1.2-8 \[\]
1.2-11 \^
1.3-8 \[\]
‣ IsComplexesCategory( C ) | ( filter ) |
Returns: true or false
GAP-categories of the chain or cochain complexes category.
‣ IsComplexesCategoryByChains( C ) | ( filter ) |
Returns: true or false
GAP-categories of the chain complexes category.
‣ IsComplexesCategoryByCochains( C ) | ( filter ) |
Returns: true or false
GAP-category of the cochain complexes category.
‣ ComplexesCategoryByCochains( A ) | ( attribute ) |
Returns: a CAP category
Creates the complexes category by cochains \mathcal{C}^b(A) of an additive category A.
‣ ComplexesCategoryByChains( A ) | ( attribute ) |
Returns: a CAP category
Creates the complexes category by chains \mathcal{C}^b(A) of an additive category A.
‣ UnderlyingCategory( C ) | ( attribute ) |
Returns: a CAP category
The input is a complexes category by (co)chains C:=\mathcal{C}^b(A). The outout is A.
‣ IsChainOrCochainComplex( C ) | ( filter ) |
Returns: true or false
GAP-categories of the chain or cochain complexes.
‣ IsChainComplex( C ) | ( filter ) |
Returns: true or false
GAP-categories of the chain complexes.
‣ IsCochainComplex( C ) | ( filter ) |
Returns: true or false
GAP-categories of the cochain complexes.
‣ CreateComplex( C, L ) | ( operation ) |
Returns: a CAP object
The input is a complexes category \mathcal{C}^b(A) by chains or cochains and a list L with 4 entries: L[1] and L[2] are \mathbb{Z}-functions and L[3] and L[4] are integers or \pm\infty. The output is an object C\in \mathcal{C}^b(A) whose object at i\in\mathbb{Z} is C^i:=L[1][i], differential at i\in\mathbb{Z} is \partial_{C}^i:=L[2][i]. Its lower and upper bounds are L[3] resp. L[4].
‣ CreateComplex( C, L ) | ( operation ) |
Returns: a CAP object
The input is a complexes category \mathcal{C}^b(A) by chains or cochains, a dense-list L of morphisms in A and an integer \ell. The output is an object C\in\mathcal{C}^b(A) whose differentials are \partial_{C}^{\ell}:=L[1], \partial_{C}^{\ell+1}:=L[2], etc.
‣ Objects( C ) | ( attribute ) |
Returns: a \mathbb{Z}-function
Returns the objects of the complex as a \mathbb{Z}-function.
‣ ObjectAt( C, i ) | ( operation ) |
Returns: a CAP object
Returns the object C^i of the complex C at the index i\in\mathbb{Z}.
1.2-8 \[\]‣ \[\]( C, i ) | ( operation ) |
Returns: a CAP object
Delegates to ObjectAt(C,i).
‣ Differentials( C ) | ( attribute ) |
Returns: a \mathbb{Z}-function
Returns the differentials of the complex as a \mathbb{Z}-function.
‣ DifferentialAt( C, i ) | ( operation ) |
Returns: a CAP morphism
Returns the differential of the complex C at the index i\in\mathbb{Z}.
1.2-11 \^‣ \^( C, i ) | ( operation ) |
Returns: a CAP object
Delegates to DifferentialAt(C,i).
‣ LowerBound( C ) | ( attribute ) |
Returns: integer or infinity
Returns the lower bound \ell of C. I.e., the objects C^i are zero for all i\prec\ell.
‣ UpperBound( C ) | ( attribute ) |
Returns: integer or infinity
Returns the upper bound u of C. I.e., the objects C^i are zero for all i\succ u.
‣ ObjectsSupport( C, m, n ) | ( operation ) |
Returns: a list of integers
The input is a complex C and two integers m,n. The output is the list of indices m\preceq i \preceq n where the objects of C are non-zero.
