Let $\mu\_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu\_n=\ker[{\mathbb{C}}^\*\overset{n}{\longrightarrow}{\mathbb{C}}^\*]$.
We set
$$ \mu=\varinjlim\_n \mu\_n\subset {\mathbb{C}}^\*,\quad {\widehat{\mathbb{Z}}}(1)=\varprojlim\_n \mu\_n,\quad {\mathbb{A}}(1)={\widehat{\mathbb{Z}}}(1)\otimes\_{{\mathbb{Z}}} {\mathbb{Q}}.$$
I would like to have a canonical isomorphism
$$ \mu \overset{\sim}{\longrightarrow} {\mathbb{A}}(1)/{\widehat{\mathbb{Z}}}(1)$$
similar to the canonical isomorphism
$$\mathbb{Q}/\mathbb{Z}\to {\mathbb{A}}/{\widehat{\mathbb{Z}}} .$$
Is it possible to construct such a canonical isomorphism?



Adeles and twisted adeles
Doesn't this boil down to:


$$ \mathbb{A}(1)/\hat{\mathbb{Z}}(1) \cong \mathbb{Q}/\mathbb{Z} \otimes \hat{\mathbb{Z}}(1) \cong (\mathrm{colim}\, \mathbb{Z}/n\mathbb{Z}) \otimes \hat{\mathbb{Z}}(1) \cong \mathrm{colim}\, (\mathbb{Z}/n\mathbb{Z} \otimes \hat{\mathbb{Z}}(1)) \cong \mathrm{colim}\, \mu\_{n} = \mu$$


Here I use that tensoring commutes with colimits, because it is a left-adjoint:
$$ \mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, \mathrm{Hom}(B, C)) $$
(This is the universal property of the tensor product.)


It is a general fact that left (resp. right) adjoint functors commute with colimits (resp. limits). See <https://en.wikipedia.org/wiki/Adjoint_functors#Limit_preservation>.


