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I am about to write something at jet bundle and elsewhere about the general abstract perspective.
In chapter 2 of Beilinson-Drinfeld’s Chiral Algebras they have the nice characterization of the Jet bundle functor as the right adjoint to the forgetful functor $F : Scheme_{\mathcal{D}}(X) \to Scheme(X)$ from D-schemes over $X$ to just schemes over $X$.
Now, since D-modules on $X$ are quasicoherent modules on the de Rham space $\Pi_{inf}(X)$, I guess we can identify
$Scheme_{\mathcal{D}}(X)$with
$Schemes/\Pi_{inf}(X)$and hence the forgetful functor above is the pullback functor
$\array{ F(E) &\to& E \\ \downarrow && \downarrow \\ X &\to& \Pi_{inf}(X) }$aling the lower canonical morphism (“constant infinitesimal path inclusion”).
This would mean that we have the following nice general abstract characterization of jet bundles:
let $\mathbf{H}$ be a cohesive (infinity,1)-topos equipped with infinitesimal cohesion $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$. For any $X \in \mathbf{X}$ we then have the canonical morphism $i : X \to \mathbf{\Pi}_{inf}(X)$.
The Jet bundle functor is then simply the corresponding base change geometric morphism
$Jet := (i^* \dashv i_*) : \mathbf{H}/X \to \mathbf{H}/\mathbf{\Pi}(X)$or rather, if we forget the $\mathcal{D}$-module structure on the coherent sheaves on the jet bundle, it is the comonad $i^* i_*$ induced by that.
Does that way of saying it ring a bell with anyone?
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