‣ 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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