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.
generated by GAPDoc2HTML