jet comonad (changes) in nLab

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Idea

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

Jet X≔i *i * Jet_X \coloneqq i^\ast i_\ast

for base change along the XX-component of the unit of ℑ\Im

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

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

{T ∞X ⟶p 1 X ↓ p 2 (pb) ↓ i X X ⟶i X ℑX}⇒((p 2) *(p 1) *≃(i X) *(i X) *). \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 ∞XT^\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 XX. The co-Kleisli category of the jet comonad has the interpretation as being the category of bundles over XX with differential operators between them as morphisms (Marvan 86, Marvan 89).

• 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 VV 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 𝒟\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 ,

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

  Formalizing Cartan Geometry in Modal Homotopy Type Theory , 2017

• Felix Cherubini: Synthetic GG-jet-structures in modal homotopy type theory, Mathematical Structures in Computer Science (2024) 1–35 [[doi:10.1017/S0960129524000355](https://doi.org/10.1017/S0960129524000355), arXiv:1806.05966]

For more references see at jet bundle.