# nLab cartesian differential category

Contents

category theory

## Applications

#### 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)

# Contents

## 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…)