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It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class field theory. To attempt increasing resemblance, one may dualize and speak about correspondence between characters/one dimensional representations of the fundamental group over some field and first cohomology with coefficients in the multiplicative group of that field; in sufficiently nice cases this should in particular give correspondence between appropriate brand of coverings, line bundles and their Chern classes. One may even try to imitate ramification by considering branched coverings and not necessarily locally free coherent sheaves or I don't know what (Question zero - has this actually been made more rigorous and if yes, where?).

I am not aware of any relevant higher-dimensional stuff, relating, say, 2-dimensional representations of the fundamental group to characteristic classes of the corresponding vector bundles in any context - say, over $\mathbb C$ even, let alone other coefficient fields or higher dimensional non-locally-free coherent sheaves. Are there any texts about this?

The question What is the precise relationship between Langlands and Tannakian formalism? seems to be related but I am after something much more simple-minded

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    $\begingroup$ Just a remark (which you probably already know about): "abelianization of the fundamental group is the first homology group, as some remote relative of class field theory" is a statement used in the cohomological approach to class field theory. $\endgroup$
    – skd
    Commented Aug 23, 2018 at 5:12
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    $\begingroup$ Semi-related: mathoverflow.net/questions/7283/topological-langlands $\endgroup$
    – Drew Heard
    Commented Aug 23, 2018 at 7:41
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    $\begingroup$ The cohomological approach to class field theory is discussed in Milne's notes jmilne.org/math/CourseNotes/CFT.pdf; the essential idea is that $\mathrm{Gal}(K^\mathrm{ab}/K)$ can be thought of $\mathrm{H}_1(\mathrm{Gal}(K^\mathrm{sep}/K; \mathbf{Z})$, where $K$ is a nonarchimedean local field. In any case, I would say that Tate's theorem (II.3.11 in Milne) is the "most important" part of the proof, not this result. This is because Tate's theorem implies that if $L/K$ is a finite extension with Galois group $G$, then... $\endgroup$
    – skd
    Commented Aug 23, 2018 at 19:11
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    $\begingroup$ ...there is an isomorphism $\hat{\mathrm{H}}^{-2}(G; \mathbf{Z}) \to \hat{\mathrm{H}}^0(G; L^\times)$; the left hand side is $G_\mathrm{ab} = \mathrm{H}_1(G; \mathbf{Z})$, and the right hand side is $K^\times/\mathrm{N}(L^\times)$. This defines the local Artin homomorphism. Also: the link Drew pointed to is very relevant. I don't know anything about Langlands, but as far as I know (from high-level discussions with people) the n=2 case of Langlands is very closely related to the theory of modular forms. It should be no surprise that chromatic homotopy theory at height 2 also... $\endgroup$
    – skd
    Commented Aug 23, 2018 at 19:17
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    $\begingroup$ ... admits a deep relationship with the theory of modular forms (via the $\mathbf{E}_\infty$-ring of topological modular forms, and its numerous variants). Moreover, I've heard some people say that there should be a geometric interpretation of TMF cocycles in terms of "2-vector bundles", but I don't know what these are (this paper seems relevant: arxiv.org/abs/1805.04146). tl;dr: the discussion in the link Drew provided probably isn't the simplest analogue of Langlands, but it definitely seems to be the "right" one. $\endgroup$
    – skd
    Commented Aug 23, 2018 at 19:21

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