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?
(there was a section with my (non-working) ideas on this, which I removed after the answers came in.)
Best Answer
I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me:
The product and the co-product of categories are best defined by an universal mapping property.
The adeles $\mathbb{A}$ are the inductive limit of all $S$-adeles $$\mathbb{A}(S) = \mathbb{R} \times \prod\limits_{p \in S} \mathbb{Q}_p \times \prod\limits_{p \notin S} \mathbb{Z}_p$$ for a finite set of places. The universal property is described as follows: If you are given a map $$\phi : \mathbb{A} \rightarrow X$$ to some topological space $X$, then there exists for every large enough set $S$ a unique $\phi_S : \mathbb{A}(S) \rightarrow X$ such that $\phi_S = \phi$ on $\mathbb{A}(S)$.
Remark: For this to work, it is important that all but finitely many compact subrings ($\mathbb{Z}_p$ in the above context) are actually open subrings, since you have to choose the characteristic functions of this set at the places $p \notin S$ to construct $\phi_S$. So I doubt that the restricted product can be defined in the category of locally compact groups/rings/spaces in such a fashion. On the other hand, the restricted product can be defined for a family of pairs of topological spaces $(A_i \supset B_i)_i$ with almost all $B_i$ are open in $A_i$.