next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Complexes :: Complexes

Complexes -- Homological data types and routines

Description

This package provides routines for studying homology in a general algebraic setting.

Authors

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)

Menu