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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.

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According to Liebermann's Introduction to the theory of semi-holonomic jets 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...

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