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 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 only found discussion of this in arXiv:1305.4550).

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?

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?

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 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 only found discussion of this in arXiv:1305.4550).

Questions:

• What are some other references discussing the tensor product of $\mathbb{E}_\infty$-spaces?
• What is the unit of this tensor product? (In particular, is it related to the classifying space (without group completion) $|\mathrm{N}_\bullet(\mathbb{F})|$ of the groupoid of finite sets and permutations?)
• Finally, what are some concrete examples of it?

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 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 only found discussion of this in arXiv:1305.4550).

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?