Anders Kock's two texts on 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" 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.