‣ IsChainOrCochainMorphism( phi ) | ( filter ) |
‣ IsBoundedBelowChainOrCochainMorphism( phi ) | ( filter ) |
‣ IsBoundedAboveChainOrCochainMorphism( phi ) | ( filter ) |
‣ IsBoundedChainOrCochainMorphism( phi ) | ( filter ) |
‣ IsChainMorphism( phi ) | ( filter ) |
‣ IsBoundedBelowChainMorphism( phi ) | ( filter ) |
‣ IsBoundedAboveChainMorphism( phi ) | ( filter ) |
‣ IsBoundedChainMorphism( phi ) | ( filter ) |
‣ IsCochainMorphism( phi ) | ( filter ) |
‣ IsBoundedBelowCochainMorphism( phi ) | ( filter ) |
‣ IsBoundedAboveCochainMorphism( phi ) | ( filter ) |
‣ IsBoundedCochainMorphism( phi ) | ( filter ) |
Returns: true or false
Gap-categories for chain and cochains morphisms.
‣ ChainMorphism( C, D, l ) | ( operation ) |
Returns: a chain morphism
The input is two chain complexes \(C,D\) and a \(\mathbb{Z}\)-function \(l\). The output is the chain morphism \(\phi:C\rightarrow D\) defined by \(\phi_i :=l[i]\).
‣ ChainMorphism( C, D, l, k ) | ( operation ) |
Returns: a chain morphism
The input is two chain complexes \(C,D\), dense list \(l\) and an integer \(k\). The output is the chain morphism \(\phi:C\rightarrow D\) such that \(\phi_{k}=l[1]\), \(\phi_{k+1}=l[2]\), etc.
‣ ChainMorphism( c, m, d, n, l, k ) | ( operation ) |
Returns: a chain morphism
The output is the chain morphism \(\phi:C\rightarrow D\), where \(d^C_m = c[ 1 ], d^C_{m+1} =c[ 2 ],\) etc. \(d^D_n = d[ 1 ], d^D_{n+1} =d[ 2 ],\) etc. and \(\phi_{k}=l[1]\), \(\phi_{k+1}=l[2]\), etc.
‣ CochainMorphism( C, D, l ) | ( operation ) |
Returns: a cochain morphism
The input is two cochain complexes \(C,D\) and a \(\mathbb{Z}\)-function \(l\). The output is the cochain morphism \(\phi:C\rightarrow D\) defined by \(\phi_i :=l[i]\).
‣ CochainMorphism( C, D, l, k ) | ( operation ) |
Returns: a chain morphism
The input is two cochain complexes \(C,D\), dense list \(l\) and an integer \(k\). The output is the cochain morphism \(\phi:C\rightarrow D\) such that \(\phi^{k}=l[1]\), \(\phi^{k+1}=l[2]\), etc.
‣ CochainMorphism( c, m, d, n, l, k ) | ( operation ) |
Returns: a cochain morphism
The output is the cochain morphism \(\phi:C\rightarrow D\), where \(C^m = c[ 1 ], C^{m+1} =c[ 2 ],\) etc. \(D^n = d[ 1 ], D^{n+1} =d[ 2 ],\) etc. and \(\phi^{k}=l[1]\), \(\phi^{k+1}=l[2]\), etc.
‣ StalkChainMorphism( f, n ) | ( operation ) |
‣ StalkCochainMorphism( f, n ) | ( operation ) |
Returns: a (co)chain morphism
The input is a morphism \(f:a\rightarrow b\) in a category \(A\). The output is chain (resp. cochain) morphism \(f_{\bullet}\in\mathrm{Ch}_\bullet(A)(f^{\bullet}\in\mathrm{Ch}^\bullet(A))\) where \(f^{\bullet}_n=f( f_{\bullet}^n=f)\) and \(f^{\bullet}_i=0(f_{\bullet}^i=0)\) whenever \(i\neq n\).
‣ HomologyFunctorialAt( phi, n ) | ( operation ) |
‣ CohomologyFunctorialAt( phi, n ) | ( operation ) |
Returns: a morphism
The input is a (co)chain complex morphism \(\phi:C\to D\) and an integer \(n\). The outout is the associated morphism between the (co)homology objects of \(C\) resp. \(D\) at index \(n\).
‣ Morphisms( phi ) | ( attribute ) |
Returns: \(\mathbb{Z}\)-function
The output is morphisms of the chain or cochain morphism as a \(\mathbb{Z}\)-function.
‣ AsChainMorphism( phi ) | ( attribute ) |
Returns: a chain morphism
The input is a cochain morphism \(\phi\) and the output is the associated chain morphism.
‣ AsCochainMorphism( phi ) | ( attribute ) |
Returns: a cochain morphism
The input is a chain morphism \(\phi\) and the output is the associated cochain morphism.
‣ MappingCone( phi ) | ( attribute ) |
Returns: complex
The input a chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is its mapping cone chain (resp. cochain) complex \(\mathrm{Cone}(\phi )\).
‣ NaturalInjectionInMappingCone( phi ) | ( attribute ) |
Returns: chain (resp. cochain) morphism
The input a chain (resp. cochain) morphism \(\phi:C\rightarrow D\). The output is the natural injection \(i:D\rightarrow \mathrm{Cone}(\phi )\).
‣ NaturalProjectionFromMappingCone( phi ) | ( attribute ) |
Returns: chain (resp. cochain) morphism
The input a chain ( resp. cochain) morphism \(\phi:C\rightarrow D\). The output is the natural projection \(\pi:\mathrm{Cone}(\phi ) \rightarrow C[u]\) where \(u=-1\) if \(\phi\) is chain morphism and \(u=1\) if \(\phi\) is cochain morphism.
‣ MappingCylinder( phi ) | ( attribute ) |
Returns: complex
The input a chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is its mapping cylinder chain (resp. cochain) complex
‣ NaturalInjectionOfSourceInMappingCylinder( phi ) | ( attribute ) |
Returns: morphism
The input a chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is the natural embedding \(C\rightarrow \mathrm{Cyl}(\phi )\). I.e., the composition
‣ NaturalInjectionOfRangeInMappingCylinder( phi ) | ( attribute ) |
Returns: morphism
The input a chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is the natural embedding \(D \rightarrow \mathrm{Cyl}(\phi )\). I.e., the composition This morphism can be proven to be quasi-isomorphism. See Weibel, page 21.
‣ NaturalMorphismFromMappingCylinderInRange( phi ) | ( attribute ) |
Returns: morphism
The input a chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is the natural morphism \(\mathrm{Cyl}(\phi )\rightarrow D\). It can be shown that \(D\) and \(\mathrm{Cyl}(\phi )\) are homotopy equivalent. This homotopy equivalence is given by the two morphisms
\[\mathrm{NaturalInjectionOfRangeInMappingCylinder}(\phi):D \rightarrow \mathrm{Cyl}(\phi )\]
and
\[\mathrm{NaturalMorphismFromMappingCylinderInRange}(\phi):\mathrm{Cyl}(\phi ) \rightarrow D.\]
See Weibel, page 21.
‣ NaturalMorphismFromMappingCylinderInMappingCone( phi ) | ( attribute ) |
Returns: morphism
The input a chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is the natural morphism \(\mathrm{Cyl}(\phi )\rightarrow \mathrm{Cone}(\phi )\). It can be shown that
\[0 \rightarrow C\xrightarrow[]{\alpha} \mathrm{Cyl}(\phi ) \xrightarrow[]{\beta} \mathrm{Cone}(\phi )\rightarrow 0\]
where
\[\alpha := \mathrm{NaturalInjectionOfSourceInMappingCylinder}(\phi)\]
and
\[\beta := \mathrm{NaturalMorphismFromMappingCylinderInMappingCone}(\phi),\]
is a short exact sequence. See Weibel, page 21.
