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integration axiom

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differential geometry

synthetic differential geometry

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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,≤) compatible with its ring structure (R,+,⋅) in the obvious way.

Then for any a,b∈R write

[a,b]:={x∈R∣a≤x≤b}[a,b] := \{x \in R | a \leq x \leq b\}

We say that (𝒯,(R,+,⋅,≤)) 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:

∀f∈R [a,b]:∃!∫ a −f∈R [a,b]:(∫ a −f)(a)=0∧(∫ a −f)′=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

appendix 3 of

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