UOR Cohomology

IRI: https://uor.foundation/cohomology/
Prefix: cohomology:
Space: bridge
Comment: Cochain complexes, sheaf cohomology, and local-to-global obstruction detection.

Imports

https://uor.foundation/schema/
https://uor.foundation/homology/
https://uor.foundation/resolver/
https://uor.foundation/observable/
https://uor.foundation/partition/

Classes

Name	IRI	Subclass Of	Comment
CochainGroup	`https://uor.foundation/cohomology/CochainGroup`	`http://www.w3.org/2002/07/owl#Thing`	A cochain group: the dual of a chain group, maps chains to coefficients.
CoboundaryOperator	`https://uor.foundation/cohomology/CoboundaryOperator`	`http://www.w3.org/2002/07/owl#Thing`	The coboundary operator δ^k: C^k → C^{k+1}. Satisfies δ² = 0.
CochainComplex	`https://uor.foundation/cohomology/CochainComplex`	`http://www.w3.org/2002/07/owl#Thing`	A cochain complex: a sequence of cochain groups connected by coboundary operators.
CohomologyGroup	`https://uor.foundation/cohomology/CohomologyGroup`	`http://www.w3.org/2002/07/owl#Thing`	The k-th cohomology group H^k = ker(δ^k) / im(δ^{k-1}). Measures k-dimensional obstructions.
Sheaf	`https://uor.foundation/cohomology/Sheaf`	`http://www.w3.org/2002/07/owl#Thing`	A sheaf F over a simplicial complex: assigns data to each simplex with restriction maps.
Stalk	`https://uor.foundation/cohomology/Stalk`	`http://www.w3.org/2002/07/owl#Thing`	A stalk F_σ: the local data of a sheaf at a simplex σ.
Section	`https://uor.foundation/cohomology/Section`	`http://www.w3.org/2002/07/owl#Thing`	A global section of a sheaf: a consistent choice of local data across all simplices.
LocalSection	`https://uor.foundation/cohomology/LocalSection`	`https://uor.foundation/cohomology/Section`	A local section: a consistent choice of data over a subcomplex.
RestrictionMap	`https://uor.foundation/cohomology/RestrictionMap`	`http://www.w3.org/2002/07/owl#Thing`	A restriction map ρ_{σ,τ}: maps data from a simplex to a face.
GluingObstruction	`https://uor.foundation/cohomology/GluingObstruction`	`http://www.w3.org/2002/07/owl#Thing`	A gluing obstruction: a cohomology class that detects when local sections fail to glue.

