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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