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Let $X\to Y$ be a $C^\infty$ submersion. Consider the following two sheaves.

  1. The sheaf on $Y$ comprised of jets of sections of $X\to Y$.
  2. The sheaf on $X$ given by the quotient of $\Delta_{X/Y}^{-1}C^\infty_{X\times_YX}$ by the second power of the ideal of $C^\infty$ maps about the diagonal vanishing on the diagonal.

Consider the following three definitions of a connection on the submersion $X\to Y$.

  1. Let $J^1_{X/Y}\to Y$ be the bundle of 1-jets of sections of $X\to Y$. Retaining the constant term of a Taylor expansion gives a bundle map $J^1_{X/Y}\to X$ over $Y$. A connection on $X\to Y$ is a section of this bundle map.
  2. Write $\Delta^{(1)}_{Y}$ for the locally ringed space structure on the underlying topological space of $Y$ with structure sheaf given by the above quotient of $\Delta_{Y}^{-1}C^\infty_{Y\times Y}$. It comes with two canonical locally ringed space projections $p_1,p_2:\Delta_{Y}^{(1)}\rightrightarrows Y$. A Grothendieck connection on $X\to Y$ is an isomorphism between the pullbacks (in the category of locally ringed spaces) of $X\to Y$ along $p_1,p_2$ which restricts to the identity on the diagonal.
  3. An Ehresmann connection on $f:X\to Y$ is a section of the differential $\mathrm df:\mathrm TX\to f^\ast \mathrm TY$ (in the category of $C^\infty$ vector bundles over $X$).

I understand the equivalence of the first and third definitions. For instance, starting with a connection in the sense of the first definition, an Ehresmann connection can be constructed using derivatives of local sections of $X\to Y$.

Question. How are Grothendieck connections related/equivalent to the first/third definition?

Added. For what it's worth, an explicit construction of pullbacks in the category of locally ringed spaces is sketched in this answer. In particular, it seems that on stalks, a Grothendieck connection would give an automorphism of the stalk $(C_{X,x}^\infty\otimes_{C^\infty_{Y,y}}\mathcal O_{\Delta^{(1)}_Y,y})_\mathfrak q$ where $fx=y$ and $\mathfrak q$ is a prime ideal which pulls back to the maximal ideals of $C_{X,x}^\infty,\mathcal O_{\Delta^{(1)}_Y,y}$. The structure sheaf of the first neighborhood of the diagonal of $Y$ consists of 1-jets of real $C^\infty$ map on $Y$, so in coordinates its stalk is isomorphic to the ring of dual numbers $\mathbb R[\varepsilon_1,\dots \varepsilon _{\dim Y}]$. Even on this level of stalks I don't know how to relate to a section of $\mathrm df:\mathrm TX\to X\times_Y\mathrm TY$, though the explicit construction of the pullback in $\mathsf{LRS}$ hints at viewing tangents as derivations.

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  • $\begingroup$ I suppose $k=1$? $\endgroup$ Feb 29, 2020 at 13:58
  • $\begingroup$ Yes. Removed that bit. $\endgroup$
    – Arrow
    Feb 29, 2020 at 14:34
  • $\begingroup$ It might be useful to look at this question and its answers. mathoverflow.net/questions/68305/… $\endgroup$ Feb 29, 2020 at 17:25
  • $\begingroup$ Dear @TomGoodwillie I spent a while there but wasn't able to relate to Ehresmann connections. $\endgroup$
    – Arrow
    Feb 29, 2020 at 17:51

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