Anders Kock's two texts on [synthetic differential geometry](https://ncatlab.org/nlab/show/synthetic+differential+geometry) (SDG) are a great place to get geometric intuition, especially when it comes to jets. Unfortunately, he doesn't seem to cover semi-holonomic jets in his text, and I find the usual definition of semi-holonomic jets to be difficult to interpret geometrically.


For the benefit of anyone unfamiliar with the definition of non-holonomic jets in SDG, I will repeat it here.


Recall that on formal manifolds we have the reflexive symmetric (but non-transitive) infinitesimal neighbourhood relation $\sim\_k$, and we say that $x$ and $y$ are $k$th order infinitesimal neighbours if $x \sim\_k y$. Though non-transitive, the infinitesimal neighbourhood relations satisfy the property that if $x \sim\_{k\_1} x\_1 \sim\_{k\_2} \cdots \sim\_{k\_r} x\_r$ then $x \sim\_{k\_1+ \cdots + k\_r} x\_r$.


We can define the **infinitesimal $k$-neighbourhood**, $k$-monad, or $k$-**halo** of a point $x \in M$ in the obvious way:
 $$N\_k(x) :=\{y \in M \mid x \sim\_k y \}.$$
Note that Anders Kock uses the term $k$-monad, which is also used in nonstandard analysis, but I personally think this conflicts with the more widely known meaning of the term in category theory and algebra.


Given a bundle $p: E \to M$ over a formal manifold, we can define a section $k$-jet $s \in J^k\_xE$ to be a local section $s: N\_k(x) \to E$. Thus, it is easy to see that jet prolongation of a bundle section $\sigma: M \to E$ just assigns to that section its restrictions to infinitesimal neighbourhoods: $$j^k\_x \sigma = \sigma|\_{N\_k(x)}.$$


Given a sequence of positive integers $(k\_1, \cdots, k\_r)$ we can construct the **non-holonomic jet bundle** $J^{(k\_1, \cdots, k\_r)}E :=J^{k\_1}\cdots J^{k\_r}E$. In SDG a non-holonomic $E$-valued jet $s \in J^{(k\_1, \cdots, k\_r)}\_x E$ is just a map which assigns to a sequence of infinitesimal neighbours $x \sim\_{k\_1} x\_1 \sim\_{k\_2} \cdots \sim\_{k\_r} x\_r$ an element of the fibre $s(x, x\_1, \cdots, x\_r) \in E\_{x\_r}$. From this point of view, the justification for the term "[non-holonomic](https://en.wikipedia.org/wiki/Nonholonomic_system)" is evident, since the element of the fibre $E\_{x\_r}$ we obtain by applying the non-holonomic jet depends in general on infinitesimal path taken from $x$ to $x\_r$. 


In the case where the jet $s \in J^{(k\_1, \cdots, k\_r)}\_x E$ does not depend on the infinitesimal path, but only on the endpoint $x\_r$, then it is simply a local section $s: N\_{k\_1 + \cdots + k\_r}(x) \to E$, and hence a holonomic $k\_1 + \cdots + k\_r$-jet. Thus, we have a natural embedding 
$J^{k\_1 + \cdots +k\_r}E \hookrightarrow J^{(k\_1, \cdots, k\_r)}E$



> 
> **Question:** How should semi-holonomic jets be defined in synthetic differential geometry? What is the "correct" geometric intuition for non-holonomic jets in this setting?
> 
> 
> 


**My thoughts**: Since a non-holonomic jet depends on the infinitesimal path taken, and a holonomic jet does not (depending only on the endpoint of the infinitesimal path) a semi-holonomic jet should admit an interpretation somewhere in between these two. The path dependence for semi-holonomic jets should be less strict in some sense.



Semi-holonomic jets in synthetic differential geometry
According to Liebermann's [*Introduction to the theory of semi-holonomic jets*](https://eudml.org/doc/226286) p.177:



> 
> a local section $s:U\subset M\to J^1E$ is said to be *adapted at $x\in U$* if $s(x)=j^1\_x(\beta \circ s)$ where $\beta:J^1 E\to E$ is the target map; then the jet $j^1\_x(s)$ is called *semi-holonomic*.
> 
> 
> 


Assuming that with the *target map* $\beta$ she means the usual projection sending a jet $q:N\_1(x) \to E$ to $q(x)\in E$, we can translate this definition into SDG language as follows:


Since the condition of being adapted only depends on the first jet of the section $s$ at $x$, and at the end of the definition we also only care for the first jet of $s$, we might as well start directly with a section $s$ defined only on the first infinitesimal nbh $s:N\_1(x)\to J^1E$, i.e. with a non-holonomic jet $s\in (J^1J^1 E)\_x$ (I write $E\_x$ for the fiber of a bundle over the base point $x$). Then the adaptedness condition simplifies to 
$$s(x)=\beta \circ s$$ wich is at the same time the condition on $s$ to be semi-hilonomic. The left and right side of this eq. are elements of $(J^1E)\_x$, hence both are sections of type $N\_1(x)\to E$ and we can apply them to an element $x\_1\in N\_1(x)$ to obtain the condition $s(x)(x\_1)=s(x\_1)(x\_1)$.


Summarising and borrowing some notation from dependent type theory: a non-holonimic jet $s\in (J^1 J^1 E)\_x$ at $x\in M$ is the same thing as a dependent map $s : \Pi\_{x\_1 \in N\_1(x)} \Pi\_{x\_2\in N\_1(x\_1)}E\_{x\_2}$, and such a map is semi-holonomic if 
$$
s(x)(x\_1)=s(x\_1)(x\_1)
$$
for all $x\_1\in N\_1(x)$.


Intuitively this condition means the following: think of a non-holonomic jet $s\in (J^1 J^1 E)\_x$ as specifying a family of jets $s(x\_1)\in (J^1E)\_{x\_1}$, one for every $x\_1 \sim\_1 x$. For an arbitrary non-holonomic jet these $s(x\_1)$ are unrelated, while for a holonomic jet they all agree on the intersections $N\_1(x\_1)\cap N\_1(x)$. For a semi-holonomic jet the $s(x\_1)$ only agree with the "central" jet $s(x)$ at their base point $x\_1$. I wish I could draw a picture of this here...


