UOR Homology

IRI: https://uor.foundation/homology/
Prefix: homology:
Space: bridge
Comment: Simplicial complexes, chain complexes, boundary operators, and homology groups for structural reasoning.

Imports

https://uor.foundation/schema/
https://uor.foundation/type/
https://uor.foundation/resolver/
https://uor.foundation/partition/
https://uor.foundation/observable/
https://uor.foundation/op/

Classes

Name	IRI	Subclass Of	Comment
Simplex	`https://uor.foundation/homology/Simplex`	`http://www.w3.org/2002/07/owl#Thing`	A k-simplex: a finite set of k+1 vertices drawn from constraint objects.
SimplicialComplex	`https://uor.foundation/homology/SimplicialComplex`	`http://www.w3.org/2002/07/owl#Thing`	A simplicial complex: a set of simplices closed under taking faces.
FaceMap	`https://uor.foundation/homology/FaceMap`	`http://www.w3.org/2002/07/owl#Thing`	A face map d_i: removes vertex i from a simplex, producing a face.
ChainGroup	`https://uor.foundation/homology/ChainGroup`	`http://www.w3.org/2002/07/owl#Thing`	A free abelian group generated by k-simplices (the k-th chain group C_k).
BoundaryOperator	`https://uor.foundation/homology/BoundaryOperator`	`http://www.w3.org/2002/07/owl#Thing`	The boundary operator ∂_k: C_k → C_{k-1}. Satisfies ∂² = 0.
ChainComplex	`https://uor.foundation/homology/ChainComplex`	`http://www.w3.org/2002/07/owl#Thing`	A chain complex: a sequence of chain groups connected by boundary operators.
HomologyGroup	`https://uor.foundation/homology/HomologyGroup`	`http://www.w3.org/2002/07/owl#Thing`	The k-th homology group H_k = ker(∂_k) / im(∂_{k+1}). Measures k-dimensional holes.
NerveFunctor	`https://uor.foundation/homology/NerveFunctor`	`http://www.w3.org/2002/07/owl#Thing`	The nerve functor N: maps a set of constraints to a simplicial complex.
ChainFunctor	`https://uor.foundation/homology/ChainFunctor`	`http://www.w3.org/2002/07/owl#Thing`	The chain functor C: maps a simplicial complex to a chain complex.
KanComplex	`https://uor.foundation/homology/KanComplex`	`https://uor.foundation/homology/SimplicialComplex`	A simplicial set satisfying the Kan extension condition. The constraint nerve, when promoted from a SimplicialComplex to a KanComplex, carries a full homotopy type — not just its homology groups.
HornFiller	`https://uor.foundation/homology/HornFiller`	`http://www.w3.org/2002/07/owl#Thing`	A witness that a given horn (an incomplete simplex boundary) can be filled, certifying the Kan condition at a specific dimension and position.
PostnikovTruncation	`https://uor.foundation/homology/PostnikovTruncation`	`http://www.w3.org/2002/07/owl#Thing`	The k-th Postnikov truncation τ≤k of the constraint nerve: a KanComplex whose homotopy groups πj vanish for j > k.
KInvariant	`https://uor.foundation/homology/KInvariant`	`http://www.w3.org/2002/07/owl#Thing`	The k-invariant κk that classifies the extension from the (k−1)-truncation to the k-truncation of the Postnikov tower. Trivial κk means the truncation splits as a product.
DeformationComplex	`https://uor.foundation/homology/DeformationComplex`	`https://uor.foundation/homology/ChainComplex`	The deformation complex of a CompleteType T: a chain complex whose H⁰ = automorphisms, H¹ = first-order deformations, H² = obstructions to extending deformations.

