Return to Question

My ideas

via the tensor product description

The description using a tensor product can be a hint: the coproduct in the category of $R$-algebras is the tensor product over $R$. For simplicity, one might try to solve the problem first for function fields, so the infinite places disappear. So the Adeles are a coproduct of a product $\mathbb{A}_K = (\prod_\nu \mathcal{O}_\nu) \otimes_\mathcal{O} K$, and the desired topology contains $\prod \mathcal{O}_\nu$ as open subset.

How to topologize tensor products of topological $\mathcal{O}$-algebras canonically and categorically? The right definition would consist of a universal property. There seems to be some research around this, for example Alb & Ursul: Tensor products of compact rings, but I was unable to obtain the article. Wikipedia offers only definition for Hilbert or Banach spaces, and I'm unable to adapt these definition to this case.

more generally

A drawback of the tensor product description is that is doesn't generalize immediately to a restricted product of locally compact topological groups $G_n$ with open compact subgroups $K_n$:

${\prod_{n=0}^\infty}' G_n := \{ (g_n)_n \in \prod_{n=0}^\infty G_n\ |\ \text{ all but finitely many } g_n \in K_n\}$

You can write down a dirty description that replaces the tensor product description:

$ \left( (\prod K_n) \times (\bigoplus G_n) \right)/\sim$

where $\sim$ identifies elements just as a tensor product would do.

via a strange category

My first attempt to tackle the problem was to craft the "right" category. I failed, but here are my failed ideas:

So, one might craft a category where the restricted product corresponds to the categorical (co)product. My try: objects are pairs of locally compact groups $(G,H)$ with $H$ a compact open subgroup of $G$, morphisms $(G,H) \to (G',H')$ are continuous group homomorphisms $G \to G'$ that carry $H$ into $H'$. It contains the category of compact groups $H$ as the subcategory of pairs $(H,H)$.

Sadly, the product of this category is just the ordinary product of groups and the coproduct is too small to be isomorphic to the Adèles. The category lacks either a finiteness condition (to get the right product) or something else (to get the right coproduct).

another strange category

Fix a sequence of locally compact groups $G_\nu$ with compact open subgroups $H_\nu$

One could take as objects all topological rings which admit continuous monomorphisms from all $G_\nu$ as well as from $\prod_\nu H_\nu$. The restricted product could be initial in this category. However, I don't see how this should give the right topology...

using the group structure to define the topology

One could define the topology to be minimal such that the compact subgroup $\prod H_\nu$ and all translates of it are open, and its subspace topology is the usual product topology. How to do that "categorically"?

(there was a section with my (non-working) ideas on this, which I removed after the answers came in.)

another strange category

Fix a sequence of locally compact groups $G_\nu$ with compact open subgroups $H_\nu$

One could take as objects all topological rings which admit continuous monomorphisms from all $G_\nu$ as well as from $\prod_\nu H_\nu$. The restricted product could be initial in this category. However, I don't see how this should give the right topology...

using the group structure to define the topology

One could define the topology to be minimal such that the compact subgroup $\prod H_\nu$ and all translates of it are open, and its subspace topology is the usual product topology. How to do that "categorically"?