As I understand it from [Kostecki's notes](https://www.google.co.nz/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiOwN6d1YPKAhXkL6YKHQVLA_IQFggbMAA&url=http%3A%2F%2Fwww.fuw.edu.pl%2F~kostecki%2Fsdg.pdf&usg=AFQjCNHJONLnGglncX47CO1tLlAwrJ6_WA&sig2=rrk9jClR8Y6FXHZugM6wew), the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D\_k(n)} : {(R^n)}^{D\_k(n)} \to (R^m)^{D\_k(n)},$$
where $$D\_k(n) = \{(x\_1, \ldots, x\_n)\in R^n \mid x\_{i\_1}\cdot\ldots \cdot x\_{i\_{k+1}}=0 ~\text{for all $k$-tuples}~ (i\_1, \ldots, i\_{k+1})\}$$
Thus we have that $$J^k(R^n, R^m) := \hom \left((R^n)^{D\_k(n)}, (R^m)^{D\_k(n)} \right),$$
where the hom here is the internal one. But can we define $J^k(M,N)$ for arbitrary objects $M$ and $N$, which are not necessarily manifolds?



Jets in synthetic differential geometry
It's correct that a jet $u\in J^k(M,N)$ can be defined as the equivalence class of maps $f:M\to N$ whose $k$th Taylor expansions agree at a point $x\in M$. A more "synthetic" way to think of $u$ is as a map from the $k$th infinitesimal nbhd of $x\in M$ to $N$. That's the approach Kock takes in his book *[Synthetic Geometry of Manifolds](http://home.math.au.dk/kock/SGM-final.pdf)* (section 2.7). 


The theme of the book is in fact the $k$th neighbourhood relation which Kock considers mainly for smooth finite dimensional manifolds. He partly addresses the question of how to generalise the infinitesimal nbhd relation to arbitrary objects (p.41). The issue there is, that there are at least two generalisations.


A more general approach to treat [infinitesimal neighbourhoods](https://ncatlab.org/nlab/show/infinitesimal+neighborhood) might be using Urs Schreibers [differential cohesion](https://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos).


Concerning your definition in the question: notice that a jet (as defined above) from $R^n$ to $R^m$ can be composed on the left with a map from $D\_k(n)$ to give rise to the objects you consider. But your objects are more general since they don't have to be invariant under the action of the automorphism group of $D\_k(n)$, while jets will be.


