nLab
integration axiom

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

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

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

[a,b]:={xR|axb} [a,b] := \{x \in R | a \leq x \leq b\}

We say that (𝒯,(R,+,,))(\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:

fR [a,b]:! a fR [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 July 31, 2016 20:28:18 by Anonymous Coward (68.2.242.233)