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In SGA1, Theoreme 5.1 (Riemann's Existence Theorem) states:

Let $X$ be a $\mathbb{C}$-scheme locally of finite type, $X^{\operatorname{an}}$ the associated analytical space. Then the functor which associates to a finite etale cover $X'$ of $X$ the finite etale cover $X'^{\operatorname{an}}$ of $X^{\operatorname{an}}$ is an equivalence of categories.

Here the notion of "analytic space" (espace analytique) is that defined in Serre's GAGA. Namely, it is a locally ringed space, locally given as the vanishing of holomorphic functions. As far as I can tell, in SGA the notion of etale cover of analytic spaces is simply flat and unramified.

However, Riemann's Existence Theorem is often cited as an equivalence of categories between etale covers of $X$ and topological covering spaces of $X^{\operatorname{an}}$. Thus, I am missing a step here. How does one prove an equivalence of categories between topological covering spaces of $X^{\operatorname{an}}$ and etale covers of it in the category of analytic spaces? Is there a reference for this? Is it obvious? Does hold even if $X$ is singular?

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    $\begingroup$ @WilliamChen: For any complex-analytic space $X$, the functor $(Y \rightarrow X) \rightsquigarrow (|Y|\rightarrow|X|)$ from finite etale $X$-spaces to finite-degree covering spaces of the topological space $|X|$ is an equivalence. (By "etale" I mean "local analytic isomorphism".) Full faithfulness implies the rest (as it permits to work locally on $X$!), and an $X$-morphism $Y' \rightarrow Y$ is a section to the finite etale $p_2:Y' \times_X Y\rightarrow Y'$, so a clopen in $Y'\times_X Y$ that maps bijectively onto $Y'$. Since $|\cdot|$ commutes with analytic fiber products (!), we're done. $\endgroup$
    – nfdc23
    Commented Nov 28, 2015 at 20:55
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    $\begingroup$ @Amy: An "etale cover" is not the notion you intend; you need to impose a condition akin to being a covering space, whereas "etale cover" means "surjective local analytic isomorphism". So you can focus on "finite etale" maps (where "finite" in the analytic setting is classified via coherent sheaves of algebras, in the style of the algebraic setting once one has the theory of coherent analytic sheaves in hand, as is taken for granted in that discussion in SGA1; the book "Coherent Analytic Sheaves" provides such background, as do Seminaire Cartan lectures by Houzel). $\endgroup$
    – nfdc23
    Commented Nov 28, 2015 at 20:59
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    $\begingroup$ @BenWebster: The rest of my comment gives the proof! $\endgroup$
    – nfdc23
    Commented Nov 28, 2015 at 21:00
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    $\begingroup$ There is no good notion of "unramified" for morphisms of general (locally) ringed spaces, as it involves a notion of $\Omega^1$ specific to each setting (schemes, formal schemes, rigid-analytic spaces, complex-analytic spaces, adic spaces, etc.). In each setting it is a hard theorem that "flat + unramified" (+ locally of finite presentation in some cases) is equivalent to other concepts (such as an infinitesimal criterion, or local analytic isomorphism in the complex-analytic case), and those are collectively called "etale". My first comment proves the categorical equivalence you want. $\endgroup$
    – nfdc23
    Commented Nov 29, 2015 at 5:53
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    $\begingroup$ And yes, I was also pointing out that the discussion there in SGA1 is focusing on finite etale maps, rather than general etale maps (but I forgot to require in my first comment that the finite etale $X$-space has surjective structure map onto $X$, in order for it to match with covering spaces having finite fibers on the topological side). $\endgroup$
    – nfdc23
    Commented Nov 29, 2015 at 5:54

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As William Chen points out, a topological cover of a complex analytic space inherits a canonical analytic structure. Any structure defined by being locally isomorphic to a particular ringed space is inherited under topological covers. This just isn't interesting for schemes, since covers in the Zariski topology are not very interesting (are there any non-trivial ones?).

The interesting direction of this statement is when any étale cover is a topological one. This is a famous statement, so famous it has a name: the inverse function theorem. For real and complex manifolds, this theorem holds, so étale covers are topological covers and vice versa. For complex analytic spaces...I'm not totally sure, but this paper made me think probably not. It states a version of the inverse function theorem for analytic spaces which sure didn't seem like it involved actually finding the inverse of a function, but people are free to correct me if I'm wrong.

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    $\begingroup$ What is your definition of "etale" in the complex-analytic setting? The usual definition is "local analytic isomorphism", and there is an "inverse function theorem" allowing arbitrary singularities: a map of complex-analytic spaces $X \rightarrow Y$ is etale near $x$ if it is flat at $x$ and $x$ is an isolated reduced point in $X_{f(x)}$ (for smooth $X$ and $Y$, in case of an isomorphism on tangent spaces we get an isomorphism of completed local rings). The main content in the proof is to show that if $x$ is isolated in its fiber then $f$ is finite between opens around $x$ and $f(x)$. $\endgroup$
    – nfdc23
    Commented Nov 28, 2015 at 21:17
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    $\begingroup$ In the link you give, Corollary 2.5.2 is the local structure theorem for analytic maps near a point isolated in its fiber, the "Grauert-Remmert theorem" on finite morphisms alluded to in the Introduction, and it has been known for a very long time (well-documented in the book of G&R on coherent analytic sheaves, for example; look up their notion of "locally quasi-finite" morphism). The CM hypothesis there ensures flatness by Theorem 23.1 in Matsumura's book "Commutative Ring Theory". If you also require $x$ to be reduced in its fiber then it becomes what I called "inverse function theorem". $\endgroup$
    – nfdc23
    Commented Nov 28, 2015 at 21:27

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