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?