Why the Kock-Lawvere axiom holds, and also doesn’t, in the classifier of integral rigs

Abstract

The Kock-Lawvere axiom has two formulations that are equivalent in the usual models of Synthetic Differential Geometry. We show that, in the classifier of integral rigs, and some of its pre-cohesive subtoposes, the generic model satisfies one version of the axiom but not the other.