6 - Internal logic of a topos

Summary

When in Riemannian geometry we prove “the existence theorem for geodesics” or “the existence theorem for geodesic coordinates”, the word “existence” does not have its ordinary meaning: “existence” means here “existence on a neighborhood of each point”, which does not necessarily imply “global existence”. Analogous observations can be made in analysis for the “existence theorem for the solutions of a differential equation”, and so on. In those situations, mathematics itself imposes a local character on the various results we want to prove.

As proved in 2.5.7, every sheaf F on a topological space X is a sheaf of continuous sections of some étale map p: Y → X. This topological situation suggests that, as in geometry or analysis, we should pay special attention to those properties of the sheaf F “which hold on a neighborhood of each point x ∈ X”. This is precisely the spirit of the “internal logic of sheaves”.

As an example, let us go back to section 2.11. Given a ring R, we constructed an étale map p: Y → X, with X the spectrum of the ring, so that the ring R is isomorphic to the ring of global continuous sections of p. In classical logic, given an element r ∈ R, r either is or is not invertible. But considering the continuous global section σ: X → Y corresponding to r…, σ can be globally invertible or locally invertible (i.e., invertible on a neighborhood of each point).