# 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 $(\mathcal{T}, R)$ 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 …

# 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 30, 2013 21:13:40 by Tim Porter (2.26.41.83)