‣ HomotopyMorphisms( phi ) | ( attribute ) |
Returns: A ZFunction
The input is a null-homotopic chain (resp. cochain) morphism \(\phi:C \rightarrow D\). The output is the homotopy morphisms given as a \(\mathbb{Z}\)-function \((h_i:C_i \rightarrow D_{i+1})\) ( resp. \((h_i:C_i \rightarrow D_{i-1})\) ). Note that this method can be called only if it is added to the category, see \(\texttt{AddHomotopyMorphisms}\).
‣ IsQuasiIsomorphism( phi ) | ( property ) |
Returns: true or false
The input a chain ( resp. cochain) morphism \(\phi:C\rightarrow D\). The output is true if \(\phi\) is quasi-isomorphism and false otherwise. If \(\phi\) is not bounded an error is raised.
‣ IsNullHomotopic( phi ) | ( property ) |
Returns: true or false
The input is a chain or cochain morphism \(\phi\) and output is true if \(\phi\) is null-homotopic and false otherwise.
‣ SetUpperBound( phi, n ) | ( operation ) |
Returns: a side effect
The command sets an upper bound to the morphism \(\phi\). An upper bound of \(\phi\) is an integer \(u\) with \(\phi_{i}= 0\) for \(i>n\). The integer \(u\) will be called active upper bound of \(\phi\). If \(\phi\) already has an active upper bound, say \(u^\prime\), then \(u^\prime\) will be replaced by \(u\) only if \(u^\prime>u\).
‣ SetLowerBound( phi, n ) | ( operation ) |
Returns: a side effect
The command sets an lower bound to the morphism \(\phi\). A lower bound of \(\phi\) is an integer \(l\) with \(\phi_{i}= 0\) for \(l>i\). The integer \(l\) will be called active lower bound of \(\phi\). If \(\phi\) already has an active lower bound, say \(l^\prime\), then \(l^\prime\) will be replaced by \(l\) only if \(l>l^\prime\).
‣ HasActiveUpperBound( phi ) | ( operation ) |
Returns: true or false
The input is chain or cochain morphism \(\phi\). The output is true if an upper bound has been set to \(\phi\) and false otherwise.
‣ HasActiveLowerBound( phi ) | ( operation ) |
Returns: true or false
The input is chain or cochain morphism \(\phi\). The output is true if a lower bound has been set to \(\phi\) and false otherwise.
‣ ActiveUpperBound( phi ) | ( operation ) |
Returns: an integer
The input is chain or cochain morphism. The output is its active upper bound if such has been set to \(\phi\). Otherwise we get error.
‣ ActiveLowerBound( phi ) | ( operation ) |
Returns: an integer
The input is chain or cochain morphism. The output is its active lower bound if such has been set to \(\phi\). Otherwise we get error.
‣ MorphismAt( phi, n ) | ( operation ) |
Returns: a morphism
The input is chain (resp. cochain) morphism and an integer \(n\). The output is the component of \(\phi\) in index \(n\), i.e., \(\phi_n\)(resp. \(\phi^n\)).
‣ CyclesFunctorialAt( phi, n ) | ( operation ) |
Returns: a morphism
The input is chain (resp. cochain) morphism and an integer \(n\). The output is the morphism between the kernels in index \(n\).
3.5-9 \[\]‣ \[\]( phi, n ) | ( operation ) |
Returns: an integer
The input is chain (resp. cochain) morphism and an integer \(n\). The output is the component of \(\phi\) in index \(n\), i.e., \(\phi_n\)(resp. \(\phi^n\)).
‣ IsQuasiIsomorphism( phi, n ) | ( operation ) |
Returns: an integer
The input is chain (resp. cochain) morphism and an integer \(n\). The output is the component of \(\phi\) in index \(n\), i.e., \(\phi_n\)(resp. \(\phi^n\)).
‣ MorphismsSupport( phi, m, n ) | ( operation ) |
The command gives back the list of indices between \(m\) and \(n\) where the complex morphism is not zero.
‣ Display( phi, m, n ) | ( operation ) |
The command displays the components of the morphism between \(m\) and \(n\).
‣ IsWellDefined( true, or, false ) | ( operation ) |
The command checks if the morphism is well defined between \(m\) and \(n\).
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