Goto Chapter: Top 1 2 Ind

### 1 Constructors

#### 1.1 Constructing categories

##### 1.1-1 IsComplexesCategory
 ‣ IsComplexesCategory( C ) ( filter )

Returns: true or false

GAP-categories of the chain or cochain complexes category.

##### 1.1-2 IsComplexesCategoryByChains
 ‣ IsComplexesCategoryByChains( C ) ( filter )

Returns: true or false

GAP-categories of the chain complexes category.

##### 1.1-3 IsComplexesCategoryByCochains
 ‣ IsComplexesCategoryByCochains( C ) ( filter )

Returns: true or false

GAP-category of the cochain complexes category.

##### 1.1-4 ComplexesCategoryByCochains
 ‣ ComplexesCategoryByCochains( A ) ( attribute )

Returns: a CAP category

Creates the complexes category by cochains \mathcal{C}^b(A) of an additive category A.

##### 1.1-5 ComplexesCategoryByChains
 ‣ ComplexesCategoryByChains( A ) ( attribute )

Returns: a CAP category

Creates the complexes category by chains \mathcal{C}^b(A) of an additive category A.

##### 1.1-6 UnderlyingCategory
 ‣ 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.

#### 1.2 Constructing objects

##### 1.2-1 IsChainOrCochainComplex
 ‣ IsChainOrCochainComplex( C ) ( filter )

Returns: true or false

GAP-categories of the chain or cochain complexes.

##### 1.2-2 IsChainComplex
 ‣ IsChainComplex( C ) ( filter )

Returns: true or false

GAP-categories of the chain complexes.

##### 1.2-3 IsCochainComplex
 ‣ IsCochainComplex( C ) ( filter )

Returns: true or false

GAP-categories of the cochain complexes.

##### 1.2-4 CreateComplex
 ‣ 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].

##### 1.2-5 CreateComplex
 ‣ 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.

##### 1.2-6 Objects
 ‣ Objects( C ) ( attribute )

Returns: a \mathbb{Z}-function

Returns the objects of the complex as a \mathbb{Z}-function.

##### 1.2-7 ObjectAt
 ‣ 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).

##### 1.2-9 Differentials
 ‣ Differentials( C ) ( attribute )

Returns: a \mathbb{Z}-function

Returns the differentials of the complex as a \mathbb{Z}-function.

##### 1.2-10 DifferentialAt
 ‣ 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).

##### 1.2-12 LowerBound
 ‣ 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.

##### 1.2-13 UpperBound
 ‣ 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.

##### 1.2-14 ObjectsSupport
 ‣ 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.

##### 1.2-15 ObjectsSupport
 ‣ 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.

##### 1.2-16 DifferentialsSupport
 ‣ 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.

##### 1.2-17 DifferentialsSupport
 ‣ 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.

##### 1.2-18 CocyclesAt
 ‣ 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.

##### 1.2-19 CocyclesEmbeddingAt
 ‣ 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.

##### 1.2-20 CoboundariesAt
 ‣ 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}.

##### 1.2-21 CoboundariesEmbeddingAt
 ‣ 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}.

##### 1.2-22 CohomologyAt
 ‣ 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.

##### 1.2-23 CohomologySupport
 ‣ 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.

##### 1.2-24 CohomologySupport
 ‣ 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.

##### 1.2-25 CyclesAt
 ‣ 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.

##### 1.2-26 CyclesEmbeddingAt
 ‣ 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.

##### 1.2-27 BoundariesAt
 ‣ 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}.

##### 1.2-28 BoundariesEmbeddingAt
 ‣ 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}.

##### 1.2-29 HomologyAt
 ‣ 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.

##### 1.2-30 HomologySupport
 ‣ 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.

##### 1.2-31 HomologySupport
 ‣ 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.

##### 1.2-32 IsExact
 ‣ 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.

##### 1.2-33 IsExact
 ‣ 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.

##### 1.2-34 AsChainComplex
 ‣ AsChainComplex( C ) ( attribute )

Returns: a CAP object

Convert a cochain complex into a chain complex.

##### 1.2-35 AsCochainComplex
 ‣ AsCochainComplex( C ) ( attribute )

Returns: a CAP object

Convert a chain complex into a cochain complex.

#### 1.3 Constructing morphisms

##### 1.3-1 IsChainOrCochainMorphism
 ‣ IsChainOrCochainMorphism( phi ) ( filter )

Returns: true or false

GAP-categories of the complex morphisms.

##### 1.3-2 IsChainMorphism
 ‣ IsChainMorphism( phi ) ( filter )

Returns: true or false

GAP-categories of the chain complex morphisms.

##### 1.3-3 IsCochainMorphism
 ‣ IsCochainMorphism( phi ) ( filter )

Returns: true or false

GAP-categories of the cochain complex morphisms.

##### 1.3-4 CreateComplexMorphism
 ‣ 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].

##### 1.3-5 CreateComplexMorphism
 ‣ 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.

##### 1.3-6 Morphisms
 ‣ Morphisms( phi ) ( attribute )

Returns: a \mathbb{Z}-function

Returns the morphisms as a \mathbb{Z}-function.

##### 1.3-7 MorphismAt
 ‣ 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).

##### 1.3-9 LowerBound
 ‣ LowerBound( phi ) ( attribute )

Returns: integer or infinity

Returns an integer \ell with S^i=R^i=0 for all i\prec\ell.

##### 1.3-10 UpperBound
 ‣ UpperBound( phi ) ( attribute )

Returns: integer or infinity

Returns an integer u with S^i=R^i=0 for all i\succ u.

##### 1.3-11 MorphismsSupport
 ‣ 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.

##### 1.3-12 MorphismsSupport
 ‣ 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.

##### 1.3-13 CyclesFunctorialAt
 ‣ 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.

##### 1.3-14 CocyclesFunctorialAt
 ‣ 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.

##### 1.3-15 CohomologyFunctorialAt
 ‣ 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.

##### 1.3-16 HomologyFunctorialAt
 ‣ 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.

##### 1.3-17 IsQuasiIsomorphism
 ‣ 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.

##### 1.3-18 IsHomotopicToZeroMorphism
 ‣ IsHomotopicToZeroMorphism( phi ) ( property )

Returns: true or false

Returns wheather \phi is homotopic to the zero morphism.

##### 1.3-19 WitnessForBeingHomotopicToZeroMorphism
 ‣ 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}.

##### 1.3-20 AsChainComplexMorphism
 ‣ AsChainComplexMorphism( phi ) ( attribute )

Returns: a CAP morphism

Convert a chain morphism into a cochain morphism.

##### 1.3-21 AsCochainComplexMorphism
 ‣ AsCochainComplexMorphism( phi ) ( attribute )

Returns: a CAP morphism

Convert a cochain morphism into a chain morphism.

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