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

infinitesimal 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 14:07:10. See the history of this page for a list of all contributions to it.