Examples of $\mathbb{E}_{k}$-semiring spaces

Question

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

1. The natural numbers semiring $(\mathbb{N},+,\cdot,0,1)$, the monoidal unit of $(\mathsf{Semirings},\otimes_{\mathbb{N}},\mathbb{N})$;
2. The Boolean semiring $\mathbb{B}=\{0,1\}$, with semiring structure characterised by $1+1=1$;
3. 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;
4. The semiring $(\mathrm{Idl}(R),+,\cdot,(0),R)$ of ideals of a ring $R$.

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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.

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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 $\mathsf{Ring}_{\mathbb{E}_{\infty}}(\mathcal{S})\cong\mathsf{RingSp}_{\geq0}$.

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Questions:

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

Answer 1

(as a sidenote on terminology, as Jonathan points out in the comments, "spectrum" is not really a good name for your objects precisely because you expect them not to be spectra)

If you look at theorem 8.8 in the GGN paper you cited, you'll see that categories with a nicely behaved tensor product yield examples by taking their groupoid core.

Here are some explicit classical examples:

• For a commutative ring spectrum, $\mathrm{Proj}_R^\simeq$ is a commutative semiring space, whose ring completion is $K(R)$ when $R$ is connective. More generally, if $R$ is $E_k$, this semiring space "is" $E_{k-1}$.

• There is an equivariant version of $\mathbb F$, namely $\mathrm{Fin}_G^\simeq$ for a (pro)finite group $G$; as well as an equivariant version of the previous example, where for a commutative ring $G$-spectrum $R$, you call "projective" any $R$-module which is a summand of a direct sum of $R\otimes G/H_+$'s (no shifts allowed)

Here's an example which doesn't directly fit into the context of [Thm 8.8, GGN] (although it can be made to, via e.g. the condensed/pyknotic approach):

• $\mathrm{Vect}_\mathbb R^\simeq$, $\mathrm{Vect}_\mathbb C^\simeq$, the groupoids of finite dimensional vector spaces over $\mathbb{R,C}$ respectively, but where you take into account the topology on the hom-sets. Note that this is not literally $(\mathrm{Vect}_\mathbb K)^\simeq$ for some $\infty$-category of vector spaces and all linear maps obtained by applying the nerve to the topological category - indeed, if you do that then all mapping spaces are contractible and so you get trivial categories; so you have to restrict to isomorphisms before passing to $\infty$-categories. Their ring completions are $\mathrm{ko, ku}$ respectively.

• Again, the above example has an equivariant version.

(the equivariant versions can be made into semiring $G$-spaces, but let me not get into that here)

There are probably tons of other examples.

Answer 2

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 $\Omega^\infty K(\mathcal{C})$.

1. 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 $K(\mathcal{F}_G/X)=\left(\Sigma^\infty_+X\right)^G$.
2. 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 $K(\mathcal{F}_G^f)=\Sigma^\infty_+BG$.
3. 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.
4. 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 $\bigvee_nSP^n(S)\to SP^\infty(S)=H$.