Module 11: Sheaves and Local-to-Global Reasoning
A sheaf is a structure that guarantees local consistency implies global consistency. This module formalizes the sheaf condition, develops sheaf cohomology as an algebraic obstruction detector, and connects the theory to the spin glass problem: frustrated systems are precisely those with nonzero first cohomology H¹. The module also develops symmetric delta lenses as the engineering implementation of sheaf-based contradiction detection, providing a principled way to synchronize heterogeneous models and surface structural inconsistencies rather than silently resolving them.
Learning Objectives
- Define presheaves as contravariant functors and give three concrete examples.
- State the sheaf gluing condition and explain why it guarantees local-to-global consistency.
- Compute Čech cohomology for a simple open cover of a topological space.
- Interpret H¹(U, F) ≠ 0 as an algebraic obstruction to the existence of a global section.
- Connect spin glass frustration to nonzero H¹: locally consistent spin assignments that cannot be globally extended.
- Implement a symmetric lens between two heterogeneous models and demonstrate contradiction detection.
Materials
Key Concepts
- Presheaf: A contravariant functor F: C^op → Set; assigns data to each object (open set) and restriction maps to each morphism; does not yet require consistency on overlaps.
- Sheaf Condition: A presheaf F is a sheaf if local sections on overlapping patches that agree on intersections glue uniquely to a global section; locality + gluing axioms.
- Čech Cohomology: Cohomology computed from a cover {U_i}; H⁰ is the set of global sections, H¹ measures the obstruction to gluing locally consistent data globally.
- Sheaf Cohomology H¹(U, F): The first cohomology group of a sheaf; zero if and only if every locally consistent collection of sections extends to a global section; nonzero signals frustration.
- Frustration: In a spin glass, locally consistent spin configurations that cannot be extended globally due to conflicting constraints; the physical realization of H¹ ≠ 0.
- Symmetric Lens: A bidirectional data transformation (get, put) satisfying coherence laws in both directions; surfaces contradictions between two models rather than silently picking one.
Central Concepts from Prerequisites
- Modules 2–7: categories, functors, adjunctions, data migration, lenses
- Module 10: topology, homology (direct prerequisite)
- Abstract algebra: group cohomology helpful for intuition