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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 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 $A_i$.

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 $S$ 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 $A_i$.

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 $\mathbb{A}(S) = \mathbb{R} \times \prod\limits_{p \in S} \mathbb{Q}_p$ 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 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 $A_i$.

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 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 $A_i$.

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 of the category are usually defined by an universal property.

2. The adeles $\mathbb{A}$ are inductive limit of all $S$-ad{'e}les $\mathbb{A}(S)$ for a finite set of places, so they universal property is given as: If you are given a map $\phi : \mathbb{A} \rightarrow X$ to some topological ring, that 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 all but finitely many compact rings are actually open, since you have to choose the characteristic functions at these points 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 spaces $(A_i \supset B_i)_i$ with almost all $B_i$ are open in $A_i$.

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 $\mathbb{A}(S) = \mathbb{R} \times \prod\limits_{p \in S} \mathbb{Q}_p$ 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 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 $A_i$.