nLab cartesian differential category

Idea

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.

Definition

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 $U$.

Definition

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

  $$L(X)=(L_0(X),+_X\colon L_0(X)\times L_0(X)\to L_0(X),0_X\colon 1\to L_0(X))$$

  (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)

$$D(f)\circ (a +_X b,c)=D(f)\circ (a,c) +_X D(f)\circ (b,c)$$

and

$$D(f)(0_X,a)=0_X$$

(where $a,b,c\colon A\to L_0(X)$ are morphisms in $C$);

• (to be finished…)

References

• 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

Related concepts

• differential category