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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
A differential equation is an equation involving terms that are derivatives (or differentials). One sometimes distinguishes partial differential equations (which involve partial derivatives) from ordinary differential equations (which don't).
Analogous to how ordinary equations determine and are determined by their spaces of solutions – the corresponding schemes – accordingly differential equations $F(\partial^i x)$ on sections of a bundle $E \to X$ determine and are determined by their solution spaces, which are sub-D-schemes of the jet bundle D-scheme:
In fact there is an equivalence of categories between the Eilenberg-Moore category of the jet comonad over $X$ and the category of partial differential equations with variables in $X$ (Marvan 86).
At least in nice cases for differential equations on functions on $X$ this is equivalently modeled by sub-Lie algebroids of tangent Lie algebroids.
see exterior differential system for details
A perspective on differential equations from the nPOV of synthetic differential geometry is given in
William Lawvere, Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)
(on the description of differential equations in terms of synthetic differential geometry)
See also the appendix of
In a smooth topos with $X$ and $A$ any two objects and $D = \{x \in R | x^2 = 0\}$ the abstract tangent vector, a diagram, of the form
may be read as
$f$ is a smooth function on $X$ with values in $A$;
whose derivative along the tangent vector $v \in T_x X \subset T X = X^D$…
…is the tangent vector $v(f) \in T_{f(x)} A \subset T A = A^D$.
Accordingly, the diagram
may be read as encoding the differential equation $v(f)_x = \alpha$ (at one point) whose solutions $f \in A^X$ are the extensions that complete this diagram.
To get differential equations in the more common sense that they impose a condition on the derivative of $f$ at each single point of $X$ and varying smoothly with $X$, we think of $f : X \to A$ in terms of the exponential map
and consider
In this diagram now
$v : D \to X^X$ is the adjunct of a vector field $X \to T X = X^D$ on $X$;
(the commutativity of the top right triangle ensures that indeed this $X \to T X$ is a section of the tangent bundle projection $T X \to X$);
$\alpha : D \to A^X$ is the adjunct of a map $X \to T A = A^D$ that sends each point $x \in X$ to a tangent vector in $T_{f(x)} A$
(enforced by the commutativity of the left bottom triangle)
and the commutativity of the right lower triangle is the differential equation
General accounts include
Sergiu Klainerman, PDE as a unified subject 2000 (pdf)
Boris Kruglikov, Valentin Lychagin, Geometry of differential equations, pdf
Arthemy Kiselev, The twelve lectures in the (non)commutative geometry of differential equations, preprint IHES M/12/13 pdf
Yves Andre, Solution algebras of differential equations and quasi-homogeneous varieties, arXiv:1107.1179
Mikhail Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. x+363 pp.
Discussion of the spaces of solutions is in
Characterization of the category of partial differential equations as the Eilenberg-Moore category of coalgebras over the jet comonad is due to
In the language of D-modules and hence for the special case of linear differential equations, this appears as prop. 3.4.1.1 in
A domain specific programming language for differential equations is presented in
See also