Let $K$ be a global field. As explained in this nLab page, Artin reciprocity is an isomorphism between two abelian groups $$K^\times \backslash \mathbb{I}_K / \mathcal{O} \xrightarrow{\sim} \operatorname{Gal}(K^{ab}/K)$$ where $K^{ab}$ is the maximal abelian extension of $K$ (so $\operatorname{Gal}(K^{ab}/K)$ is an abelian group), $\mathbb{I}_K$ is the group of ideles, some large group which has something to do with the primes in the ring of integers in $K$, and $\mathcal{O}$ is the subgroup of connected component of $1$ in $\mathbb{I}_K$.
I've been hearing people say that one way to think of class field theory is that field extensions behave like covering spaces. From that point of view, I should think of $K^{ab}$ as the largest cover so that the deck transformation group is abelian and the right hand side makes sense in topology. My question is if there is some way to understand the left hand side along this analogue as well?
Edit: To make this question more clear, let me mention that there is a notion of fundamental group for a general topos. See for example the nLab page Galois theory. This unifies the notion of fundamental group in topology, étale fundamental group in algebraic geometry, and hence Galois group. Maybe one way to ask this question is whether there is a general notion of ideles in topos theory?
I think this is what you are asking about, and anyway it's too long for a comment, so I'll write it as an answer.
Suppose $X$ is a topological space with universal cover $\widetilde X$, and with fundamental group $\Gamma = \pi_1(X)$ acting as the group of deck transformations of the universal covering map $\widetilde X \mapsto X$.
Let $\Gamma'=[\Gamma,\Gamma]$ be the commutator subgroup of $\Gamma$, and let $\text{ab}(\Gamma) = \Gamma / [\Gamma,\Gamma]$ be the abelianization.
By restricting the deck transformation action to the commutator subgroup, one can form an orbit space $\widetilde X^{ab} = \widetilde X / \Gamma'$ with an induced regular covering map $\widetilde X^{ab} \mapsto X$ whose deck transformation group is identified with $\text{ab}(\Gamma)$.
This covering map $\widetilde X^{ab} \mapsto X$ is what you allude to in your post. It is known as the universal abelian covering map, because if $Y \mapsto X$ is another regular covering map with abelian deck transformation group then there is a covering map factorization $\widetilde X^{ab} \mapsto Y \mapsto X$ with good properties (relating all of the deck transformation groups and fundamental groups to each other in a good way).
So perhaps the construction $\widetilde X^{ab} = \widetilde X / \Gamma'$ is some kind of analogue to the left hand side of your isomorphism.
A good example of this is when $X$ is the figure 8 space, i.e. the wedge of two circles. In this case we can visualize $$\widetilde X^{ab} = (\mathbb Z \times \mathbb R) \cup (\mathbb R \times \mathbb Z) \subset \mathbb R \times \mathbb R $$
This kind of thing is part of the theory of covering spaces, which you can find in a good topology book such as Munkres. There you will see the theory of regular covering spaces carefully worked out, together with the correspondence between quotient groups of the fundamental group and deck transformation groups of regular covering spaces.