2.4-1 \[\]
2.4-2 \^
‣ IsChainOrCochainComplex( C ) | ( filter ) |
‣ IsChainComplex( C ) | ( filter ) |
‣ IsCochainComplex( C ) | ( filter ) |
‣ IsBoundedBelowChainOrCochainComplex( C ) | ( filter ) |
‣ IsBoundedAboveChainOrCochainComplex( C ) | ( filter ) |
‣ IsBoundedChainOrCochainComplex( C ) | ( filter ) |
‣ IsBoundedBelowChainComplex( C ) | ( filter ) |
‣ IsBoundedAboveChainComplex( C ) | ( filter ) |
‣ IsBoundedChainComplex( C ) | ( filter ) |
‣ IsBoundedBelowCochainComplex( C ) | ( filter ) |
‣ IsBoundedAboveCochainComplex( C ) | ( filter ) |
‣ IsBoundedCochainComplex( C ) | ( filter ) |
Returns: true or false
Gap-categories for chain and cochain complexes.
‣ ChainComplex( A, diffs ) | ( operation ) |
‣ CochainComplex( A, diffs ) | ( operation ) |
Returns: a chain complex
The input is category A and a \mathbb{Z}-function diffs. The output is the chain (resp. cochain) complex M_{\bullet}\in \mathrm{Ch}(A) (M^{\bullet}\in \mathrm{Ch}^\bullet(A)) where d^M_{i}=\mathrm{diffs}[ i ](d_M^{i}=\mathrm{diffs}[ i ]).
‣ ChainComplex( diffs, n ) | ( operation ) |
‣ CochainComplex( diffs, n ) | ( operation ) |
Returns: a (co)chain complex
The input is a finite dense list diffs and an integer n . The output is the chain (resp. cochain) complex M_{\bullet}\in \mathrm{Ch}(A) (M^{\bullet}\in \mathrm{Ch}^\bullet(A)) where d^M_{n}:= \mathrm{diffs}[ 1 ](d_M^n :=\mathrm{diffs}[ 1 ]),d^M_{n+1}=\mathrm{diffs}[ 2 ](d_M^{n+1}:=\mathrm{diffs}[ 2 ]), etc.
‣ ChainComplex( diffs ) | ( operation ) |
‣ CochainComplex( diffs ) | ( operation ) |
Returns: a (co)chain complex
The same as the previous operations but with n=0.
‣ StalkChainComplex( M, n ) | ( operation ) |
‣ StalkCochainComplex( M, n ) | ( operation ) |
Returns: a (co)chain complex
The input is an object M\in A. The output is chain (resp. cochain) complex M_{\bullet}\in\mathrm{Ch}_\bullet(A)(M^{\bullet}\in\mathrm{Ch}^\bullet(A)) where M_n=M( M^n=M) and M_i=0(M^i=0) whenever i\neq n.
‣ ChainComplexWithInductiveSides( d, G, F ) | ( operation ) |
Returns: a chain complex
The input is a morphism d\in A and two functions F,G. The output is chain complex M_{\bullet}\in\mathrm{Ch}_\bullet(A) where d^{M}_{0}=d and d^M_{i}=G^{i}(d) for all i\leq -1 and d^M_{i}=F^{i}(d ) for all i \geq 1.
‣ ChainComplexWithInductiveSides( n, d, G, F ) | ( operation ) |
Returns: a chain complex
The input is an integer n, a morphism d\in A and two functions F,G. The output is chain complex M_{\bullet}\in\mathrm{Ch}_\bullet(A) where d^{M}_{n}=d and d^M_{i}=G^{i}(d) for all i\leq -1 and d^M_{i}=F^{i}(d ) for all i \geq 1.
‣ CochainComplexWithInductiveSides( d, G, F ) | ( operation ) |
Returns: a cochain complex
The input is a morphism d\in A and two functions F,G. The output is cochain complex M^{\bullet}\in\mathrm{Ch}^\bullet(A) where d_{M}^{0}=d and d_M^{i}=G^{i}( d) for all i\leq -1 and d_M^{i}=F^{i}( d ) for all i \geq 1.
‣ ChainComplexWithInductiveNegativeSide( d, G ) | ( operation ) |
Returns: a chain complex
The input is a morphism d\in A and a functions G. The output is chain complex M_{\bullet}\in\mathrm{Ch}_\bullet(A) where d^{M}_{0}=d and d^M_{i}=G^{i}( d ) for all i\leq -1 and d^M_{i}=0 for all i \geq 1.
