# nLab diffiety

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

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Variational calculus

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 $\mathcal{E}\hookrightarrow J^\infty_\Sigma(E)$ of a partial differential equation $D \Phi = 0$ regarded as an ordinary equation $\tilde D = 0$ on the jet bundle $J^\infty_\Sigma(E)$ of some bundle $E \overset{fb}{\to} \Sigma$.

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

$D \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)$

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

$\tilde D \colon J^\infty_\Sigma(E) \to F$

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

$\,$

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

For instance in Lagrangian field theory the bundle in question is a field bundle $E \overset{fb}{\to} \Sigma$, the partial differential equation is the Euler-Lagrange equation $\delta_{EL}\mathbf{L} = 0$, and its diffiety solution locus $\mathcal{E}$ inside the jet bundle $J^\infty_\Sigma(E)$ 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 $\Sigma$ of free variables). This statement generalizes to the synthetic differential geometry of the Cahiers topos $\mathbf{H}$ (Khavkine-Schreiber 17), where the jet comonad is realized as the comonad corresponding to base change along the de Rham shape projection $\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma$. By comonadic descent this implies that over formally smooth base spaces $\Sigma$ formally integrable diffieties are equivalently the bundles over the de Rham shape $\Im \Sigma$:

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

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