Let $X$ be a nice scheme (additional assumptions could be added), and let $ET(X)$ be its (Artin-Mazur) etale homotopy type. I am looking for a/the scheme $Y$ over $X$ whose etale homotopy type $ET(Y)$ will be the topological universal cover of $ET(X)$. By definition $ET(X)$ is the geometric realization of a simplicial set, and it was pointed out to me that if $Y \rightarrow X$ is the universal cover of a simplicial set $X$, then the geometric realization $|Y|$ is the topological universal cover of $|X|$.

What is the meaning of "Universal cover of a simplicial set"; Is there a reference for that, and also for the second assertion?. How could we apply this to find $Y$?

Let $X$ be a nice scheme (additional assumptions could be added), and let $Et(X)$ be its (Artin-Mazur) etale homotopy type. I am looking for a/the scheme $Y$ over $X$ whose etale homotopy type $Et(Y)$ will be the topological universal cover of $Et(X)$. By definition $Et(X)$ is the geometric realization of a simplicial set, and it was pointed out to me that if $R \rightarrow S$ is the universal cover of a simplicial set $S$, then the geometric realization $|R|$ is the topological universal cover of $|S|$.

What is the meaning of "Universal cover of a simplicial set"; Is there a reference for that, and also for the second assertion?. How could we apply this to find $Y$?

Edit: If we want to avoid the simplicial method. Suppose that there is a scheme $Y$ over $X$ such that

1. The first etale homotopy group $\pi_1^{et}(Y)$ is trivial.

2. For all $n \geq 2$ we have $\pi_n^{et}(Y) \simeq \pi_n^{et}(X)$.

Are these conditions sufficient to state that $Et(Y)$ is the topological universal cover of $Et(X)$?