‣ ChainComplexWithInductivePositiveSide( d, F ) | ( operation ) |
Returns: a chain complex
The input is a morphism d\in A and a functions F. The output is chain complex M_{\bullet}\in\mathrm{Ch}_\bullet(A) where d^{M}_{0}=d and d^M_{i}=F^{i}( d ) for all i\geq 1 and d^M_{i}=0 for all i \leq 1.
‣ CochainComplexWithInductiveNegativeSide( d, G ) | ( operation ) |
Returns: a cochain complex
The input is a morphism d\in A and a functions G. The output is cochain complex M^{\bullet}\in\mathrm{Ch}^\bullet(A) where d_{M}^{0}=d and d_M^{i}=G^{i}( d ) for all i\leq -1 and d_M^{i}=0 for all i \geq 1.
‣ CochainComplexWithInductivePositiveSide( d, F ) | ( operation ) |
Returns: a cochain complex
The input is a morphism d\in A and a functions F. The output is cochain complex M^{\bullet}\in\mathrm{Ch}^\bullet(A) where d_{M}^{0}=d and d_M^{i}=F^{i}( d ) for all i\geq 1 and d_M^{i}=0 for all i \leq 1.
‣ Differentials( C ) | ( attribute ) |
Returns: a \mathbb{Z}-function
The command returns the differentials of the chain or cochain complex as a \mathbb{Z}-function.
‣ Objects( C ) | ( attribute ) |
Returns: a \mathbb{Z}-function
The command returns the objects of the chain or cochain complex as a \mathbb{Z}-function.
‣ AsChainComplex( C ) | ( attribute ) |
Returns: a chain complex
If the input is a cochain complex C, then the output is the associated chain complex. Otherwise, the output is C.
‣ AsCochainComplex( C ) | ( attribute ) |
Returns: a cochain complex
If the input is a chain complex C, then the output is the associated cochain complex. Otherwise, the output is C.
‣ IsExact( C ) | ( property ) |
Returns: a boolian
The input is a bounded chain (resp. cochain) complex C and two integers m,n. The output is true when C is an exact complex, otherwise the output is false.
‣ IsContractable( C ) | ( property ) |
Returns: a boolian
The input is a bounded chain (resp. cochain) complex C and two integers m,n. The output is true when C is a contractible complex, otherwise the output is false.
2.4-1 \[\]‣ \[\]( C, i ) | ( operation ) |
Returns: an object
The command returns the object of the chain or cochain complex in index i.
2.4-2 \^‣ \^( C, i ) | ( operation ) |
Returns: a morphism
The command returns the differential of the chain or cochain complex in index i.
‣ CyclesAt( C, n ) | ( operation ) |
Returns: a morphism
The input is a chain or cochain complex C and an integer n. The output is the kernel embedding of the differential in index n.
‣ BoundariesAt( C, n ) | ( operation ) |
Returns: a morphism
The input is a chain (resp. cochain) complex C and an integer n. The output is the image embeddin of i+1'th ( resp. i-1'th) differential of C.
‣ GeneralizedEmbeddingOfHomologyAt( C, n ) | ( operation ) |
Returns: a generalized morphism
The input is a chain complex and an integer n. The output is the generalized embedding (defined by span) of the homology object at index n.
‣ GeneralizedProjectionOntoHomologyAt( C, n ) | ( operation ) |
Returns: a generalized morphism
The input is a chain complex and an integer n. The output is the generalized embedding (defined by span) on the homology object at index n.
‣ GeneralizedEmbeddingOfCohomologyAt( C, n ) | ( operation ) |
Returns: a generalized morphism
The input is a chain complex and an integer n. The output is the generalized embedding (defined by span) of the cohomology object at index n.
‣ GeneralizedProjectionOntoCohomologyAt( C, n ) | ( operation ) |
Returns: a generalized morphism
The input is a chain complex and an integer n. The output is the generalized projection (defined by span) on the cohomology object at index n.
‣ DefectOfExactnessAt( C, n ) | ( operation ) |
‣ CohomologyAt( C, n ) | ( operation ) |
‣ HomologyAt( C, n ) | ( operation ) |
Returns: an object
The input is a chain (resp. cochain) complex C and an integer n. The outout is the homology (resp. cohomology) object of C at index n. The input is a (co)chain complex C and an integer n. The outout is the (co)homology object of C at index n.
