In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. 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, 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, 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.
- Sagave–Schlichtkrull, Diagram spaces and symmetric spectra, arXiv: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.
- Lind, Diagram spaces, diagram spectra, and spectra of units, arXiv: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?
Best Answer
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. 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.