Chapter 37: Sheaf Coherence from ω-Germ Compatibility

A presheaf assigns data to open sets and restricts it to smaller ones. A sheaf GunningRossi1965,Hormander1990 is a presheaf with two additional properties: local data that agrees on overlaps can be glued into global data (the gluing axiom), and a global section that vanishes locally vanishes globally (the locality axiom). Classical complex analysis proves the sheaf property for holomorphic functions via analytic continuation and the identity theorem. In Category τ, the presheaf of holomorphic functions on τ³ is defined using cylinder domains (II.D10) as the “open sets,” and the sheaf axioms are verified from ω-germ stagewise compatibility and the ultrametric topology. the relevant theorem (II.T32): the presheaf O_τ satisfies both sheaf axioms, with locality following from ultrametric separation and gluing from tower coherence. The **Gluing Lemma (II.L06) provides the explicit construction. The ultrametric topology makes gluing easier than in classical analysis: cylinder domains are clopen, overlaps are clean, and no partition of unity is needed.