I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. Apologies if this is unreasonable. We'll start with some definitions and then the questions. (I think I rephrased Def. 4 correctly but I'm not certain.)

Definition 1. [0.1] Let $\begin{smallmatrix}Y\\\downarrow\\\operatorname{Spec}k\end{smallmatrix}$ be a morphism of schemes and consider a $k$-algebra $R$. Two $R$-points $y,y^\prime:\operatorname{Spec}R\to X$ over $\operatorname{Spec}k$ are said to be infinitesimally close if they have the same image under the map $X(R)\to X(R/\sqrt 0)$.

Definition 2. [0.3] Let $\begin{smallmatrix}Y\\\downarrow\\\operatorname{Spec}k\end{smallmatrix}$ be a morphism of schemes. A crystal of quasi-coherent sheaves on this morphism consists of a quasi-coherent sheaf $F$ on $Y$ along with a family of isomorphisms with some properties. Namely, for every pair of infinitesimally close $R$-valued points of $\begin{smallmatrix}Y\\\downarrow\\\operatorname{Spec}k\end{smallmatrix}$, i.e commutative triangles $y,y^\prime:\operatorname{Spec}R\to Y$ over $\operatorname{Spec}k$, we have an isomorphism $\eta_{y,y^\prime}:y^\ast Y\cong y^{\prime\ast}Y$. These isomorphisms are required to behave well with respect to base change and, crucially, they must satisfy the cocycle condition: $\eta_{x,z}=\eta_{y,z}\circ\eta_{x,y}$.

Definition 3. [0.5] Let $\begin{smallmatrix}Y\\\downarrow\\\operatorname{Spec}k\end{smallmatrix}$ be a morphism of schemes. A crystal of schemes on this morphism consists of a scheme morphism $\begin{smallmatrix}X\\\downarrow\\Y\end{smallmatrix}$ along with a family of isomorphisms with some properties. Namely, for every pair of infinitesimally close $R$-valued points of $y,y^\prime$ of $\begin{smallmatrix}Y\\\downarrow\\\operatorname{Spec}k\end{smallmatrix}$ we have an isomorphism of $R$-schemes $y^\ast\begin{smallmatrix}X\\\downarrow\\Y\end{smallmatrix}\cong y^{\prime\ast}\begin{smallmatrix}X\\\downarrow\\Y\end{smallmatrix}$

Definition 4. [2.2,2.3] Let $\begin{smallmatrix}Y\\\downarrow\\S\end{smallmatrix}$ be a morphism of schemes. Write $p_1,p_2$ for the projections $\Delta_{Y/S}^{(1)}\rightrightarrows Y$ from the first neighborhood of the diagonal and note the inclusion $\jmath_0:Y\hookrightarrow \Delta^{(1)}_{Y/S}$ is a common section of both $p_1,p_2$. Fix some fibration over the category of schemes of interest and let $M$ be an object in the fiber over $Y$. A connection on $M$ is an isomorphism $\eta:p_1^\ast M\cong p_2^\ast M$ such that $\jmath_0^\ast (\eta)=1_M$. It is integrable if the pullbacks of the next level of the Čech nerve satisfy the cocycle condition (see 2.3 in the link).

Definition 5. [nlab] Let $\begin{smallmatrix}X\\\;\downarrow p\\Y\end{smallmatrix}$ be a morphism of schemes. A $p$-connection is a section of the induced differential $\mathrm T_X\to p^\ast \mathrm T_Y$. A $p$-connection is flat if its transpose map $\mathrm T_Y\to p_\ast\mathrm T_X$ commutes with brackets of vector fields.

Question 1. In our setting, if $y,y^\prime$ are infinitesimally close and $y^\prime,y^{\prime\prime}$ are infinitesimally close, then $y,y^{\prime\prime}$ are infinitesimally close. However, in his work on synthetic differential geometry, Kock stresses that the "first order neighborhood relation" is not transitive. Is there anything to be said here about the apparent conceptual contradiction? In Combinatorial Differential Forms Breen and Messing specifically stress the significance of Kock's work, so I would like to understand the gap.

Question 2. Definition 4 seems very conceptual. It's basically descent data for a fibration. In the special case of the codomain fibration, a connection w.r.t $\begin{smallmatrix}Y\\\downarrow\\S\end{smallmatrix}$ on a "bundle" $\begin{smallmatrix}X\\\downarrow\\Y\end{smallmatrix}$ does indeed give a "parallel transport between fibers of infinitesimally close points". However, I don't really have a feel for the role of the bundle $\begin{smallmatrix}Y\\\downarrow\\S\end{smallmatrix}$ here. The absolute case is familiar from differential geometry, where a connection on a bundle is absolute w.r.t its codomain. What's the geometric role of $\begin{smallmatrix}Y\\\downarrow\\S\end{smallmatrix}$ here?

Question 3. In [0.1.2], it is stressed that the motivation for Definitions 2,3 come from considering a flat connection on a bundle. The motivating example is that of a flat connection on a bundle of smooth manifolds. As a manifold, the base is locally simply connected, and so locally on the base there's a single homotopy class of paths between two points. Hence by flatness of the connection there's only one parallel transport map between each pair of fibers. On the other hand, flatness/integrability seems to be a property of a connection in Definition 4. What is the relationship between Definitions 2,3 and Definition 4?

Question 4. Definition 5 is precisely the notion of an Ehresmann connection from differential geometry. A complete Ehresmann connection on a submersion induces global parallel transport diffeomorphisms. Without completeness, I don't know of anything that can be said. How is the notion of Definition 5 related to Definitions 2,3,4? It's also strange to me that Definition 5 defines a connection on a bundle in an absolute fashion, namely without reference to another bundle. Again I would like a feel for the picture and usefulness of the relative notions above.

Question 5. Evidently Definition 4 can be recast for a fibration over many categories. Suppose we consider a fibration of submersions/fiber bundles/vector bundles over smooth manifolds. In this setting, what notion of connection does Definition 4 recover? At least when $S=\bf 1$, does it recover the usual notion(s) of Ehresmann connection on a bundle?

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    $\begingroup$ Question 1: Infinitesimally close is not the same as square-zero close. You have a neighborhood relation of arbitrary finite order. $\endgroup$ – S. Carnahan Dec 22 '18 at 5:15
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    $\begingroup$ Question 2: $S$ is a base for a family of connections with no infinitesimal relation between fibers. We only have infinitesimal gluing in the "vertical direction" of $Y \to S$. $\endgroup$ – S. Carnahan Dec 22 '18 at 5:18

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