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I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me:

1. The product and the co-product of categories are best defined by an universal mapping property.

2. 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 $S$ an 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 $A_i$.

3 added 79 characters in body

I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me:

1. The product and the co-product of categories are best defined by an universal mapping property.

2. The adeles $\mathbb{A}$ are the inductive limit of all $S$-adeles $\mathbb{A}(S)$\mathbb{A}(S) = \mathbb{R} \times \prod\limits_{p \in S} \mathbb{Q}_p$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$$to some topological space X, then there exists for every large enough set S an unique \phi_S : \mathbb{A}(S) \rightarrow X such that \phi_S = \phi on \mathbb{A}(S). Remark: For this to work, it is important that all but finitely many compact subrings (\mathbb{Z}_p in the above context) are actually open subrings, since you have to choose the characteristic functions of this set at these the places p \notin S to construct \phi_S. So I doubt that the restricted product can be defined in the category of locally compact groups/rings/spaces in such a fashion. But on On the other hand, the restricted product can be defined for a family of pairs of topological spaces (A_i \supset B_i)_i with almost all B_i are open in A_i. 2 added 155 characters in body I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me: 1. The product and the coproduct co-product of the category categories are usually best defined by an universal mapping property. 2. The adeles \mathbb{A} are the inductive limit of all S-ad{\'e}les S-adeles \mathbb{A}(S) \mathbb{A}(S) = \mathbb{R} \times \prod\limits_{p \in S} \mathbb{Q}_p for a finite set of places, so they . The universal property is given described as follows: If you are given a map \phi \phi : \mathbb{A} \rightarrow X X$$ to some topological ring, that space$X$, then there exists for every large enough set$S$an 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 rings subrings ($\mathbb{Z}_p$in the above context) are actually open subrings, since you have to choose the characteristic functions of this set at these points places to construct$\phi_S$. So I doubt that the restricted product can be defined in the category of locally compact groups/rings/spaces. But 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\$.

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