Background on the Adèles

The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a restricted product of the complete local fields $K_\nu$, where $\nu$ ranges over all places of $K$. The restricted product is usually defined as the subset of $\prod_\nu K_\nu$ given by

$\mathbb{A}_K := \prod_\nu' K_\nu := \{ (x_\nu)_\nu \in \prod_\nu K_\nu\ |\ \text{ all but finitely many } x_\nu \in \mathcal{O}_\nu\}$

where $\mathcal{O}_\nu$ is the ring of integers in $K_\nu$.

Tensor product description

An alternative description, for the sake of concreteness given for the rationals $K=\mathbb{Q}$ can be made by using the tensor product:

$\mathbb{A}_\mathbb{Q} = \left(\left(\prod_p \mathbb{Z}_p\right) \otimes_\mathbb{Z} \mathbb{Q}\right) \times \mathbb{R}.$

This is the same, because $\mathbb{Z}_p \otimes \mathbb{Q} = \mathbb{Q}_p$ and the tensor product captures the finiteness condition. As there are always only finitely many infinite places, this description can be given for any number field as well (and of course function fields, since they don't have infinite places at all).

The topology on the Adèles

The restricted product comes with a restricted product topology, which is not the subspace topology from the ordinary product (despite its name), but the topology whose subbasis sets are

$V_{\eta,U_\eta} := \{(x_\nu)_\nu \in \prod_\nu K_\nu\ |\ x_\nu \in \mathcal{O}_\nu \text{ for } \nu \neq \eta, \text{ and } x_\eta \in U_\eta\}$

with $\eta$ a place and $U_\eta \subseteq K_\eta$ any open subset. The subspace topology from the product differs from this by requiring only $x_\nu \in \mathcal{O}_\nu$ for all but finitely many places, which are not fixed uniformly for a subbasis set.

Given a subset $U$ of $\mathbb{A}_K$ which is open in the ordinary subspace topology from the ordinary product, for every place $\nu$ there might be an $x \in U$ such that $x_\nu \notin \mathcal{O}_\nu$. If instead $U$ is open in the restricted product topology, there is a fixed finite set of places $\{\nu_1,...,\nu_m\}$ such that for every $x \in U$ and every other place $\nu \neq \nu_i$ we have $x_\nu \in \mathcal{O}_\nu$.

Nice properties of this topology are: You get again a locally compact group with compact open subgroup $\prod_\nu \mathcal{O}_\nu$ and that the Haar measure on $\mathbb{A}_K$ gives the quotient $\mathbb{A}_K/K$ a finite measure (with $K$ embedded diagonally by the maps $K \to K_\nu$).

The question: how to describe the Adèles categorically?

More specifically, I'd like to understand the restricted topology as well. The ordinary product is a limit, and as such it carries the initial topology. Any subspace carries the initial topology as well, but this gives the wrong topology, not the restricted product topology but the topology restricted from the product.

• Is it impossible to give a categorical description?
• Would it even be useful to have a categorical description?
• Does one have to apply a limit-colimit procedure or might a single limit or colimit suffice?
• There are some similarities with ultraproducts, which are classically not defined in a categorical way, but it is possible. The restricted product is somewhat dual to an ultraproduct. Could that help?
• Is there a good canonical way to topologize the tensor product of topological algebras over a topological ring? Would that solve my problem?
• Which (universal) properties do the Adèles satisfy?

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"?