Complexes -- Homological data types and routines
Description
This package provides routines for studying homology in a general algebraic setting.
Version
This documentation describes version 0.3 of Complexes.
Source code
The source code from which this documentation is derived is in the file
Complexes.m2.
Exports
Types
- Complex -- the class of all chain or cochain complexes
- ComplexMap -- the class of all homomorphisms of complexes
Functions and commands
- canonicalMap (missing documentation)
- complex -- make a chain or cochain complex
- concentration -- indices on which a complex may be non-zero
- cylinder (missing documentation)
- freeResolution (missing documentation)
- homotopic (missing documentation)
- isComplexMorphism (missing documentation)
- isNullHomotopyOf (missing documentation)
- isQuasiIsomorphism (missing documentation)
- liftMapAlongQuasiIsomorphism (missing documentation)
- minimize (missing documentation)
- naiveTruncation (missing documentation)
- nullHomotopic (missing documentation)
- nullHomotopy (missing documentation)
- randomComplexMap (missing documentation)
- resolutionMap (missing documentation)
Methods
- - ComplexMap (missing documentation)
- betti(Complex) -- display of degrees in a complex
- canonicalMap(Complex,Complex) (missing documentation)
- coimage(ComplexMap) (missing documentation)
- cokernel(ComplexMap) (missing documentation)
- Complex ** Complex -- tensor product of complexes
- Complex ** Module, see Complex ** Complex -- tensor product of complexes
- Module ** Complex, see Complex ** Complex -- tensor product of complexes
- Complex ** ComplexMap (missing documentation)
- Complex ** Ring (missing documentation)
- Complex == Complex -- whether two complexes are equal
- Complex == ZZ, see Complex == Complex -- whether two complexes are equal
- ZZ == Complex, see Complex == Complex -- whether two complexes are equal
- Complex ^ Array, see Complex _ Array -- the canonical inclusion or projection map of a direct sum
- Complex _ Array -- the canonical inclusion or projection map of a direct sum
- Complex ^ ZZ, see Complex _ ZZ -- access individual object in a complex
- Complex _ ZZ -- access individual object in a complex
- Complex Array -- shift a complex
- ComplexMap * ComplexMap -- composition of homomorphisms of complexes
- Number * ComplexMap, see ComplexMap * ComplexMap -- composition of homomorphisms of complexes
- RingElement * ComplexMap, see ComplexMap * ComplexMap -- composition of homomorphisms of complexes
- ComplexMap ** Complex (missing documentation)
- ComplexMap ** ComplexMap (missing documentation)
- ComplexMap ** Module (missing documentation)
- ComplexMap ** Ring (missing documentation)
- ComplexMap + ComplexMap (missing documentation)
- ComplexMap + Number (missing documentation)
- ComplexMap + RingElement (missing documentation)
- ComplexMap ++ ComplexMap (missing documentation)
- ComplexMap - ComplexMap (missing documentation)
- ComplexMap - Number (missing documentation)
- ComplexMap - RingElement (missing documentation)
- ComplexMap == ComplexMap (missing documentation)
- ComplexMap == ZZ (missing documentation)
- ComplexMap ^ Array (missing documentation)
- ComplexMap ^ ZZ (missing documentation)
- ComplexMap _ Array (missing documentation)
- ComplexMap _ ZZ (missing documentation)
- ComplexMap | ComplexMap (missing documentation)
- ComplexMap || ComplexMap (missing documentation)
- ComplexMap Array (missing documentation)
- components(Complex) -- list the components of a direct sum
- components(ComplexMap) (missing documentation)
- concentration(Complex), see concentration -- indices on which a complex may be non-zero
- concentration(ComplexMap) (missing documentation)
- cone(ComplexMap) (missing documentation)
- cylinder(ComplexMap) (missing documentation)
- degree(ComplexMap) (missing documentation)
- Complex ++ Complex, see directSum(Complex) -- direct sum of complexes
- directSum(Complex) -- direct sum of complexes
- directSum(ComplexMap) (missing documentation)
- dual(Complex) -- makes the dual of a complex
- dual(ComplexMap) (missing documentation)
- extend(Complex,Complex,Matrix) (missing documentation)
- gradedModule(Complex) (missing documentation)
- HH Complex -- homology of a complex
- HH ComplexMap (missing documentation)
- HH^ZZ ComplexMap (missing documentation)
