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The usual notion of trivial covering morphism is in a sense intrinsic to the adjunction $\Pi_0\dashv H$ between connected components and discrete spaces: a continuous map $f$ is a trivial covering morphism if and only if its naturality square is a pullback square. The naturality square being a pullback encodes being an identity on each connected component.

Is there an adjunction that gives an intrinsic notion of formal étaleness? I know that infinitesimal cohesion might have something to do with this, but I find it very hard to fill in the blanks in the ideas sketched by Lawvere. On the nlab there are fascinating entries on differential cohesion which unfortunately I cannot digest. I struggle with the $\infty$-groupoids and the latter.

So, rephrasing, what is the "down to earth" adjunction that yields formally étale morphisms? Is there any place to read about it at the 1 or 2-categorical level? What is this adjunction for commutative ring spectra?

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  • $\begingroup$ Your characterization of etaleness seems to only make sense for finitely generated schemes over the complex numbers. It may be worth fixing this issue before considering formal etaleness. $\endgroup$
    – js21
    Dec 5, 2016 at 13:28
  • $\begingroup$ @js21 sorry, what do you mean? I don't offer any characterization of étaleness in my question. Is there anything wrong about what I wrote for trivial covers? $\endgroup$
    – Arrow
    Dec 5, 2016 at 14:18
  • $\begingroup$ Sorry, I misread your first paragraph. $\endgroup$
    – js21
    Dec 5, 2016 at 14:34
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    $\begingroup$ An example is perhaps an adjunction (or more than one adjunction pair) between reduced and unreduced schemes? When you have non-reduced schemes you can use the dual numbers to talk about infinitesimal neighbourhoods, and ask that these lift uniquely. Formally étale is I believe equivalent to this. $\endgroup$
    – David Roberts
    Dec 6, 2016 at 1:30

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