Relative tensor products of $\infty$-categories of local systems

The Lurie tensor product is part of a symmetric monoidal structure on the $\infty$ -category $\mathrm{Pr}^\mathrm{L}$ of presentable $\infty$ -categories.

It enjoys many good formal properties, and is often computable. An example of this computability is that if $C$ is a small $\infty$ -category and $\mathcal{D,E}$ are presentable, then $\mathcal D^C \otimes \mathcal E \simeq (\mathcal D\otimes \mathcal E)^C$ . A special case of this is the following : given spaces $A$ and $B$ , and using $\mathcal S$ to denote the $\infty$ -category of spaces, we have $\mathcal S^A \otimes \mathcal S^B\simeq \mathcal S^{A\times B}$ .

More generally, given a presentably symmetric monoidal $\infty$ -category $\mathcal C$ , it follows that $\mathcal C^A\otimes_{\mathcal C}\mathcal C^B\simeq \mathcal C^{A\times B}$ , where we use the relative tensor product.

The goal of this post is to adress the natural question that arises from this : given a pullback $P\simeq A\times_B C$ , do the natural maps induce an equivalence $\mathcal C^A \otimes_{\mathcal C^B} \mathcal C^C \xrightarrow{\simeq} \mathcal C^P$ ?

Here, restriction along $A \to B$ (resp. $C\to B$ ) induces a symmetric monoidal colimit preserving functor $\mathcal C^B \to \mathcal C^A$ (resp. $\mathcal C^B \to \mathcal C^C$ ) and thus we get a natural map as indicated.

The special case where $B$ is a point was discussed above, and this more general question was raised by several people. We first show that the answer to this question is no and discuss a few variants later.

Theorem

Fot any non-contractible connected pointed space $(A,a)$ , the canonical map $\mathcal S \otimes_{\mathcal S^A} \mathcal S \to \mathcal S^{\Omega(A,a)}$ associated to the pullback $\Omega(A,a) \simeq a\times_A a$ is not an equivalence.

We begin by recalling a lemma, whose proof is left as an exercise:

Lemma

Let $\mathcal C \in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})$ and $\mathcal{D,E}$ be a right (resp. left) $\mathcal C$ -module. Then $\mathcal D\otimes_{\mathcal C}\mathcal E$ is generated under colimits by the image of $\mathcal D \times\mathcal E$ under the canonical map (so-called pure tensors).

Hint : There is a proof that an ordinary tensor product of abelian groups is generated by pure tensors using only the universal property. The same proof works here, slightly categorified.

Proof of the theorem :

We apply the lemma : if the map were an equivalence, $\mathcal S^{\Omega(A,a)}$ would be generated under colimits by the image of $\mathcal S\times \mathcal S$ , which consists of constant local systems.

Now we claim that if $\mathcal S^X$ is generated under colimits by constant local systems, then $X$ is empty or contractible. Indeed, it implies that any non-empty local system has a non-empty limit (as it receives some maps from constant local systems). However, by the un/straightening equivalence, for any $x\in X$ , the limit of $\hom_X(-,x)$ is the space of sections of the map $x\to X$ , which is empty if $X$ is not contractible.

Remark

Whenever $\mathcal C^{\Omega(A,a)}$ is not generated under colimits by constant local systems, the argument above works for $\mathcal C$ . In particular, this also provides stable counterexamples. The condition that $\mathcal C^X$ be generated under colimits by local systems is related to $\mathcal C$ -affineness of $X$ .

Remark

There are simpler counter-examples : if $\mathcal C$ is a (presentable) $1$ -category, then $\mathcal C^A \simeq \mathcal C$ whenever $A$ is simply-connected. In particular, taking $A = B^2L$ for some abelian group $L$ , we find that in the pullback case, the map in question is a map $\mathcal C \to \mathcal C^{BL}$ , so this being an equivalence means that every local system over $BL$ is constant. We gave the proof above because it generalizes more easily, cf. the remark above; and among other things can yield stable counterexamples. The simpler proof was pointed out to me by Shachar Carmeli.

For completeness, we mention one of these stable counterexamples: consider the stable $\infty$ -category $\mathrm{Mod}_\mathbb Q$ of rational spectra. Then, for any simply-connected $\pi$ -finite space $X$ , the “trivial local system” functor induces an equivalence $\mathrm{Mod}_\mathbb Q\to \mathrm{Mod}_\mathbb Q^X$ . The same clearly does not hold for a general connected $\pi$ -finite space $X$ (consider the case of $X=BG$ ), and so the same argument as above applies. There are generalizations of this at “higher heights”, see Ambidexterity and height by Carmeli, Schlank and Yanovski for an extended discussion.

We conclude this post by mentioning that a solution to the “problem” that $\mathcal C^\bullet$ does not send pullbacks to pushouts of commutative algebras would be to consider the cospan $A\to B\leftarrow C$ not as three spaces, but as two “spaces over $B$ ”, and work in a setting parametrized over $B$ ; i.e. considering, instead of $\mathrm{Mod}_{\mathcal C^B}(\mathrm{Pr}^\mathrm{L})$ , the $\infty$ -category $\mathrm{Mod}_\mathcal C(\mathrm{Pr}^\mathrm{L})^B$ .

Thuses

Relative tensor products of $\infty$ -categories of local systems