TITLE: Examples of $\mathbb{E}_{k}$-semiring spaces
QUESTION [5 upvotes]: 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 $\mathrm{Idl}(R)$?

REPLY [4 votes]: Here are some interesting examples of symmetric bimonoidal groupoids $\mathcal{C}$.  In each case, the resulting classifying space $B\mathcal{C}$ is an $E_\infty$ ring space, whose group completion is $\Omega^\infty K(\mathcal{C})$.

Fix a finite group $G$, and let $\mathcal{F}_G$ be the groupoid of finite $G$-sets.   Then $K(\mathcal{F}_G)=\bigvee_{(H)}\Sigma^\infty_+BW_G(H)$, where $H$ runs over the conjugacy classes of subgroups of $G$, and $W_G(H)=N_G(H)/H$  is the Weyl group.  This can also be described as the Lewis-May fixed point spectrum $(S_G)^G$ of the $G$-equivariant sphere spectrum.  More generally, if $X$ is any finite $G$-set and $\mathcal{F}_G/X$ is the groupoid of finite $G$-sets equipped with a map to $X$, then $K(\mathcal{F}_G/X)=\left(\Sigma^\infty_+X\right)^G$.
Now let $\mathcal{F}_G^f$ be the subgroupoid of free $G$-sets, or equivalently, the groupoid of finite sets equipped with a $G$-torsor.  Then $B(\mathcal{F}_G^f)\simeq\coprod_nB(G\wr \Sigma_n)$, which is the total extended power of $BG$.  We also have $K(\mathcal{F}_G^f)=\Sigma^\infty_+BG$.
Fix a field $F$, and let $\mathcal{V}_F$ be the groupoid of finite-dimensional vector spaces over $F$.  Then $B(\mathcal{V}_F)\simeq\coprod_nBGL_n(F)$ and $K(\mathcal{V}_F)$ is just the algebraic $K$-theory spectrum $K(F)$.  By a trellis in $V$ I mean an unordered set of one-dimensional subspaces of $V$ whose direct sum is $V$.  Let $\mathcal{T}_F$ be the groupoid of finite-dimensional vector spaces over $F$ equipped with a trellis; this has a fairly evident symmetric bimonoidal structure.  Then $B(\mathcal{T}_F)\simeq\coprod_nB(GL_1(F)\wr\Sigma_n)$ and $K(\mathcal{T}_F)=\Sigma^\infty_+BGL_1(F)$.  The forgetful functor $\mathcal{T}_F\to\mathcal{V}_F$ gives rise to a well-known map $\Sigma^\infty_+BGL_1(F)\to K(F)$ of ring spectra.
By a multiset I mean a finite set $X$ equipped with a multiplicity function $m_X\colon X\to\mathbb{N}$.  A morphism of multisets is a function $f\colon X\to Y$ that is bijective up to multiplicity in the sense that $\sum_{f(x)=y}m_X(x)=m_Y(y)$ for all $y\in Y$.  We write $\mathcal{M}$ for the category of multisets, which has a fairly evident symmetric bimonoidal structure.  There is an adjunction linking $\mathcal{M}$ to the discrete category $\mathbb{N}$, so $B(\mathcal{M})\simeq\mathbb{N}$ and $K(\mathcal{M})\simeq H$.  The Eilenberg-MacLane spectrum $H$ can also be interpreted as the infinite symmetric product $SP^\infty(S)$, so it is filtered by subspectra $SP^k(S)$, which have been important for a number of applications.  We can define $\mathcal{M}_k=\{X\in\mathcal{M}\;:\;m_X(x)\leq k\text{ for all }x\}$, then it turns out that $K(\mathcal{M}_k)\simeq SP^k(S)$.  This is a reinterpretation of work of Kathryn Lesh, from which one can also extract some information about $B(\mathcal{M}_k)$.  The product functor $\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ sends $\mathcal{M}_p\times\mathcal{M}_q$ to $\mathcal{M}_{pq}$.  Using this, we can make $\coprod_n\mathcal{M}_n$ into a symmetric bimonoidal category with a symmetric bimonoidal functor to $\mathcal{M}$.  This corresponds to a well-known ring map $\bigvee_nSP^n(S)\to SP^\infty(S)=H$.