nLab integration axiom

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

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

appendix 3 of

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis