9
$\begingroup$

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?
$\endgroup$
2
  • 6
    $\begingroup$ 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.) $\endgroup$ Aug 7 '21 at 10:48
  • $\begingroup$ @RuneHaugseng Thanks, this is excellent! $\endgroup$
    – Théo
    Aug 7 '21 at 20:37
10
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ This is absolutely amazing, thank you so much! $\endgroup$
    – Théo
    Aug 8 '21 at 20:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.