Let $\mathbb C[[x]]$ be the ring of formal power series in one indeterminate $x$. Let $\mathbb C((x))$ be the field of formal Laurent power series. We know that that $\mathbb C((x))$ is the fraction field of $\mathbb C[[x]]$. We also know that every localization $S^{-1}A$ of a commutative ring $A$ with respect to a multiplicative subset $S$ can be expressed as the colimit
$$S^{-1}A=colim_{s\in S}A_s,$$
where $A_s$ is the localization of $A$ with respect to the multiplicative set $\{s^i:i\geq0\}\subseteq S\subset A$.
In light of the above information, is the correct colimit expression of the fraction field $\mathbb C((x))$ the following:
$$\mathbb C((x)):=T^{-1}\mathbb C[[x]]=colim_{x^m\in T}\mathbb C[[x]]_{x^{m}}=colim_{m\geq0}\mathbb C[[x]]x^{-m},$$
where $T:=\{x^m:m\geq0\}$, $\mathbb C[[x]]_{x^{m}}$ is the localization of $\mathbb C[[x]]$ with respect to the multiplicative subset $\{x^{im}:i\geq0\}$? Does this make any sense considering the fact that $\mathbb C[[x]]T^{-1}=\mathbb C[[x]]_{x}$? Do we view $\mathbb C[[x]]x^{-m}$ rank-$1$ free modules over $\mathbb C[[x]]$?