nLab integration axiom

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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

Last revised on December 14, 2022 at 23:19:22. See the history of this page for a list of all contributions to it.