Cohomology
Definition
Cohomology is the algebraic dual of homology. Where chain groups map downward via boundary operators, cochain groups map upward via coboundary operators. The CochainGroup C^k is the dual of the chain group C_k, consisting of linear functionals on k-chains.
The coboundary operator δ^k : C^k → C^{k+1} satisfies δ² = 0, dual to
the boundary property ∂² = 0. This is encoded as the identity
coboundarySquaredZero.
The cohomology group H^k = ker(δ^k) / im(δ^{k-1}) measures obstructions to extending local data to global data.
Sheaf Theory
A Sheaf assigns algebraic data to each open set of the constraint topology, subject to restriction and gluing axioms. The key components are:
| Class | Role |
|---|---|
| Stalk | Local data at a single point (constraint) |
| Section | Compatible assignment of data over an open set |
| GluingObstruction | Failure of local sections to assemble globally |
A local section is a section defined over a small open neighborhood. The restriction maps ensure that sections agree on overlaps.
De Rham Duality
The identity deRhamDuality establishes a natural pairing between homology
and cohomology:
H_k × H^k → R
This pairing is non-degenerate: every homology class can be detected by
evaluating a cohomology class on it. The identity sheafCohomologyBridge
connects sheaf cohomology to the abstract cochain definition.
Local-to-Global Principle
The identity localGlobalPrinciple captures when local sections can be glued
into global sections. Specifically:
- If H^1 = 0 (no gluing obstructions), every compatible family of local sections extends to a unique global section.
- If H^1 is nontrivial, the GluingObstruction classes in H^1 classify the distinct ways that gluing can fail.
This principle is the cohomological foundation of the resolution pipeline: local constraint satisfaction does not always imply global resolution. See Sheaf Semantics for worked examples of gluing obstructions and their resolution-theoretic interpretation.