integration axiom

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 July 31, 2016 20:28:18
by Anonymous Coward
(68.2.242.233)