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I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate.

Here is a list of definitions for an arrow $f$ to be étale.

  1. SDG I - Kock's SDG book, section I.17, and Kostecki's notes, definition 8.10, "specialized" to our case (take the small object to be just $D=\operatorname{Spec}_RR[\epsilon]$), ask for the square below to be a pullback $$\require{AMScd} \begin{CD} TM @>{df}>> TN\\ @V{\pi}VV @VV{\pi^\prime}V\\ M @>>{f}> N \end{CD}$$ Geometrically, this seems like the most intuitive - identifying the tangent bundle of $M$ with the tangents of $N$ with basepoints in the image of $f$. By pullback pasting this implies also $df:T_xX\cong T_{f(x)}Y$.
  2. SDG II - Rephrasing the pullback as a strong left lifting property, $f:M\rightarrow N$ is étale if it has the unique right lifting property w.r.t to the point $0:\mathsf{1}\rightarrow D$: $$\array{ \mathsf{1} &\longrightarrow& M \\ \downarrow &\nearrow& \downarrow \\ D &\longrightarrow& N }$$ This doesn't look intuitive to me, but whatever if it's formally equivalent to 1.
  3. Algebraic Geometry I - a morphism of affine schemes $f:\operatorname{Spec}B\rightarrow\operatorname{Spec}A$ is étale if it has the unique right lifting property against infinitesimal extensions, i.e (opposites of) regular epimorphisms $\hat R\twoheadrightarrow R$ with nilpotent kernel: $$\array{ \operatorname{Spec}R &\longrightarrow& \operatorname{Spec}B \\ \downarrow &\nearrow& \downarrow \\ \operatorname{Spec}\hat R &\longrightarrow& \operatorname{Spec}A }$$ This definition seems conceptual, but I don't really understand why it should be related to the much more intuitive notion of local diffeomorphism. Why should local diffeomorphisms "extend to infinitesimal neighborhoods"?

Are definitions 1 and 3 equivalent or related in some useful setting? How to move between them?

Added. If we do not specialize 1 to second-order infinitesimals and instead take the definition asking the square below to be a pullback for generalized tangent space ($W$ is a Weil algebra), are 1 and 3 equivalent? $$\require{AMScd} \begin{CD} M^{D(W)} @>>> N^{D(W)}\\ @VVV @VVV\\ M @>>{f}> N \end{CD}$$

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    $\begingroup$ 1 is the special case of 3 with $\hat R=\operatorname{Spec}R[\epsilon]/(\epsilon^2)$. $\endgroup$
    – Marc Hoyois
    Commented May 8, 2016 at 14:52
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    $\begingroup$ With respect to definition (3): you should think of $\textrm{Spec}\hat R$ as some kind of tubular neighbourood of $\textrm{Spec}R$, so the property is saying that you can lift (small enough) tubular neighbouroods along local diffeomorphisms. If you think about it for a while you'll see that this is more or less equivalent to being a local diffeomorphism $\endgroup$
    – Denis Nardin
    Commented May 8, 2016 at 14:55
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    $\begingroup$ In algebraic geometry, the morphism $f$ needs to be locally of finite presentation in order to be étale. Without that condition, you have the notion called "formally étale". $\endgroup$
    – S. Carnahan ♦
    Commented May 8, 2016 at 15:18
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    $\begingroup$ One thing that helped me concetualizing this is the following analogies: formally smooth ~ submersion, formally unramified ~ immersion, formally etale ~ local isomorphism. To get rid of the formally we just add "locally of finite presentation" which is equivalent to requiring that upon restricting to open affines we get that one affine is a compact object in the category of algebras over the other (using commutation with filtered colimits we can lift morphisms from local rings to open sets - this is the main application I've seen to this finiteness condition). $\endgroup$
    – Saal Hardali
    Commented May 8, 2016 at 18:37
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    $\begingroup$ I think people are confused (at least, I am) as to what exactly you're asking for. Do you agree that (1) and (2) are equivalent for $D=\mathrm{Spec}R[\varepsilon]$? Do you agree that (2) and (3) are equivalent for $R$-algebras with $\hat R=R[\varepsilon]$ (so that $\mathrm{Spec}R=1$ and $\mathrm{Spec}\hat R=D$)? I think that's what @MarcHoyois was pointing out to you. So I thought your question was about how to lift to higher order infinitesimals (and I tried to clarify this in my comment), but now it seems that it's not... $\endgroup$
    – Gro-Tsen
    Commented May 9, 2016 at 17:04

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