- HH^ZZ Complex, see HH_ZZ Complex -- homology or cohomology module of a complex
- HH_ZZ Complex -- homology or cohomology module of a complex
- HH_ZZ ComplexMap (missing documentation)
- Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Complex,Module), see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Module,Complex), see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Complex,ComplexMap) (missing documentation)
- Hom(ComplexMap,Complex) (missing documentation)
- Hom(ComplexMap,ComplexMap) (missing documentation)
- Hom(ComplexMap,Module) (missing documentation)
- Hom(Module,ComplexMap) (missing documentation)
- homomorphism'(ComplexMap) (missing documentation)
- homomorphism(ComplexMap) (missing documentation)
- homomorphism(ZZ,Matrix,Complex) (missing documentation)
- homotopic(ComplexMap,ComplexMap) (missing documentation)
- id _ Complex (missing documentation)
- image(ComplexMap) (missing documentation)
- inducedMap(Complex,Complex) (missing documentation)
- isCommutative(ComplexMap) (missing documentation)
- isComplexMorphism(ComplexMap) (missing documentation)
- isDirectSum(Complex) (missing documentation)
- isHomogeneous(Complex) -- whether a complex is homogeneous
- isHomogeneous(ComplexMap) (missing documentation)
- isNullHomotopyOf(ComplexMap,ComplexMap) (missing documentation)
- isQuasiIsomorphism(ComplexMap) (missing documentation)
- isWellDefined(Complex) -- whether a complex is well-defined
- isWellDefined(ComplexMap) (missing documentation)
- kernel(ComplexMap) (missing documentation)
- length(Complex) -- length of a complex
- liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) (missing documentation)
- map(Complex,Complex,ComplexMap) (missing documentation)
- map(Complex,Complex,Function) (missing documentation)
- map(Complex,Complex,HashTable) (missing documentation)
- map(Complex,Complex,List) (missing documentation)
- map(Complex,Complex,ZZ) (missing documentation)
- max(Complex) (missing documentation)
- min(Complex) (missing documentation)
- minimalPresentation(Complex) (missing documentation)
- minimalPresentation(ComplexMap) (missing documentation)
- minimize(Complex) (missing documentation)
- Module ** ComplexMap (missing documentation)
- naiveTruncation(Complex,InfiniteNumber,ZZ) (missing documentation)
- naiveTruncation(Complex,Nothing,ZZ) (missing documentation)
- naiveTruncation(Complex,Sequence) (missing documentation)
- naiveTruncation(Complex,ZZ,InfiniteNumber) (missing documentation)
- naiveTruncation(Complex,ZZ,Nothing) (missing documentation)
- naiveTruncation(Complex,ZZ,ZZ) (missing documentation)
- naiveTruncation(ComplexMap,Sequence) (missing documentation)
- naiveTruncation(ComplexMap,Sequence,Sequence) (missing documentation)
- nullHomotopic(ComplexMap) (missing documentation)
- nullHomotopy(ComplexMap) (missing documentation)
- Number + ComplexMap (missing documentation)
- Number - ComplexMap (missing documentation)
- poincare(Complex) (missing documentation)
- poincareN(Complex) (missing documentation)
- prune(Complex) (missing documentation)
- prune(ComplexMap) (missing documentation)
- randomComplexMap(Complex,Complex) (missing documentation)
- regularity(Complex) (missing documentation)
- resolution(Complex) (missing documentation)
- resolutionMap(Complex) (missing documentation)
- ring(Complex) -- access the ring of a complex
- ring(ComplexMap) (missing documentation)
- RingElement + ComplexMap (missing documentation)
- RingElement - ComplexMap (missing documentation)
- RingMap Complex (missing documentation)
- RingMap ComplexMap (missing documentation)
- source(ComplexMap) (missing documentation)
- sum(Complex) (missing documentation)
- sum(ComplexMap) (missing documentation)
- Symbol ^ Complex (missing documentation)
- Symbol _ Complex (missing documentation)
- target(ComplexMap) (missing documentation)
- tensor(Complex,Complex) (missing documentation)
- tensorAssociativity(Complex,Complex,Complex) (missing documentation)
- ZZ == ComplexMap (missing documentation)
Symbols
- Boundary (missing documentation)
- Base, see complex -- make a chain or cochain complex
- Cycle (missing documentation)
- homotopy (missing documentation)
- HomWithComponents (missing documentation)
- InternalDegree (missing documentation)
- isSemifreeComplex (missing documentation)
- ResolutionMap (missing documentation)
- UseTarget (missing documentation)