In the $\infty$-world, [connective spectra play the role of abelian groups](https://ncatlab.org/nlab/show/abelian+infinity-group), while [$\mathbb{E}\_\infty$-spaces play that of commutative monoids](https://ncatlab.org/nlab/show/E-infinity+space). This may be rephrased by saying that we may identify the $\infty$-categories of spectra and $\mathbb{E}\_\infty$-spaces with the $\infty$-categories $\mathsf{Mon}\_{\mathbb{E}\_\infty}(\mathcal{S})$ and $\mathsf{Grp}\_{\mathbb{E}\_{\infty}}(\mathcal{S})$ of $\mathbb{E}\_\infty$-monoids/groups in the $\infty$-category of anima $\mathcal{S}$.


Now, the $1$-categories $\mathsf{Ab}$ and $\mathsf{CMon}$ come equipped with tensor products $\otimes\_{\mathbb{Z}}$ and $\otimes\_{\mathbb{N}}$. These correspond in homotopy theory to the tensor products $\otimes\_{\mathbb{S}}$ and $\otimes\_{\mathbb{F}}$ of connective spectra and $\mathbb{E}\_\infty$-spaces.


While the tensor product of connective spectra is widely discussed in the literature, I'm finding it a bit difficult to find references for that of $\mathbb{E}\_\infty$-spaces. So far I've found the following:


* Gepner–Groth–Nikolaus, *Universality of multiplicative infinite loop space machines*, [arXiv:1305.4550](https://arxiv.org/abs/1305.4550), which establishes in Theorem 5.1 a universal property for the tensor product $\otimes\_{\mathbb{F}}$ as the unique functor making the free $\mathbb{E}\_\infty$-monoid functor
$$
\mathcal{S}\_\*\to\mathsf{Mon}\_{\mathbb{E}\_{\infty}}(\mathcal{S})
$$
into a symmetric monoidal functor.
* Blumberg–Cohen–Schlichtkrull, *Topological Hochschild homology of Thom spectra and the free loop space*, [arXiv:0811.0553](https://arxiv.org/abs/0811.0553), which establishes a point-set model for $\mathbb{E}\_{\infty}$-spaces, called *$\*$-modules*, rectifying $\mathbb{E}\_\infty$-spaces to strict monoids in $\*$-modules. See also [MO 92866](https://mathoverflow.net/questions/92866/).
* Sagave–Schlichtkrull, *Diagram spaces and symmetric spectra*, [arXiv:1103.2764](https://arxiv.org/abs/1103.2764), which establishes another point-set model for $\mathbb{E}\_{\infty}$-spaces, called *$\mathcal{I}$-spaces*, similarly rectifying $\mathbb{E}\_\infty$-spaces to strict monoids in $\mathcal{I}$-spaces. See also [arXiv:1111.6413](https://arxiv.org/abs/1111.6413).
* Lind, *Diagram spaces, diagram spectra, and spectra of units*, [arXiv:0908.1092](https://arxiv.org/abs/0908.1092), which proves that $\mathcal{I}$-spaces and $\*$-modules define equivalent homotopy theories.


**Questions:**


* What are some other references discussing the tensor product of $\mathbb{E}\_\infty$-spaces?
* What is the unit of this tensor product?
* Finally, what are some concrete examples of it?



Tensor products of $\mathbb{E}_\infty$-spaces
The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E\_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful point of view that does not appear in loc. cit. is that this tensor product comes from the Lawvere theory of commutative monoids. To explain this, consider the $(2,1)$-category $\mathrm{Span}(\mathrm{Fin})$ whose objects are finite sets and whose morphisms are spans $I\leftarrow K\rightarrow J$. It has the following universal property: for any $\infty$-category $\mathcal C$ with finite products, there is an equivalence
$$
\mathrm{CMon}(\mathcal C) = \mathrm{Fun}^\times(\mathrm{Span}(\mathrm{Fin}),\mathcal C),
$$
where $\mathrm{Fun}^\times$ is the $\infty$-category of functors that preserve finite products. Since $\mathrm{Span}(\mathrm{Fin})$ is self-dual, this means that $E\_\infty$-spaces are finite-product-preserving presheaves on $\mathrm{Span}(\mathrm{Fin})$:
$$
\mathrm{CMon}(\mathcal S) = \mathcal P\_\Sigma(\mathrm{Span}(\mathrm{Fin})).
$$
This was first studied in the [thesis of J. Cranch](https://arxiv.org/pdf/1011.3243.pdf). From this perspective, the direct sum and tensor product are the Day convolutions of $\sqcup$ and $\times$ on $\mathrm{Span}(\mathrm{Fin})$ (here $\times$ means the usual product of finite sets, which is not the categorical product in $\mathrm{Span}(\mathrm{Fin})$; the latter is the same as the categorical coproduct, i.e., the disjoint union $\sqcup$). For example, $E\_\infty$-semirings can be described as right-lax symmetric monoidal functors $(\mathrm{Span}(\mathrm{Fin}),\times)\to(\mathcal S,\times)$ that preserve finite products.


