synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
symmetric monoidal (∞,1)-category of spectra
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
for base change along the $X$-component of the unit of $\Im$
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.)
$T^\infty X$ is the infinitesimal disk bundle.
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 Borger's absolute geometry a similar base change as above is interpreted as the arithmetic jet space construction.
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
(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:
Discussion in differential cohesion:
Discussion in differentially cohesive modal homotopy type theory:
For more references see at jet bundle.
Last revised on March 28, 2023 at 10:38:26. See the history of this page for a list of all contributions to it.