‣ ObjectsSupport( C ) | ( attribute ) |
Returns: a list of integers
The input is a complex C whose lower and upper bounds are integers. The output is the list of indices where the objects of C are non-zero.
‣ DifferentialsSupport( C, m, n ) | ( operation ) |
Returns: a list of integers
The input is a complex C and two integers m,n. The output is the list of indices m\preceq i \preceq n where the differentials of C are non-zero.
‣ DifferentialsSupport( C ) | ( attribute ) |
Returns: a list of integers
The input is a complex C whose lower and upper bounds are integers. The output is the list of indices where the differentials of C are non-zero.
‣ CocyclesAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the kernel object of \partial_{C}^i.
‣ CocyclesEmbeddingAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the kernel embedding of \partial_{C}^i.
‣ CoboundariesAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the image object of the differential \partial_{C}^{i-1}.
‣ CoboundariesEmbeddingAt( C, i ) | ( operation ) |
Returns: a CAP morphism
The input is a cochain complex C. The output is the image embedding of the differential \partial_{C}^{i-1}.
‣ CohomologyAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the cohomology object of C at i.
‣ CohomologySupport( C, m, n ) | ( operation ) |
Returns: a list of integers
The input is a complex C and two integers m,n. The output is the list of indices m\preceq i \preceq n where the cohomology objects of C are non-zero.
‣ CohomologySupport( C ) | ( attribute ) |
Returns: a list of integers
The input is a complex C whose lower and upper bounds are integers. The output is the list of indices where the cohomology objects of C are non-zero.
‣ CyclesAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the kernel object of \partial_{C}^i.
‣ CyclesEmbeddingAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the kernel embedding of \partial_{C}^i.
‣ BoundariesAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the image object of the differential \partial_{C}^{i+1}.
‣ BoundariesEmbeddingAt( C, i ) | ( operation ) |
Returns: a CAP morphism
The input is a cochain complex C. The output is the image embedding of the differential \partial_{C}^{i+1}.
‣ HomologyAt( C, i ) | ( operation ) |
Returns: a CAP object
The input is a cochain complex C. The output is the homology object of C at i.
‣ HomologySupport( C, m, n ) | ( operation ) |
Returns: a list of integers
The input is a complex C and two integers m,n. The output is the list of indices m\preceq i \preceq n where the homology objects of C are non-zero.
‣ HomologySupport( C ) | ( attribute ) |
Returns: a list of integers
The input is a complex C whose lower and upper bounds are integers. The output is the list of indices where the homology objects of C are non-zero.
‣ IsExact( C ) | ( property ) |
Returns: a list of integers
The input is a complex C whose lower and upper bounds are integers. The output is wheather the (co)homology support is empty.
‣ IsExact( C, m, n ) | ( operation ) |
Returns: a list of integers
The input is a complex C and two integers m,n. The output is wheather the (co)homology support between m and n is empty.
‣ AsChainComplex( C ) | ( attribute ) |
Returns: a CAP object
Convert a cochain complex into a chain complex.
‣ AsCochainComplex( C ) | ( attribute ) |
Returns: a CAP object
Convert a chain complex into a cochain complex.
‣ IsChainOrCochainMorphism( phi ) | ( filter ) |
Returns: true or false
GAP-categories of the complex morphisms.
‣ IsChainMorphism( phi ) | ( filter ) |
Returns: true or false
GAP-categories of the chain complex morphisms.
‣ IsCochainMorphism( phi ) | ( filter ) |
Returns: true or false
GAP-categories of the cochain complex morphisms.
‣ CreateComplexMorphism( C, S, L, R ) | ( operation ) |
Returns: a CAP morphism
The input is a complexes category \mathcal{C}^b(A) by chains or cochains, two objects S and R and a list L with 3 entries: L[1] is a \mathbb{Z}-function; L[2] and L[3] are integers or \pm\infty. The output is the morphism \phi:S \to R in \mathcal{C}^b(A) whose morphism at i\in\mathbb{Z} is L[1][i] and its lower and upper bounds are L[2] resp. L[3].
