As I understand it from Kostecki's notes, 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?
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4$\begingroup$ The definition you gave makes sense for arbitrary objects in a topos where $D_k(n)$ exists. Maybe it is not well behaved for your purposes? Also a small remark: the thing you defined as $J^k(R^n,R^m)$ is not what people usually denote with that symbol, but rather the space of (non-holonomic) sections of the space of jets. Also, have you seen the nice book by Anders Kock "Synthetic Geometry of Manifolds"? It also talks about jets in the synthetic context home.math.au.dk/kock/SGM-final.pdf $\endgroup$– Michael BächtoldCommented Dec 30, 2015 at 14:23
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2$\begingroup$ Upon further thought, your definition of $J^k(R^n,R^m)$ is not even the space of sections of non holonomic jets, but something more complicated, and your definition of $j^kf $ is also not what people usually mean by that. I'll write an answer if I have time. But Kock's book should contain an answer. $\endgroup$– Michael BächtoldCommented Dec 30, 2015 at 20:11
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$\begingroup$ I thought that $J^k(R^n, R^m)$ usually denotes the equivalence classes of maps $R^n \to R^m$ which agree up to their $k$th order Taylor expansion? $\endgroup$– ಠ_ಠCommented Dec 31, 2015 at 0:39
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$\begingroup$ What precisely is $R$ here? $\endgroup$– Dmitri ZaitsevCommented Nov 16, 2017 at 11:09
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$\begingroup$ @DmitriZaitsev $R$ is the line object in some smooth topos, which plays the role of the real number line in SDG. $\endgroup$– ಠ_ಠCommented Nov 16, 2017 at 11:14
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1 Answer
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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 (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 might be using Urs Schreibers differential cohesion.
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.
answered Dec 31, 2015 at 9:53