# Tensor products of $\mathbb{E}_\infty$-spaces

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?
• The unit is the free commutative monoid on the point, which is $\coprod_{n=0}^{\infty} B\Sigma_n$, or the groupoid of finite sets and bijections. More generally, Gepner-Groth-Nikolaus show that the free commutative monoid functor $F$ is symmetric monoidal (for the cartesian product of spaces), so for spaces $X,Y$ you have $F(X) \otimes F(Y) \simeq F(X \times Y)$. (In fact, since any commutative monoid is a (simplicial) colimit of free ones, this in a sense determines the tensor product, since it also preserves colimits in each variable.) Aug 7 '21 at 10:48
• @RuneHaugseng Thanks, this is excellent!
– Théo
Aug 7 '21 at 20:37

## 1 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.

• This is absolutely amazing, thank you so much!
– Théo
Aug 8 '21 at 20:09