Properties

Name	Kind	Functional	Domain	Range	Comment
dimension	Datatype	true	`https://uor.foundation/homology/Simplex`	`http://www.w3.org/2001/XMLSchema#integer`	The dimension k of a simplex (number of vertices minus one).
vertex	Object	false	`https://uor.foundation/homology/Simplex`	`https://uor.foundation/type/Constraint`	A vertex of this simplex, drawn from the set of constraint objects.
vertexCount	Datatype	true	`https://uor.foundation/homology/Simplex`	`http://www.w3.org/2001/XMLSchema#positiveInteger`	The number of vertices in this simplex (dimension + 1).
isFaceOf	Object	false	`https://uor.foundation/homology/Simplex`	`https://uor.foundation/homology/Simplex`	Indicates that this simplex is a face of another simplex.
pinIntersection	Object	false	`https://uor.foundation/homology/Simplex`	`https://uor.foundation/partition/SiteIndex`	A site coordinate in the partition whose intersection pins this simplex.
hasSimplex	Object	false	`https://uor.foundation/homology/SimplicialComplex`	`https://uor.foundation/homology/Simplex`	A simplex belonging to this simplicial complex.
maxDimension	Datatype	true	`https://uor.foundation/homology/SimplicialComplex`	`http://www.w3.org/2001/XMLSchema#integer`	The maximum dimension of any simplex in this simplicial complex.
eulerCharacteristic	Datatype	true	`https://uor.foundation/homology/SimplicialComplex`	`http://www.w3.org/2001/XMLSchema#integer`	The Euler characteristic of this simplicial complex: the alternating sum of simplex counts by dimension.
simplicialVertexCount	Datatype	true	`https://uor.foundation/homology/SimplicialComplex`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The total number of vertices (0-simplices) in this simplicial complex.
removesVertex	Datatype	true	`https://uor.foundation/homology/FaceMap`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The index i of the vertex removed by this face map d_i.
sourceSimplex	Object	true	`https://uor.foundation/homology/FaceMap`	`https://uor.foundation/homology/Simplex`	The source simplex of this face map.
targetFace	Object	true	`https://uor.foundation/homology/FaceMap`	`https://uor.foundation/homology/Simplex`	The target face (result simplex) of this face map.
degree	Datatype	true	`https://uor.foundation/homology/ChainGroup`	`http://www.w3.org/2001/XMLSchema#integer`	The degree k of this chain group (the dimension of its generating simplices).
rank	Datatype	true	`—`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The rank of a free abelian group (number of generators).
generatedBy	Object	false	`https://uor.foundation/homology/ChainGroup`	`https://uor.foundation/homology/Simplex`	A simplex that generates this chain group.
sourceGroup	Object	true	`https://uor.foundation/homology/BoundaryOperator`	`https://uor.foundation/homology/ChainGroup`	The source chain group C_k of this boundary operator.
targetGroup	Object	true	`https://uor.foundation/homology/BoundaryOperator`	`https://uor.foundation/homology/ChainGroup`	The target chain group C_{k-1} of this boundary operator.
satisfiesBoundarySquaredZero	Datatype	true	`https://uor.foundation/homology/BoundaryOperator`	`http://www.w3.org/2001/XMLSchema#boolean`	Whether this boundary operator satisfies the fundamental property ∂² = 0.
hasChainGroup	Object	false	`https://uor.foundation/homology/ChainComplex`	`https://uor.foundation/homology/ChainGroup`	A chain group belonging to this chain complex.
hasBoundary	Object	false	`https://uor.foundation/homology/ChainComplex`	`https://uor.foundation/homology/BoundaryOperator`	A boundary operator belonging to this chain complex.
homologyDegree	Datatype	true	`https://uor.foundation/homology/HomologyGroup`	`http://www.w3.org/2001/XMLSchema#integer`	The degree k of this homology group H_k.
bettiNumber	Datatype	true	`https://uor.foundation/homology/HomologyGroup`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The Betti number β_k = rank(H_k): the rank of this homology group.
kanWitness	Object	false	`https://uor.foundation/homology/KanComplex`	`https://uor.foundation/homology/HornFiller`	A horn filler witnessing the Kan condition for this complex.
hornDimension	Datatype	true	`https://uor.foundation/homology/HornFiller`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The dimension of the horn that this filler completes.
hornPosition	Datatype	true	`https://uor.foundation/homology/HornFiller`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The position (missing face index) of the horn that this filler completes.
truncationLevel	Datatype	true	`https://uor.foundation/homology/PostnikovTruncation`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The truncation level k of this Postnikov truncation τ≤k.
truncationSource	Object	true	`https://uor.foundation/homology/PostnikovTruncation`	`https://uor.foundation/homology/KanComplex`	The KanComplex from which this Postnikov truncation is derived.
kInvariant	Object	true	`https://uor.foundation/homology/PostnikovTruncation`	`https://uor.foundation/homology/KInvariant`	The k-invariant classifying the extension at this truncation level.
kInvariantTrivial	Datatype	true	`https://uor.foundation/homology/KInvariant`	`http://www.w3.org/2001/XMLSchema#boolean`	True iff this k-invariant is trivial, meaning the Postnikov truncation splits as a product.
deformationBase	Object	true	`https://uor.foundation/homology/DeformationComplex`	`https://uor.foundation/type/CompleteType`	The CompleteType whose deformation complex this is.
tangentDimension	Datatype	true	`https://uor.foundation/homology/DeformationComplex`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The dimension of the tangent space H¹(Def(T)): the number of first-order deformations.
obstructionDimension	Datatype	true	`https://uor.foundation/homology/DeformationComplex`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The dimension of the obstruction space H²(Def(T)): the number of independent obstructions to extending deformations.
groundedIn	Object	true	`https://uor.foundation/observable/BettiNumber`	`https://uor.foundation/homology/HomologyGroup`	The homology group that grounds this Betti number observable.
laplacianOf	Object	true	`https://uor.foundation/observable/SpectralGap`	`https://uor.foundation/homology/ChainComplex`	The chain complex whose Laplacian determines this spectral gap.
homologicalAnalysis	Object	true	`https://uor.foundation/resolver/ResolutionState`	`https://uor.foundation/homology/ChainComplex`	The chain complex used for homological analysis of a resolution state.

Named Individuals

Name	Type	Properties	Comment
boundarySquaredZero	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_boundarySquaredZero_lhs` `rhs`: `https://uor.foundation/schema/term_boundarySquaredZero_rhs` `forAll`: `https://uor.foundation/schema/term_boundarySquaredZero_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	∂² = 0: the boundary of a boundary is zero.
nerveFunctorN	`https://uor.foundation/homology/NerveFunctor`	—	The nerve functor N: constraints → simplicial complex.
chainFunctorC	`https://uor.foundation/homology/ChainFunctor`	—	The chain functor C: simplicial complex → chain complex.
psi_4	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_psi_4_lhs` `rhs`: `https://uor.foundation/schema/term_psi_4_rhs` `forAll`: `https://uor.foundation/schema/term_psi_4_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	ψ_4: HomologyGroups → BettiNumbers (extraction functor).
indexBridge	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/schema/term_indexBridge_lhs` `rhs`: `https://uor.foundation/schema/term_indexBridge_rhs` `forAll`: `https://uor.foundation/schema/term_indexBridge_forAll` `verificationDomain`: `https://uor.foundation/op/Topological`	Index bridge: connects Euler characteristic to alternating Betti sum.