integration axiom in nLab

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,b∈Ra, b\in R write

[a,b]:={x∈R|a≤x≤b} [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:

∀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 \,.