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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
One source of cartesian differential categories is tangent bundle categories: in such a category, the full subcategory of objects with a trivial (in the appropriate sense) tangent bundle is a cartesian differential category.
We present the revised definition of a cartesian differential category due to Geoff Cruttwell, see below.
The adjective “cartesian” refers to the existence of finite products.
A prototypical example of a carteisan differential category is the category whose objects are open subsets of finite-dimensional real vector (or affine) spaces and morphisms are smooth maps. Below, if $U\subset R^n$ is such an object, then $L_0(U)=R^n$ is the tangent space at any point of $U$.
A (generalized) cartesian differential category is a category $C$ with finite products equipped with the following data:
for any object $X\in C$ a commutative monoid
(the tangent space at any point) such that $L(L_0(X))=L(X)$ and $L(X\times Y)=L(X)\times L(Y)$;
for any morphism $f\colon X\to Y$ in $C$, a morphism $D(f):L_0(X)\times X \to L_0(Y)$ (the derivative of $f$) such that
$D(+_X)=+_X\circ p_1$, $D(0_X)=0_X\circ p_1$ (where $p_1$ projects onto the first factor of a product);
(the derivative is a linear map)
and
(where $a,b,c\colon A\to L_0(X)$ are morphisms in $C$);
Rick Blute, Robin Cockett, and R.A.G. Seely, Cartesian differential categories, Theory and Applications of Categories,22, pg. 622–672, 2008 (tac:22-23)
Geoff Cruttwell, Cartesian differential categories revisited, Mathematical Structures in Computer Science, Vol. 27 (1), pg. 70-91 (first published online in 2015), doi:10.1007/s10485-019-09572-y
Last revised on July 12, 2021 at 10:07:10. See the history of this page for a list of all contributions to it.