Properties

Name	Kind	Functional	Domain	Range	Comment
cochainDegree	Datatype	true	`https://uor.foundation/cohomology/CochainGroup`	`http://www.w3.org/2001/XMLSchema#integer`	The degree k of this cochain group C^k.
cochainRank	Datatype	true	`https://uor.foundation/cohomology/CochainGroup`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The rank (dimension) of this cochain group.
dualOf	Object	true	`https://uor.foundation/cohomology/CochainGroup`	`https://uor.foundation/homology/ChainGroup`	The chain group that this cochain group is dual to.
coboundarySource	Object	true	`https://uor.foundation/cohomology/CoboundaryOperator`	`https://uor.foundation/cohomology/CochainGroup`	The source cochain group of this coboundary operator.
coboundaryTarget	Object	true	`https://uor.foundation/cohomology/CoboundaryOperator`	`https://uor.foundation/cohomology/CochainGroup`	The target cochain group of this coboundary operator.
satisfiesCoboundarySquaredZero	Datatype	true	`https://uor.foundation/cohomology/CoboundaryOperator`	`http://www.w3.org/2001/XMLSchema#boolean`	Whether this coboundary operator satisfies δ² = 0.
hasCochainGroup	Object	false	`https://uor.foundation/cohomology/CochainComplex`	`https://uor.foundation/cohomology/CochainGroup`	A cochain group belonging to this cochain complex.
hasCoboundary	Object	false	`https://uor.foundation/cohomology/CochainComplex`	`https://uor.foundation/cohomology/CoboundaryOperator`	A coboundary operator belonging to this cochain complex.
cohomologyDegree	Datatype	true	`https://uor.foundation/cohomology/CohomologyGroup`	`http://www.w3.org/2001/XMLSchema#integer`	The degree k of this cohomology group H^k.
cohomologyRank	Datatype	true	`https://uor.foundation/cohomology/CohomologyGroup`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The rank (dimension) of this cohomology group.
sheafOver	Object	true	`https://uor.foundation/cohomology/Sheaf`	`https://uor.foundation/homology/SimplicialComplex`	The simplicial complex that this sheaf is defined over.
coefficientIn	Object	true	`https://uor.foundation/cohomology/Sheaf`	`https://uor.foundation/schema/Ring`	The coefficient ring of this sheaf.
hasStalks	Object	false	`https://uor.foundation/cohomology/Sheaf`	`https://uor.foundation/cohomology/Stalk`	A stalk belonging to this sheaf.
stalkAt	Object	true	`https://uor.foundation/cohomology/Stalk`	`https://uor.foundation/homology/Simplex`	The simplex at which this stalk is located.
restrictsFrom	Object	true	`https://uor.foundation/cohomology/RestrictionMap`	`https://uor.foundation/homology/Simplex`	The source simplex of this restriction map.
restrictsTo	Object	true	`https://uor.foundation/cohomology/RestrictionMap`	`https://uor.foundation/homology/Simplex`	The target simplex (face) of this restriction map.
hasGlobalSection	Object	false	`https://uor.foundation/cohomology/Sheaf`	`https://uor.foundation/cohomology/Section`	A global section of this sheaf.
obstructionClass	Object	true	`https://uor.foundation/cohomology/GluingObstruction`	`https://uor.foundation/cohomology/CohomologyGroup`	The cohomology class that this gluing obstruction represents.
addressesSuggestion	Object	false	`https://uor.foundation/cohomology/GluingObstruction`	`https://uor.foundation/resolver/RefinementSuggestion`	The refinement suggestion that, if applied, would resolve this gluing obstruction. Computed by the kernel when ψ₆ detects H^1 ≠ 0: the obstruction class indexes the site pair that is incompatible, and the suggestion targets that pair with a new bridging constraint.
sheafAnalysis	Object	true	`https://uor.foundation/resolver/ResolutionState`	`https://uor.foundation/cohomology/Sheaf`	The sheaf analysis associated with a resolution state.

Named Individuals

Name	Type	Properties	Comment
coboundarySquaredZero	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_coboundarySquaredZero_lhs` `rhs`: `https://uor.foundation/schema/term_coboundarySquaredZero_rhs` `forAll`: `https://uor.foundation/schema/term_coboundarySquaredZero_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	δ² = 0: the coboundary of a coboundary is zero.
deRhamDuality	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_deRhamDuality_lhs` `rhs`: `https://uor.foundation/schema/term_deRhamDuality_rhs` `forAll`: `https://uor.foundation/schema/term_deRhamDuality_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	Discrete de Rham duality: H^k ≅ Hom(H_k, R).
sheafCohomologyBridge	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_sheafCohomologyBridge_lhs` `rhs`: `https://uor.foundation/schema/term_sheafCohomologyBridge_rhs` `forAll`: `https://uor.foundation/schema/term_sheafCohomologyBridge_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	Sheaf cohomology equals simplicial cohomology for constant sheaves.
localGlobalPrinciple	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_localGlobalPrinciple_lhs` `rhs`: `https://uor.foundation/schema/term_localGlobalPrinciple_rhs` `forAll`: `https://uor.foundation/schema/term_localGlobalPrinciple_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	Local-global principle: H^1(K; F) = 0 implies all local sections glue to global sections.