I ask this question because I'm quite confused about the fundamental theorem of covering spaces ;
This theorem say that under suitable hypothesis for a topological space $X$ (semi locally simply connected) we have an equivalence of categories : $$Cov(X) \simeq [\Pi_1 (X) \to Set]$$ between the category of coverings of $X$ and the category of functors from the fundamental groupoïd of $X$ to the category of sets : we can "reverse" the point of view.
The way I understand this result is that a covering of such a space depend only on the fundamental groupoïd of the base space $X$, thus of his $2$-skeleton if $X$ is a CW-complex. In particular, Adding $n$-cells to $X$ for $n > 2$ shouldn't have any effect on a given covering (e.g. the universal one).
But on the other side, the long exact sequence of homotopy groups that is induced by a given covering of $X$ show that the higher homotopy groups (for $n > 1$) of $X$ and his covering are equal. So if we add 3-cells to "kill" $\pi_2 (X)$, this must have some effect on the universal covering, meaning that he depend on more than the 2-skeleton of $X$
What am I missing ?