nLab
diffiety

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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)

Variational calculus

Contents

Idea

The concept of diffiety (Vinogradov 81) reflects the concept of partial differential equation (generally non-linear) in analogy to how the concept of algebraic variety reflects that of polynomial equation:

A diffiety is the solution-locus J Σ (E)\mathcal{E}\hookrightarrow J^\infty_\Sigma(E) of a partial differential equation DΦ=0D \Phi = 0 regarded as an ordinary equation D˜=0\tilde D = 0 on the jet bundle J Σ (E)J^\infty_\Sigma(E) of some bundle EfbΣE \overset{fb}{\to} \Sigma.

Here Σ\Sigma is the space of free variables of the PDE, EfbΣE \overset{fb}{\to} \Sigma is the bundle of dependent variables, and a differential operator

D:Γ Σ(E)Γ Σ(F) D \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)

on the space of smooth sections of fbfb is represented by a bundle morphism

D˜:J Σ (E)F \tilde D \colon J^\infty_\Sigma(E) \to F

out of the jet bundle via jet prolongation j Σ :Γ Σ(E)Γ Σ(J Σ (E))j^\infty_\Sigma \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(J^\infty_\Sigma(E)) as DΦ=D˜(j Σ (Φ))D \Phi = \tilde D (j^\infty_\Sigma(\Phi)):

\,

diffiety jet bundle differential operator D˜=0 J Σ (E) D˜ F \array{ & \text{diffiety} && \text{jet bundle} & \array{\text{differential} \\ \text{operator}} & \\ & \mathcal{E} &\overset{\tilde D = 0}{\hookrightarrow}& J^\infty_\Sigma(E) &\overset{\tilde D}{\longrightarrow}& F }

For instance in Lagrangian field theory the bundle in question is a field bundle EfbΣE \overset{fb}{\to} \Sigma, the partial differential equation is the Euler-Lagrange equation δ ELL=0\delta_{EL}\mathbf{L} = 0, and its diffiety solution locus \mathcal{E} inside the jet bundle J Σ (E)J^\infty_\Sigma(E) is called the shell of the field theory.

In Marvan 86 it was observed that Vinogradov’s formally integrable diffieties are equivalently the coalgebras over the jet comonad acting on locally pro-manifold-bundles (over a base space Σ\Sigma of free variables). This statement generalizes to the synthetic differential geometry of the Cahiers topos H\mathbf{H} (Khavkine-Schreiber 17), where the jet comonad is realized as the comonad corresponding to base change along the de Rham shape projection Ση ΣΣ\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma. By comonadic descent this implies that over formally smooth base spaces Σ\Sigma formally integrable diffieties are equivalently the bundles over the de Rham shape Σ\Im \Sigma:

PDEs Σ(H)=Diffeties Σ(H)J Σ CoAlg(H /Σ)H /(Σ) PDEs_\Sigma(\mathbf{H}) \;=\; Diffeties_\Sigma(\mathbf{H}) \;\simeq\; J^\infty_\Sigma CoAlg\left(\mathbf{H}_{/\Sigma} \right) \;\simeq\; \mathbf{H}_{/\Im(\Sigma)}

(Khavkine-Schreiber 17, thorem 3.52, theorem 3.60)

This makes manifest how diffieties are the analog in differential geometry of concepts in algebraic geometry: For Σ\Sigma a suitable scheme then a quasicoherent module over its de Rham shape Σ\Im \Sigma (“crystal”) is called a D-module and represents an algebraic linear partial differential equation, while a relative scheme over Σ\Im \Sigma is called a D-scheme and represents a general algebraic partial differential equation. See also at D-geometry for more on this.

References

General

Review

See also

As jet coalgebras

Diffieties as coalgebras over the jet comonad are discussed in

Conferences

Last revised on December 11, 2017 at 06:26:30. See the history of this page for a list of all contributions to it.