integration axiom

**differential geometry**
**synthetic differential geometry**
## Axiomatics ##
* smooth topos
* infinitesimal space
* amazing right adjoint
* Kock-Lawvere axiom
* integration axiom
* microlinear space
* synthetic differential supergeometry
* super smooth topos
* . infinitesimal cohesion
* de Rham space
* formally smooth morphism, formally unramified morphism,
formally étale morphism
* jet bundle
## Models ##
* Models for Smooth Infinitesimal Analysis
* Fermat theory
* smooth algebra ($C^\infty$-ring)
* smooth locus
* smooth manifold, formal smooth manifold, derived smooth manifold
* smooth space, diffeological space, Frölicher space
* smooth natural numbers
* Cahiers topos
* smooth ∞-groupoid
* synthetic differential ∞-groupoid
## Concepts
* tangent bundle,
* vector field, tangent Lie algebroid;
* differentiation, chain rule
* differential forms
* differential equation, variational calculus
* Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space
* connection on a bundle, connection on an ∞-bundle
* Riemannian manifold
* isometry, Killing vector field
## Theorems
* Hadamard lemma
* Borel's theorem
* Boman's theorem
* Whitney extension theorem
* Steenrod-Wockel approximation theorem
* Poincare lemma
* Stokes theorem
* de Rham theorem
* Chern-Weil theory
## Applications
* Lie theory, ∞-Lie theory
* Chern-Weil theory, ∞-Chern-Weil theory
* gauge theory
* ∞-Chern-Simons theory
* Klein geometry, Klein 2-geometry, higher Klein geometry
* Euclidean geometry, Cartan geometry, higher Cartan geometry
* Riemannian geometry
* gravity, supergravity

In the axiomatic formulation of differential geometry given by synthetic differential geometry the standard Kock-Lawvere axiom provides a notion of differentiation. In general this need not come with an inverse operation of integration. The additional *integration axiom* on a smooth topos does ensure this.

**(integration axiom)**

Let $(\mathcal{T}, R)$ be a smooth topos and let the line object $R$ be equipped with the structure of a partial order $(R, \leq)$ compatible with its ring structure $(R, +, \cdot)$ in the obvious way.

Then for any $a, b\in R$ write

$[a,b] := \{x \in R | a \leq x \leq b\}$

We say that $(\mathcal{T},(R,+,\cdot,\leq))$ satisfies the **integration axiom** if for all such intervals, all functions on the interval arise uniquely as derivatives on functions on the interval that vanish at the left boundary:

$\forall f \in R^{[a,b]} : \exists ! \int_a^{-} f \in R^{[a,b]} :
(\int_a^{-} f)(a) = 0 \wedge (\int_a^{-} f)' = f
\,.$

… need to say more …

The axiom holds for all the smooth topos presented in MSIA, listed in appendix 2 there. See appendix 3 for the proof.

page 49 of

- Anders Kock,
*Synthetic differential geometry*(page 61 of pdf)

appendix 3 of

Revised on November 30, 2013 21:13:40
by Tim Porter
(2.26.41.83)