# nLab integration axiom

differential geometry

synthetic differential geometry

# Idea

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.

# Definition

###### Definition

(integration axiom)

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

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

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

We say that $\left(𝒯,\left(R,+,\cdot ,\le \right)\right)$ 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:

$\forall f\in {R}^{\left[a,b\right]}:\exists !{\int }_{a}^{-}f\in {R}^{\left[a,b\right]}:\left({\int }_{a}^{-}f\right)\left(a\right)=0\wedge \left({\int }_{a}^{-}f\right)\prime =f\phantom{\rule{thinmathspace}{0ex}}.$\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 …

# Examples

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

# References

page 49 of

appendix 3 of

Revised on November 24, 2009 16:16:16 by david karapetyan? (204.140.153.10)