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

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

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

We say that $(𝒯,(R,+,\cdot ,\le ))$ 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:

$$\forall f\in R^{[a,b]}:\exists !{\int }_a^{-}f\in R^{[a,b]}:({\int }_a^{-}f)(a)=0\wedge ({\int }_a^{-}f)\prime =f.$$\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, Synthtic differential geometry (pdf)

appendix 3 of

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