Relative tensor products of
-categories of local systems
The Lurie tensor product is part of a symmetric monoidal structure on the -category
of presentable
-categories.
It enjoys many good formal properties, and is often computable. An example of this computability is that if is a small
-category and
are presentable, then
. A special case of this is the following : given spaces
and
, and using
to denote the
-category of spaces, we have
.
More generally, given a presentably symmetric monoidal -category
, it follows that
, 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 , do the natural maps induce an equivalence
?
Here, restriction along (resp.
) induces a symmetric monoidal colimit preserving functor
(resp.
) and thus we get a natural map as indicated.
The special case where 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 , the canonical map
associated to the pullback
is not an equivalence.
We begin by recalling a lemma, whose proof is left as an exercise:
Lemma
Let and
be a right (resp. left)
-module. Then
is generated under colimits by the image of
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, would be generated under colimits by the image of
, which consists of constant local systems.
Now we claim that if is generated under colimits by constant local systems, then
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
, the limit of
is the space of sections of the map
, which is empty if
is not contractible.
Remark
Whenever is not generated under colimits by constant local systems, the argument above works for
. In particular, this also provides stable counterexamples. The condition that
be generated under colimits by local systems is related to
-affineness of
.
Remark
There are simpler counter-examples : if is a (presentable)
-category, then
whenever
is simply-connected. In particular, taking
for some abelian group
, we find that in the pullback case, the map in question is a map
, so this being an equivalence means that every local system over
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 -category
of rational spectra. Then, for any simply-connected
-finite space
, the “trivial local system” functor induces an equivalence
. The same clearly does not hold for a general connected
-finite space
(consider the case of
), 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 does not send pullbacks to pushouts of commutative algebras would be to consider the cospan
not as three spaces, but as two “spaces over
”, and work in a setting parametrized over
; i.e. considering, instead of
, the
-category
.