UOR Homology

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

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.

Learn more: Pipeline Overview · Homological Analysis

Class hierarchy

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	Subclass Of	Comment
`Simplex`	`Thing`	A k-simplex: a finite set of k+1 vertices drawn from constraint objects.
`SimplicialComplex`	`Thing`	A simplicial complex: a set of simplices closed under taking faces.
`FaceMap`	`Thing`	A face map d_i: removes vertex i from a simplex, producing a face.
`ChainGroup`	`Thing`	A free abelian group generated by k-simplices (the k-th chain group C_k).
`BoundaryOperator`	`Thing`	The boundary operator ∂_k: C_k → C_{k-1}. Satisfies ∂² = 0.
`ChainComplex`	`Thing`	A chain complex: a sequence of chain groups connected by boundary operators.
`HomologyGroup`	`Thing`	The k-th homology group H_k = ker(∂_k) / im(∂_{k+1}). Measures k-dimensional holes.
`NerveFunctor`	`Thing`	The nerve functor N: maps a set of constraints to a simplicial complex.
`ChainFunctor`	`Thing`	The chain functor C: maps a simplicial complex to a chain complex.
`KanComplex`	`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`	`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`	`Thing`	The k-th Postnikov truncation τ≤k of the constraint nerve: a KanComplex whose homotopy groups πj vanish for j > k.
`KInvariant`	`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`	`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	`Simplex`	`integer`	The dimension k of a simplex (number of vertices minus one).
`vertex`	Object	false	`Simplex`	`Constraint`	A vertex of this simplex, drawn from the set of constraint objects.
`vertexCount`	Datatype	true	`Simplex`	`positiveInteger`	The number of vertices in this simplex (dimension + 1).
`isFaceOf`	Object	false	`Simplex`	`Simplex`	Indicates that this simplex is a face of another simplex.
`pinIntersection`	Object	false	`Simplex`	`SiteIndex`	A site coordinate in the partition whose intersection pins this simplex.
`hasSimplex`	Object	false	`SimplicialComplex`	`Simplex`	A simplex belonging to this simplicial complex.
`maxDimension`	Datatype	true	`SimplicialComplex`	`integer`	The maximum dimension of any simplex in this simplicial complex.
`eulerCharacteristic`	Datatype	true	`SimplicialComplex`	`integer`	The Euler characteristic of this simplicial complex: the alternating sum of simplex counts by dimension.
`simplicialVertexCount`	Datatype	true	`SimplicialComplex`	`nonNegativeInteger`	The total number of vertices (0-simplices) in this simplicial complex.
`removesVertex`	Datatype	true	`FaceMap`	`nonNegativeInteger`	The index i of the vertex removed by this face map d_i.
`sourceSimplex`	Object	true	`FaceMap`	`Simplex`	The source simplex of this face map.
`targetFace`	Object	true	`FaceMap`	`Simplex`	The target face (result simplex) of this face map.
`degree`	Datatype	true	`ChainGroup`	`integer`	The degree k of this chain group (the dimension of its generating simplices).
`rank`	Datatype	true	`—`	`nonNegativeInteger`	The rank of a free abelian group (number of generators).
`generatedBy`	Object	false	`ChainGroup`	`Simplex`	A simplex that generates this chain group.
`sourceGroup`	Object	true	`BoundaryOperator`	`ChainGroup`	The source chain group C_k of this boundary operator.
`targetGroup`	Object	true	`BoundaryOperator`	`ChainGroup`	The target chain group C_{k-1} of this boundary operator.
`satisfiesBoundarySquaredZero`	Datatype	true	`BoundaryOperator`	`boolean`	Whether this boundary operator satisfies the fundamental property ∂² = 0.
`hasChainGroup`	Object	false	`ChainComplex`	`ChainGroup`	A chain group belonging to this chain complex.
`hasBoundary`	Object	false	`ChainComplex`	`BoundaryOperator`	A boundary operator belonging to this chain complex.
`homologyDegree`	Datatype	true	`HomologyGroup`	`integer`	The degree k of this homology group H_k.
`bettiNumber`	Datatype	true	`HomologyGroup`	`nonNegativeInteger`	The Betti number β_k = rank(H_k): the rank of this homology group.
`kanWitness`	Object	false	`KanComplex`	`HornFiller`	A horn filler witnessing the Kan condition for this complex.
`hornDimension`	Datatype	true	`HornFiller`	`nonNegativeInteger`	The dimension of the horn that this filler completes.
`hornPosition`	Datatype	true	`HornFiller`	`nonNegativeInteger`	The position (missing face index) of the horn that this filler completes.
`truncationLevel`	Datatype	true	`PostnikovTruncation`	`nonNegativeInteger`	The truncation level k of this Postnikov truncation τ≤k.
`truncationSource`	Object	true	`PostnikovTruncation`	`KanComplex`	The KanComplex from which this Postnikov truncation is derived.
`kInvariant`	Object	true	`PostnikovTruncation`	`KInvariant`	The k-invariant classifying the extension at this truncation level.
`kInvariantTrivial`	Datatype	true	`KInvariant`	`boolean`	True iff this k-invariant is trivial, meaning the Postnikov truncation splits as a product.
`deformationBase`	Object	true	`DeformationComplex`	`CompleteType`	The CompleteType whose deformation complex this is.
`tangentDimension`	Datatype	true	`DeformationComplex`	`nonNegativeInteger`	The dimension of the tangent space H¹(Def(T)): the number of first-order deformations.
`obstructionDimension`	Datatype	true	`DeformationComplex`	`nonNegativeInteger`	The dimension of the obstruction space H²(Def(T)): the number of independent obstructions to extending deformations.
`groundedIn`	Object	true	`BettiNumber`	`HomologyGroup`	The homology group that grounds this Betti number observable.
`laplacianOf`	Object	true	`SpectralGap`	`ChainComplex`	The chain complex whose Laplacian determines this spectral gap.
`homologicalAnalysis`	Object	true	`ResolutionState`	`ChainComplex`	The chain complex used for homological analysis of a resolution state.

Named Individuals

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

UOR Homology

Imports

Classes

Properties

Named Individuals

Related Concepts