# What is the restricted product categorically?

The restricted product is a construction for locally compact abelian topological groups.

Let $I$ be an indexing set, with $J$ some finite subset.

Let $G_i$ be a locally compact topological group for each $i$ in $I$

Let $K_i$ be an open compact subgroup of $G_i$ for each $i$ in $I-J$

Then the restricted product $\Pi 'G_i$ is defined to be a subset of the product $\Pi G_i$ where each $g_i$ is in $K_i$ for all but finitely many $i$ in $I-J$.

It is always a locally compact group.

This construction is used in defining the adele ring & the idele group for global fields.

One way of reformulating this, is the restricted product is a a subgroup of the full product such that cofinitely many projections of the restricted product are compact.

This seems analogous in some ways to the coproduct of abelian groups which is a subgroup of of the product such that cofinitely many projections are the trivial subgroup.

You can define restricted products in any category. Here's the general setup. Suppose $\mathcal{C}$ is a category, $I$ is a set, and for each $i\in I$ we choose two objects $K_i$ and $G_i$ and a morphism $K_i\to G_i$ in $\mathcal{C}$. Then we may define a restricted product of this system as follows. For each finite subset $A\subseteq I$, let $P_A$ be a product $\prod_{i\in A} G_i\times \prod_{i\in I\setminus A} K_i$. Whenever $A\subseteq B$, our maps $K_i\to G_i$ induce a natural map $P_A\to P_B$, and in this way the objects $P_A$ form a diagram in $\mathcal{C}$ indexed by the poset of finite subsets of $I$. A restricted product of our system is then a colimit of this diagram.
In your case, the category $\mathcal{C}$ is locally compact abelian groups (or just topological groups), the morphisms $K_i\to G_i$ are the inclusion maps for $i\in I\setminus J$ and for $i\in J$ we have $K_i=G_i$ and the map is the identity.