Sectoral Topology (Recognition Topology)

Sectoral Topology (VII.L36) is the architectural lemma that the recognition Grothendieck topology J_rec is a valid topology generated by recognition morphisms, inheriting pullback stability from the kernel topology J_τ. J_rec equips the four-sector decomposition (S_E, S_P, S_D, S_C) and the Logos sector S_L with a sheaf-theoretic structure: covers are recognition events, sections glue across registers, and J-closure delivers the sectoral architecture.

Categorical invariant. J_rec is a valid Grothendieck topology generated by recognition morphisms; it inherits pullback stability from J_τ. Covers correspond to recognition events; sections glue across registers; J_rec-closure of subobjects of the sector-indexed presheaf yields the structural sectoral architecture.

Primary registry anchor: VII.L36

Supporting items: VII.D11, VII.T03, VII.D76

1. I.K0 — Universe Postulate
2. VII.T03 — Sector Decomposition at E₃ — five sectors S_E/S_P/S_D/S_C and S_L
3. VII.D76 — Social Ontology — recognition morphisms generating cover relations
4. VII.L36 — Recognition Topology Generation — J_rec is a valid Grothendieck topology

Lean modules referenced: TauLib.BookVII.Social.Ontology

Sectoral Topology is instantiated whenever a content's admissibility is checked by gluing local recognition data across registers. Examples: a scientific claim accepted because it survives empirical replication AND logical proof AND community endorsement (sections glued via J_rec); a moral judgement that holds because empirical-, normative-, proof-, and commitment-readings all cohere on a J_rec-cover.

Examples:

• Scientific consensus formation: a claim holds when sections glue across labs (Reg_E covers) and theorists (Reg_D covers)
• Moral judgement coherence: a verdict holds when empirical evidence, normative warrant, proof structure, and stance all glue on a J_rec-cover
• Recognition events as covers: a peer-review acceptance, a court ruling, a community endorsement — each generates a J_rec-cover
• Sheaf-failure as fragmentation: when sections do not glue (witnesses contradict across registers), the proposition has no sectoral home

Register codomain: All five sectors (S_E, S_P, S_D, S_C, S_L) — sectoral topology is the architectural glue that lets sheaf-theoretic gluing operate across the entire register architecture

Manuscript reference: manuscript-sources/book-07/part08/ch94.tex

Status: Formalized

Module: TauLib.BookVII.Social.Ontology

Lean kind: theorem

Lean symbol: Tau.BookVII.Social.Ontology.recognition_topology_generation

Related glossary entries

• MG-A10-logos-sector-closure Logos Sector Closure
• MG-A01-ci-operator-graph CI Operator Graph
• MG-R01-empirical-register Empirical Register (Reg_E)

Referenced by

• MG-A10-logos-sector-closure Logos Sector Closure