Ab}$ in homotopy theory.

The $\infty$-categories $\mathcal{S}_*$, $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S})$, $\mathsf{Grp}_{\mathbb{E}_{\infty}}(\mathcal{S})$, and $\mathsf{Sp}$ all admit symmetric monoidal structures, determined uniquely by the requirement that the free functors from $\mathcal{S}$ to them can be equipped with a symmetric monoidal structure (see Theorem 5.1 of Gepner–Groth–Nikolaus's *Universality of multiplicative infinite loop space machines*, arXiv:1305.4550).

We can thus consider $\mathbb{E}_{k}$-monoids in these categories:

- For $(\mathsf{Sp},\otimes_{\mathbb{S}},\mathbb{S})$, we get
**$\mathbb{E}_{k}$-ring spectra**; - For $(\mathsf{Grp}_{\mathbb{E}_{\infty}}(\mathcal{S}),\otimes_{QS^0},QS^0)$, we get
**$\mathbb{E}_{k}$-ring spaces**, which are equivalent to connective $\mathbb{E}_{k}$-ring spectra; - Finally, for $(\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S}),\otimes_{\mathbb{F}},\mathbb{F})$, we obtain
**$\mathbb{E}_{k}$-semiring spaces**.

The last of these provides a partial (i.e. connective) analogue of semirings in homotopy theory.

However, examples of $\mathbb{E}_{k}$-semiring spaces (that are not $\mathbb{E}_{k}$-ring spectra) are harder to come by.

A motivating example of them is given by the $\mathbb{E}_{\infty}$-space $\mathbb{F}\overset{\mathrm{def}}{=}\coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}$―the classifying space of the groupoid of finite sets and permutations―which can be given the structure of an $\mathbb{E}_{\infty}$-semiring space, becoming the monoidal unit of the symmetric monoidal $\infty$-category $\mathsf{Semirings}_{\mathbb{E}_{\infty}}(\mathcal{S})$ of $\mathbb{E}_{\infty}$-semiring spaces. Additionally, $\mathbb{F}$ is the spectral analogue of the semiring $\mathbb{N}$ of natural numbers, and, by the multiplicative Barratt–Priddy–Quillen–Segal theorem, its $\mathbb{E}_{\infty}$-ring space completion (i.e. $QS^0\otimes_{\mathbb{F}}\mathbb{F}$) is $QS^0$, corresponding to the sphere spectrum $\mathbb{S}$ under the equivalence $\mathsf{Ring}_{\mathbb{E}_{\infty}}(\mathcal{S})\cong\mathsf{RingSp}_{\geq0}$.

**Questions:**

- What are some other examples of $\mathbb{A}_{k}$-semiring or $\mathbb{E}_{k}$-semiring spaces?
- What are some examples of homotopy associative/commutative semiring spaces, i.e. monoids or commutative monoids in $(\mathsf{Ho}(\mathcal{S}),\otimes_{\mathbb{F}},\mathbb{F})$?
- Finally, do we have semiring space analogues of $\mathbb{B}$, $\mathbb{T}$, $\mathbb{A}$, and $\mathrm{Idl}(R)$?

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