nLab
cartesian differential category

Contents

Context

Category theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 UR nU\subset R^n is such an object, then L 0(U)=R nL_0(U)=R^n is the tangent space at any point of UU.

Definition

A (generalized) cartesian differential category is a category CC with finite products equipped with the following data:

  • for any object XCX\in C a commutative monoid

    L(X)=(L 0(X),+ X:L 0(X)×L 0(X)L 0(X),0 X:1L 0(X))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)L(L_0(X))=L(X) and L(X×Y)=L(X)×L(Y)L(X\times Y)=L(X)\times L(Y);

  • for any morphism f:XYf\colon X\to Y in CC, a morphism D(f):L 0(X)×XL 0(Y)D(f):L_0(X)\times X \to L_0(Y) (the derivative of ff) such that

    • D(+ X)=+ Xp 1D(+_X)=+_X\circ p_1, D(0 X)=0 Xp 1D(0_X)=0_X\circ p_1 (where p 1p_1 projects onto the first factor of a product);

    • (the derivative is a linear map)

D(f)(a+ Xb,c)=D(f)(a,c)+ XD(f)(b,c)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 XD(f)(0_X,a)=0_X

(where a,b,c:AL 0(X)a,b,c\colon A\to L_0(X) are morphisms in CC);

  • (to be finished…)

References

Last revised on July 12, 2021 at 10:07:10. See the history of this page for a list of all contributions to it.