# Construction of the universal covering space of the etale homotopy type $Et(X)$

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 $$Et(X)$$?

• Do you have a universal cover of the multiplicative group $\mathbb G_m$? The conditions that you listed seem to be sufficient if you replace the second by the stronger condition that the map $\pi_n^{\operatorname{ét}}(Y)\to\pi_n^{\operatorname{ét}}(X)$ induced by the map $Y\to X$ is an isomorphism for every $n>1$. However, I suspect that the existence is very rare.
– Z. M
Jul 20, 2021 at 9:39
• @Z.M Thank you for your comment. Yes, the condition you mentioned is exactly the one i meant. Jul 25, 2021 at 16:29

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.

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, 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 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.

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 $$\tilde X=\operatorname{Spec}\bar k[t^{\pm 1}][t^{1/n}, n\geq 2]$$.

• Thank you for your detailed answer. Could we translate the meaning of a n-truncated morphism $Et(Y) \rightarrow Et(X)$ by $\pi_p(Et(Y)) \simeq \pi_p(Et(X))$ for all $p \geq n+2$? Jul 25, 2021 at 21:19
• Not quite, $n$-truncated means iso on $\pi_p$ for $p\geq n+2$ and mono on $\pi_{n+1}$. However, for pro-spaces one cannot quite define them this way because pro-spaces may not have enough base points (eg, the set of points could be empty). The actual definition is: a morphism in $\mathrm{Pro}(\mathcal S)$ is $n$-truncated if it is a cofiltered limit of $n$-truncated morphisms in $\mathcal S$. Jul 26, 2021 at 5:58
• I guess that the subtlety of $\operatorname{Pro}(\mathcal S)$ disappears if you pass to the profinite completion?
– Z. M
Jul 26, 2021 at 11:10
• @Z.M Indeed profinite space are much better behaved. In particular a morphism of profinite spaces is $n$-truncated iff it is so after passing to the limit, this is Prop. E.4.6.1 in Spectral Algebraic Geometry. In order to be able to check $n$-truncativity using homotopy pro-groups, I would guess that $\pi_0$ being profinite is enough, which is certainly the case for $\mathrm{Et}$ of a qcqs scheme. Jul 26, 2021 at 12:50