‣ CreateComplexMorphism( C, S, L, ell, R ) | ( operation ) |
Returns: a CAP morphism
The input is a complexes category \mathcal{C}^b(A) by chains or cochains, two objects S and R and a dense-list L of morphisms in A and an integer \ell. The output is the morphism \phi:S \to R in \mathcal{C}^b(A) whose morphism at \ell is L[1], at \ell+1 is L[2], etc. In particular, \ell is a lower bound of \phi.
‣ Morphisms( phi ) | ( attribute ) |
Returns: a \mathbb{Z}-function
Returns the morphisms as a \mathbb{Z}-function.
‣ MorphismAt( phi, i ) | ( operation ) |
Returns: a CAP morphism
Returns the morphism of \phi at the index i\in\mathbb{Z}.
1.3-8 \[\]‣ \[\]( phi, i ) | ( operation ) |
Returns: a CAP morphism
Delegates to MorphismAt(\phi, i).
‣ LowerBound( phi ) | ( attribute ) |
Returns: integer or infinity
Returns an integer \ell with S^i=R^i=0 for all i\prec\ell.
‣ UpperBound( phi ) | ( attribute ) |
Returns: integer or infinity
Returns an integer u with S^i=R^i=0 for all i\succ u.
‣ MorphismsSupport( phi, m, n ) | ( operation ) |
Returns: a list of integers
The input is a complex morphism \phi and two integers m,n. The output is the list of indices m\preceq i \preceq n where the morphisms of \phi are non-zero.
‣ MorphismsSupport( phi, i ) | ( attribute ) |
Returns: a list of integers
The input is a complex morphism \phi whose lower and upper bounds are integers. The output is the list of indices where the morphisms of \phi are non-zero.
‣ CyclesFunctorialAt( phi, i ) | ( operation ) |
Returns: a CAP morphism
The input is a complex morphism \phi:S \to R and an integer i. The output is the morphism induced by the functoriality of the cycle objects of S and R at the index i.
‣ CocyclesFunctorialAt( phi, i ) | ( operation ) |
Returns: a CAP morphism
The input is a complex morphism \phi:S \to R and an integer i. The output is the morphism induced by the functoriality of the cocycle objects of S and R at the index i.
‣ CohomologyFunctorialAt( phi, i ) | ( operation ) |
Returns: a CAP morphism
The input is a complex morphism \phi:S \to R and an integer i. The output is the morphism induced by the functoriality of the cohomology objects of S and R at the index i.
‣ HomologyFunctorialAt( phi, i ) | ( operation ) |
Returns: a CAP morphism
The input is a complex morphism \phi:S \to R and an integer i. The output is the morphism induced by the functoriality of the homology objects of S and R at the index i.
‣ IsQuasiIsomorphism( phi ) | ( property ) |
Returns: true or false
Returns wheather \phi is a quasi-isomorphism, i.e., wheather it induces isomorphisms between the (co)homology objects of S and R.
‣ IsHomotopicToZeroMorphism( phi ) | ( property ) |
Returns: true or false
Returns wheather \phi is homotopic to the zero morphism.
‣ WitnessForBeingHomotopicToZeroMorphism( phi ) | ( attribute ) |
Returns: \mathbb{Z}-function
Returns a \mathbb{Z}-function w witnessing that \phi is homotopic to the zero morphism. If \phi is a chain morphism, then w^i is a morphism from S^i to R^{i+1} and if \phi is a cochain morphism, then w^i is a morphism from S^i to R^{i-1}.
‣ AsChainComplexMorphism( phi ) | ( attribute ) |
Returns: a CAP morphism
Convert a chain morphism into a cochain morphism.
‣ AsCochainComplexMorphism( phi ) | ( attribute ) |
Returns: a CAP morphism
Convert a cochain morphism into a chain morphism.
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