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 $Et(X)$?



Construction of the universal covering space of the etale homotopy type $Et(X)$
Such an "étale universal cover" exists at least if $X$ is Noetherian and geometrically unibranch (and for all qcqs $X$ if one considers profinite étale homotopy types).


**Background.** I will regard the étale homotopy type of a scheme as an object in the $\infty$-category $\mathrm{Pro}(\mathcal S)$ of pro-spaces. In the $\infty$-category $\mathcal S$, the universal cover of a pointed space $(X,x)$ can be characterized as the initial object in the $\infty$-category of pointed $0$-truncated morphisms to $(X,x)$ (i.e., morphisms with discrete fibers). Similarly, one can define $0$-truncated morphisms in $\mathrm{Pro}(\mathcal S)$, and every pointed object admits a universal cover. References for the étale homotopy type and for $n$-truncated/$n$-connected morphisms in $\mathrm{Pro}(\mathcal S)$ are Sections E.2 and E.4.2 in [Spectral Algebraic Geometry](https://www.math.ias.edu/%7Elurie/papers/SAG-rootfile.pdf).


Let $\mathcal S\_{<\infty}$ be the $\infty$-category of truncated spaces and let $\mathrm{Et}\colon \mathrm{Sch} \to \mathrm{Pro}(\mathcal S\_{<\infty})$ be the [protruncated étale homotopy type](https://arxiv.org/pdf/1812.11637.pdf), which is the homotopy-coherent incarnation of the construction of Artin and Mazur. Under the equivalence
$$\mathrm{Pro}(\mathcal S\_{<\infty}) \simeq \mathrm{Fun}^\mathrm{acc,lex}(\mathcal S\_{<\infty},\mathcal S)^\mathrm{op},$$
$\mathrm{Et}(X)$ is the functor $\mathcal S\_{<\infty}\to\mathcal S$ sending a truncated space $K$ to the global sections $\Gamma(X\_\mathrm{et},K)$ of the constant étale sheaf on $X$ with value $K$.


**Construction.** Let $X$ be a scheme with a geometric point $x\colon \operatorname{Spec} k\to X$, with $k$ separably closed. Consider the category $\mathrm{FEt}(X,x)$ of pointed finite étale covers of $X$, that is, factorizations of $x$ through a finite étale morphism $X'\to X$. This category has finite limits and in particular is cofiltered (it is also essentially small). Thus, the limit $\tilde X$ of the forgetful functor
$$
\mathrm{FEt}(X,x) \to \mathrm{Sch}\_{/X}
$$
exists (note that $\tilde X$ depends on $x$, so the notation is abusive). This is a natural candidate for the "universal cover" of $(X,x)$.


Now, the statement that $\mathrm{Et}(\tilde X)\to\mathrm{Et}(X)$ is the universal cover of the pointed pro-space $(\mathrm{Et}(X),x)$ is equivalent to the following three conditions: 1) the map $\mathrm{Et}(\tilde X)\to\mathrm{Et}(X)$ is $0$-truncated, 2) $\mathrm{Et}(\tilde X)$ is connected, and 3) $\pi\_1^\mathrm{et}(\tilde X,x)$ is trivial.


**Results.**


1) always holds if $X$ is quasi-compact and quasi-separated.


First, I claim that the functor $\mathrm{Et}$ preserves the limit of the diagram defining $\tilde X$. Under the above equivalence of $\infty$-categories, this is the statement that for $K$ a truncated space, $\Gamma((-)\_\mathrm{et},K)$ transforms this limit into a colimit, which is a standard property of étale cohomology with respect to inverse limits of qcqs schemes. [Here it is important that $K$ is truncated, otherwise this may not be true.]


Then, if $p\colon X'\to X$ is finite étale, I claim that the morphism $\mathrm{Et}(p)\colon \mathrm{Et}(X')\to \mathrm{Et}(X)$ is $0$-truncated. In fact, it is the pullback of a morphism of groupoids $\pi\colon\Xi'\to\Xi$ with finite discrete fibers. To see this, note that the morphism of étale $\infty$-topoi induced by $p$ is itself the pullback of such a morphism $\pi$. The point is then that for any space $K$, $\Gamma(X'\_\mathrm{et}, K)$ is $\Gamma(X\_\mathrm{et},p\_\*K)$ and $p\_\*K$ is a locally constant sheaf on $X$ in a strong sense: it is the pullback of the sheaf $\pi\_\*K$ on $\Xi$ (the fact that the fibers of $\pi$ are finite spaces is used here, to commute the pushforward with the pullback). One can thus apply Proposition 2.15 in [Higher Galois theory](http://www.mathematik.ur.de/hoyois/papers/highergalois.pdf) to compute $\Gamma(X\_\mathrm{et},p\_\*K)$ in terms of the étale homotopy type of $X$. Unpacking this formula gives $\mathrm{Et}(X')=\mathrm{Et}(X)\times\_\Xi\Xi'$. [Here, $K$ need not be truncated, so $p$ induces a $0$-truncated morphism on actual étale homotopy types, and $X$ need not even be qcqs.]


**ETA:** If $X$ is locally noetherian, another proof of this claim is Lemma 2.1 in Schmidt-Stix, [Anabelian geometry with etale homotopy types](https://arxiv.org/pdf/1504.01068.pdf).


Finally, since $0$-truncated morphisms are stable under limits, $\mathrm{Et}(\tilde X)\to\mathrm{Et}(X)$ is also $0$-truncated.


2) also holds if $X$ is qcqs. In this case $\tilde X$ is connected, as any clopen subset of $\tilde X$ lifts to a clopen subset of some finite étale $X'\to X$.


3) holds if we assume moreover that the pro-group $\pi\_1^\mathrm{et}(X,x)$ is profinite, e.g., $X$ is Noetherian and geometrically unibranch. Then $\pi\_1^\mathrm{et}(\tilde X,x)$ is also profinite by 1), so 3) is equivalent to the statement that every finite étale cover of $\tilde X$ is trivial, which holds by construction.


Note that the additional assumption for 3) is not needed if we pass to the profinite completions. That is, if $X$ is qcqs, then $\mathrm{Et}(\tilde X)^\wedge \to \mathrm{Et}(X)^\wedge$ is the universal cover of the profinite étale homotopy type $(\mathrm{Et}(X)^\wedge,x)$.


**Example.** Let $X=\mathbb G\_m$ over a field $k$ of characteristic zero, pointed at $1$ by an algebraic closure $\bar k$ of $k$. Then $\tilde X=\operatorname{Spec}\bar k[t^{\pm 1}][t^{1/n}, n\geq 2]$.