**The unit.** As Rune already explained, the unit for the tensor product of $E\_\infty$-spaces is the free $E\_\infty$-space on a point, that is the groupoid $\mathrm{Fin}^\simeq$ of finite sets with the $E\_\infty$-structure given by disjoint union. This is equivalently the presheaf on $\mathrm{Span}(\mathrm{Fin})$ represented by the point, which is the unit for $\times$ on $\mathrm{Span}(\mathrm{Fin})$.


Here are a few examples I could think of. Let $E\in \mathrm{CMon}(\mathcal S)$.


**Tensoring with a free $E\_\infty$-space.** Let $X\in\mathcal S$. Then
$$
\left(\coprod\_{n\geq 0} (X^n)\_{h\Sigma\_n}\right) \otimes E = \operatorname{colim}\_X E,
$$
where the colimit is taken in $\mathrm{CMon}(\mathcal S)$. This follows from the case $X=\*$ using that $\otimes$ preserves colimits in each variable.


**Tensoring with $\mathbb S$.** Tensoring with the sphere spectrum $\mathbb S$ is the same as group-completing:
$$
\mathbb S\otimes E = E^\mathrm{gp}.
$$
For example, for a ring $R$,
$$
\mathbb S\otimes \mathrm{Proj}(R) = K(R).
$$
where $\mathrm{Proj}(R)$ is the groupoid of finitely generated projective $R$-modules, and $K(R)$ is the K-theory space.


**Tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$.** Another localization of $\mathrm{CMon}(\mathcal S)$ is obtained by inverting integers (or rather, finite sets). The inclusion of the full subcategory of $E\_\infty$-spaces on which multiplication by $n$ is invertible has a left adjoint $E\mapsto E[n^{-1}]$, which is equivalent to tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. But unlike in the cases of either abelian monoids or spectra, $\mathrm{Fin}^\simeq[n^{-1}]$ is not just the sequential colimit of multiplication by $n$ maps; it is obtained from the latter by killing suitable perfect subgroups of its fundamental groups, in the sense of Quillen's plus construction, to ensure that $n$ acts invertibly.


**Tensoring with $\mathbb N$.** Let $\mathrm{FFree}\_{\mathbb N}$ be the 1-category of finite free $\mathbb N$-modules. There is a functor
$$
\mathrm{Span}(\mathrm{Fin}) \to \mathrm{FFree}\_{\mathbb N}
$$
sending a finite set $I$ to $\mathbb N^I$, inducing an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P\_\Sigma(\mathrm{Span}(\mathrm{Fin})) \stackrel{\mathrm{str}}\rightleftarrows \mathcal P\_\Sigma(\mathrm{FFree}\_{\mathbb N}).
$$
Objects in the right-hand side are sometimes called *strictly commutative monoids* (the group-complete ones are connective $H\mathbb Z$-module spectra). Tensoring with $\mathbb N$ amounts to strictifying a commutative monoid in this sense:
$$
\mathbb N\otimes E = E^\mathrm{str}.
$$
Unlike $\mathbb S$, $\mathbb N$ is not an idempotent semiring, that is, strictifying is not a localization. Indeed, $\mathbb N\otimes\mathbb N$ is an $E\_\infty$-space whose group completion is the "integral dual Steenrod algebra".


**Tensoring with $\mathrm{Vect}\_\mathbb{C}^\simeq$.** Let $\mathrm{Vect}\_\mathbb{C}^\simeq=\coprod\_{n\geq 0} BU(n)$, where $U(n)$ is regarded as an $\infty$-group (despite the notation, this is not really the core of an $\infty$-category of vector spaces). This is an $E\_\infty$-space whose group completion is $\mathrm{ku}$. There is a related $\infty$-category $2\mathrm{Vect}\_{\mathbb C}$ whose objects are finite sets and whose morphisms are matrices of complex vector spaces. As in the previous example we get an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P\_\Sigma(\mathrm{Span}(\mathrm{Fin})) \rightleftarrows \mathcal P\_\Sigma(2\mathrm{Vect}\_{\mathbb C}).
$$
An object in the right-hand side is roughly speaking a commutative monoid such that $U(n)$ acts on the multiplication by $n$ map in a coherent way. Tensoring with $\mathrm{Vect}\_\mathbb{C}^\simeq$ gives the free commutative monoid with such structure.


