nLab jet comonad

Idea

In a context $\mathbf{H}$ of differential cohesion with $\Im$ the infinitesimal shape modality, then for any object $X\in \mathbf{H}$ the base change comonad

$$Jet_X \coloneqq i^\ast i_\ast$$

for base change along the $X$-component of the unit of $\Im$

$$\mathbf{H}_{/X} \stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}} \mathbf{H}_{/\Im(X)} \,,$$

may be interpreted as sending any bundle over $X$ to its jet bundle.

This characterization via base change is more or less implicit in (Kock 10, remark 7.3.1) (to translate from the pull-push $(p_2)_\ast p_1^\ast$ shown there, as in (Deligne 70) use that in a topos the epimorphism $X \to \Im X$ is effective and then use the Beck-Chevalley condition to get the push-pull shown above.)

$$\left\{ \array{ T^\infty X &\stackrel{p_1}{\longrightarrow}& X \\ \downarrow^{\mathrlap{p_2}} &(pb)& \downarrow^{\mathrlap{i_X}} \\ X &\stackrel{i_X}{\longrightarrow}& \Im X } \right\} \;\; \Rightarrow \;\; ((p_2)_\ast (p_1)^\ast \simeq (i_X)^\ast (i_X)_\ast) \,.$$

$T^\infty X$ is the infinitesimal disk bundle.

Properties

The Eilenberg-Moore category of coalgebras over the jet comonad has the interpretation of the category of partial differential equations with variables in $X$. The co-Kleisli category of the jet comonad has the interpretation as being the category of bundles over $X$ with differential operators between them as morphisms (Marvan 86, Marvan 89).

Related concepts

• topos of coalgebras over a comonad

• in Borger's absolute geometry a similar base change as above is interpreted as the arithmetic jet space construction.

References

In the context of differential geometry the comonad structure on the jet bundle construction, as well as the interpretation of its EM-category as that of partial differential equations, is due to

• Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)

(Proposition 1.4 in Marvan 86 needs an extra “weakened transversality” condition on the equalizer, this is fixed in (Theorem 1.3, Marvan’s thesis). The extra condition is that the equalizer must remain an equalizer after an application of the $V$ functor, which maps fibered manifolds to their vertical tangent bundles.)

• Michal Marvan, thesis, 1989 (pdf, web)

• Michal Marvan, On the horizontal cohomology with general coefficients, 1989 (web announcement, web archive)

• Michal Marvan, section 1.1 of On Zero-Curvature Representations of Partial Differential Equations, (1993) (web)

Discussion in synthetic differential geometry:

• Pierre Deligne, Equations Différentielles à Points Singuliers Réguliers, 1970

• Anders Kock, remark 7.3.1 Synthetic geometry of manifolds, Cambridge Tracts in Mathematics 180 (2010). (pdf)

In the context of algebraic geometry and D-geometry the comonad structure is observed in:

• Jacob Lurie, Notes on crystals and algebraic $\mathcal{D}$-modules (2010) [pdf]

Discussion in differential cohesion:

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)

Discussion in differentially cohesive modal homotopy type theory:

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017

For more references see at jet bundle.