Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

category theory

Applications

Algebra

higher algebra

universal algebra

Contents

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.)

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

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).

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.)

Discussion in synthetic differential geometry is in

• 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, above prop. 0.9 in Notes on crystals and algebraic D-modules (pdf), 2009

Discussion in differential cohesion is in

Discussion in differentially cohesive homotopy type theory is in

For more references see at jet bundle.