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Stream: learning: questions

Topic: Restricted product of locally compact metri[c|zable] spaces

David Michael Roberts (Mar 16 2026 at 06:39):

Here's an old MO question asking about the universal property of the adeles: https://mathoverflow.net/q/96137/4177

This question talks just about the adeles as an topological ring. But it should be a more general construction (though the Encyclopedia of Mathematics page on the concept is a bit too general: it takes the restricted product of a family of pairs of sets).

Not too long ago I asked for an explicit construction of a metric on the adeles giving the topology, and I got a really nice answer https://mathoverflow.net/a/504887/4177

Is there something general one could say here for restricted products of families of locally compact (pointed?) metric spaces? Is there a nice functor as well as an explicit construction?

John Baez (Mar 16 2026 at 07:17):

So Wikipedia talks about restricted products of topological groups... are you wanting to get away from groups?

David Michael Roberts (Mar 16 2026 at 08:17):

It shouldn't depend on the fact of starting with locally compact groups. The loc.cpt. uniformity or something should be enough. And then you get it for free for groups and other algebraic structures like rings....

Graham Manuell (Mar 16 2026 at 11:07):

This doesn't answer your question, but is the underlying additive topological group of the adeles not just the tensor product of $\mathbb{R} \times \prod_p \Z_p$ and $\mathbb{Q}$ in the category of locally compact abelian groups? So probably the adeles are the coproduct of $\mathbb{R} \times \prod_p \Z_p$ and $\mathbb{Q}$ in the category of (locally compact) topological commutative rings. This feels right to me, but I don't know why no one said this in the mathoverflow answers.

Anyway, I think the additive structure is important and the construction is not nearly as natural without it.

David Michael Roberts (Mar 16 2026 at 11:20):

Well, the answer at my question constructs a metric on the underlying set of the adeles, and the resulting metric space is apparently complete. This metric doesn't really use the algebraic structure (not additively, and barely uses division by p the p-adics)

I think perhaps supplying just the family of locally compact spaces $\mathbb{Q}_p$ isn't enough, one might need to supply the pairs $(\mathbb{Q}_p,\mathbb{Z}_p)$, but the algebraic structure isn't important to define the underlying space, merely the fact $\mathbb{Z}_p$ is compact (it's the unit ball aound 0 in the p-adic metric...).

So perhaps it's not a product-like functor on families of pointed locally compact spaces, but families of locally compact spaces equipped with a compact subspace....

Graham Manuell (Mar 16 2026 at 11:41):

I mean, just cause it's possible to define it without the additive structure doesn't mean it's natural to do so. The coproduct $\prod_{n \in \mathbb{N}} A_n$ of topological abelian groups is a restriction of the box topology, but I wouldn't say the box topology is a natural topology on products of topological spaces without the additive structure.

Graham Manuell (Mar 16 2026 at 12:09):

Though maybe there is a way to think about things that makes something like this natural in general. It's just not obvious to me that there is.

Jens Hemelaer (Mar 17 2026 at 07:42):

For the finite adeles, you can first take the product of all the $\mathbb{Z}_p$ to get the space of profinite integers. On the space of profinite integers there is then a partially defined action of the rational numbers via multiplication, or more directly, an action of the nonzero integers by multiplication. I think you can then construct the adeles as a (filtered) colimit of the functor
$D \to C \to \mathrm{Top}$
with $D$ the partial order of nonzero integers under the division relation and $C$ the category with one object and nonzero integers under multiplication as morphisms, where you first project from $D$ to $C$ and then send the unique object in $C$ to the space of profinite integers (the morphisms give the action).

In this way you get more in the domain of locally compact spaces with a group action, and the additive group structure on the space is not used anymore?