‣ HomologySupport( C, m, n ) | ( operation ) |
‣ CohomologySupport( C, m, n ) | ( operation ) |
Returns: a list
The input is a chain (resp. cochain) complex C and two integers m,n. The outout is the list of indices where the homology (resp. cohomology) objects of C are not zero.
‣ HomologySupport( C ) | ( operation ) |
‣ CohomologySupport( C ) | ( operation ) |
Returns: a list
The same as above but for bounded complexes.
‣ ObjectsSupport( C, m, n ) | ( operation ) |
‣ DifferentialsSupport( C, m, n ) | ( operation ) |
Returns: a list
The input is a chain (resp. cochain) complex C and two integers m,n. The outout is the list of indices where the objects (resp. differentials) of C are not zero.
‣ ObjectsSupport( C ) | ( operation ) |
‣ DifferentialsSupport( C ) | ( operation ) |
Returns: a list
The same as above but for bounded complexes.
‣ AsComplexOverCapFullSubcategory( A, C ) | ( operation ) |
Returns: an object
The input is a full subcategory A of some category B and a complex C in \mathrm{Ch}(B), where all objects of C actually lie in A. The output is C considered in \mathrm{Ch}(A).
‣ IsWellDefined( C, m, n ) | ( operation ) |
Returns: true or false
The input is a chain (resp. cochain) complex C and two integers m,n. The output is true when C is well defined in the interval [m,\dots,n] and false otherwise.
‣ IsWellDefined( arg ) | ( property ) |
Returns: true or false
‣ IsExactInIndex( C, n ) | ( operation ) |
Returns: true or false
The input is a chain or cochain complex C and an integer n. The outout is true if C is exact in i. Otherwise the output is false.
‣ SetUpperBound( C, n ) | ( operation ) |
Returns: Side effect
The command sets an upper bound n to the chain (resp. cochain) complex C. This means C_{i}=0(C^{i}=0) for i>n. This upper bound will be called \textit{active} upper bound of C. If C already has an active upper bound m, then m will be replaced by n only if n is better upper bound than m, i.e., m>n.
‣ SetLowerBound( C, n ) | ( operation ) |
Returns: Side effect
The command sets an lower bound n to the chain (resp. cochain) complex C. This means C_{i}=0(C^{i}=0) for n>i. This lower bound will be called \textit{active} lower bound of C. If C already has an active lower bound m, then m will be replaced by n only if n is better lower bound than m, i.e., n>m.
‣ HasActiveUpperBound( C ) | ( operation ) |
Returns: true or false
The input is chain or cochain complex. The output is true if an upper bound has been set to C and false otherwise.
‣ HasActiveLowerBound( C ) | ( operation ) |
Returns: true or false
The input is chain or cochain complex. The output is true if a lower bound has been set to C and false otherwise.
‣ ActiveUpperBound( C ) | ( operation ) |
Returns: an integer
The input is chain or cochain complex. The output is its active upper bound if such has been set to C. Otherwise we get error.
‣ ActiveLowerBound( C ) | ( operation ) |
Returns: an integer
The input is chain or cochain complex. The output is its active lower bound if such has been set to C. Otherwise we get error.
‣ DisplayComplex( C, m, n ) | ( operation ) |
Returns: nothing
The input is chain or cochain complex C and two integers m and n. The command displays all components of C between the indices m,n.
‣ ViewComplex( C, m, n ) | ( operation ) |
Returns: nothing
The input is chain or cochain complex C and two integers m and n. The command views all components of C between the indices m,n.
‣ GoodTruncationBelow( C, n ) | ( operation ) |
Returns: chain complex
‣ GoodTruncationAbove( C, n ) | ( operation ) |
Returns: chain complex
‣ GoodTruncationAbove( C, n ) | ( operation ) |
‣ GoodTruncationBelow( C, n ) | ( operation ) |
Returns: cochain complex
‣ BrutalTruncationBelow( C, n ) | ( operation ) |
Returns: chain complex
‣ BrutalTruncationAbove( C, n ) | ( operation ) |
Returns: chain complex
‣ BrutalTruncationAbove( C, n ) | ( operation ) |
Returns: chain complex
‣ BrutalTruncationBelow( C, n ) | ( operation ) |
Returns: chain complex
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