This is a bridge-space namespace in the Resolve stage of the PRISM pipeline. It provides the resolution infrastructure — queries, partitions, observables, proofs, derivations, and traces that transform inputs into certified results.
A cochain complex: a sequence of cochain groups connected by coboundary operators.
CohomologyGroup
Thing
The k-th cohomology group H^k = ker(δ^k) / im(δ^{k-1}). Measures k-dimensional obstructions.
Sheaf
Thing
A sheaf F over a simplicial complex: assigns data to each simplex with restriction maps.
Stalk
Thing
A stalk F_σ: the local data of a sheaf at a simplex σ.
Section
Thing
A global section of a sheaf: a consistent choice of local data across all simplices.
LocalSection
Section
A local section: a consistent choice of data over a subcomplex.
RestrictionMap
Thing
A restriction map ρ_{σ,τ}: maps data from a simplex to a face.
GluingObstruction
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
CochainGroup
integer
The degree k of this cochain group C^k.
cochainRank
Datatype
true
CochainGroup
nonNegativeInteger
The rank (dimension) of this cochain group.
dualOf
Object
true
CochainGroup
ChainGroup
The chain group that this cochain group is dual to.
coboundarySource
Object
true
CoboundaryOperator
CochainGroup
The source cochain group of this coboundary operator.
coboundaryTarget
Object
true
CoboundaryOperator
CochainGroup
The target cochain group of this coboundary operator.
satisfiesCoboundarySquaredZero
Datatype
true
CoboundaryOperator
boolean
Whether this coboundary operator satisfies δ² = 0.
hasCochainGroup
Object
false
CochainComplex
CochainGroup
A cochain group belonging to this cochain complex.
hasCoboundary
Object
false
CochainComplex
CoboundaryOperator
A coboundary operator belonging to this cochain complex.
cohomologyDegree
Datatype
true
CohomologyGroup
integer
The degree k of this cohomology group H^k.
cohomologyRank
Datatype
true
CohomologyGroup
nonNegativeInteger
The rank (dimension) of this cohomology group.
sheafOver
Object
true
Sheaf
SimplicialComplex
The simplicial complex that this sheaf is defined over.
coefficientIn
Object
true
Sheaf
Ring
The coefficient ring of this sheaf.
hasStalks
Object
false
Sheaf
Stalk
A stalk belonging to this sheaf.
stalkAt
Object
true
Stalk
Simplex
The simplex at which this stalk is located.
restrictsFrom
Object
true
RestrictionMap
Simplex
The source simplex of this restriction map.
restrictsTo
Object
true
RestrictionMap
Simplex
The target simplex (face) of this restriction map.
hasGlobalSection
Object
false
Sheaf
Section
A global section of this sheaf.
obstructionClass
Object
true
GluingObstruction
CohomologyGroup
The cohomology class that this gluing obstruction represents.
addressesSuggestion
Object
false
GluingObstruction
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
ResolutionState
Sheaf
The sheaf analysis associated with a resolution state.
Named Individuals
Name
Type
Comment
coboundarySquaredZero
Identity
δ² = 0: the coboundary of a coboundary is zero.
lhs: term_coboundarySquaredZero_lhs
rhs: term_coboundarySquaredZero_rhs
forAll: term_coboundarySquaredZero_forAll
verificationDomain: Topological
deRhamDuality
Identity
Discrete de Rham duality: H^k ≅ Hom(H_k, R).
lhs: term_deRhamDuality_lhs
rhs: term_deRhamDuality_rhs
forAll: term_deRhamDuality_forAll
verificationDomain: Topological
sheafCohomologyBridge
Identity
Sheaf cohomology equals simplicial cohomology for constant sheaves.
lhs: term_sheafCohomologyBridge_lhs
rhs: term_sheafCohomologyBridge_rhs
forAll: term_sheafCohomologyBridge_forAll
verificationDomain: Topological
localGlobalPrinciple
Identity
Local-global principle: H^1(K; F) = 0 implies all local sections glue to global sections.