nLab
integration axiom

**differential geometry** **synthetic differential geometry** ## Axiomatics ## * smooth topos * infinitesimal space * amazing right adjoint * Kock-Lawvere axiom * integration axiom * microlinear space * synthetic differential supergeometry * super smooth topos * . infinitesimal cohesion * de Rham space * formally smooth morphism, formally unramified morphism, formally étale morphism * jet bundle ## Models ## * Models for Smooth Infinitesimal Analysis * Fermat theory * smooth algebra ($C^\infty$-ring) * smooth locus * smooth manifold, formal smooth manifold, derived smooth manifold * smooth space, diffeological space, Frölicher space * smooth natural numbers * Cahiers topos * smooth ∞-groupoid * synthetic differential ∞-groupoid ## Concepts * tangent bundle, * vector field, tangent Lie algebroid; * differentiation, chain rule * differential forms * differential equation, variational calculus * Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space * connection on a bundle, connection on an ∞-bundle * Riemannian manifold * isometry, Killing vector field ## Theorems * Hadamard lemma * Borel's theorem * Boman's theorem * Whitney extension theorem * Steenrod-Wockel approximation theorem * Poincare lemma * Stokes theorem * de Rham theorem * Chern-Weil theory ## Applications * Lie theory, ∞-Lie theory * Chern-Weil theory, ∞-Chern-Weil theory * gauge theory * ∞-Chern-Simons theory * Klein geometry, Klein 2-geometry, higher Klein geometry * Euclidean geometry, Cartan geometry, higher Cartan geometry * Riemannian geometry * gravity, supergravity

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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]:={xRaxb} [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 November 30, 2013 21:13:40 by Tim Porter (2.26.41.83)