‣ IsAbelianCategoryWithComputableEnoughProjectives( C ) | ( property ) |
Returns: true or false
The argument is a category C. The output is whether the C is abelian and have the following methods IsProjective, SomeProjectiveObject, EpimorphismFromSomeProjectiveObject and ProjectiveLift are installed.
‣ IsAbelianCategoryWithComputableEnoughInjectives( C ) | ( property ) |
Returns: true or false
The argument is a category C. The output is whether the C is abelian and have the following methods IsInjective, SomeInjectiveObject, MonomorphismIntoSomeInjectiveObject and InjectiveColift are installed.
‣ ProjectiveResolution( C ) | ( attribute ) |
Returns: a (co)chain complex
If the input is bounded above cochain complex or bounded below chain complex then the output is projective resolution in the sense of the above definition. If the input is an object \(M\) which is not a complex and its category has enough projectives, then the output is its projective resolution in the classical sense , i.e., complex \(P^\bullet\) which is exact everywhere but in index \(0\), where \(H^0(P^\bullet)\cong M\).
‣ ProjectiveResolution( C, bool ) | ( operation ) |
Returns: a (co)chain complex
The arguments are an object C and a boolian bool. If bool = false then the ouput is ProjectiveResolution(C), otherwise the output is ProjectiveResolution(C) after trying to find its bounds.
‣ ProjectiveCochainResolution( M ) | ( attribute ) |
Returns: a cochain complex
The input is an object \(M\) in an abelian category with enough projectives. The output is ProjectiveResolution(M) as a cochain complex.
‣ ProjectiveCochainResolution( M ) | ( operation ) |
Returns: a cochain complex
The input is an object \(M\) in an abelian category with enough projectives. The output is ProjectiveResolution(M) as a cochain complex after trying to set its bounds.
‣ ProjectiveChainResolution( M ) | ( attribute ) |
Returns: a chain complex
The input is an object \(M\) in an abelian category with enough projectives. The output is ProjectiveResolution(M) as a chain complex.
‣ ProjectiveChainResolution( M ) | ( operation ) |
Returns: a chain complex
The input is an object \(M\) in an abelian category with enough projectives. The output is ProjectiveResolution(M) as a chain complex after trying to set its bounds.
‣ MorphismBetweenProjectiveResolutions( alpha ) | ( attribute ) |
Returns: a (co)chain morphism
The input is a morphism \(\alpha\) whose category is abelian with enough projectives. The output is the induced cochain morphism between the projective resolutions of the source and range of \(\alpha\). This morphism is unique up to homotopy.
‣ MorphismBetweenProjectiveResolutions( alpha, bool ) | ( operation ) |
Returns: a (co)chain morphism
The arguments are a morphism \(\alpha\) and a boolian bool. If bool = false then the ouput is MorphismBetweenProjectiveResolutions(\(\alpha\)), otherwise the output is MorphismBetweenProjectiveResolutions(\(\alpha\)) after trying to find its bounds.
‣ MorphismBetweenProjectiveCochainResolutions( alpha ) | ( attribute ) |
Returns: a cochain morphism
#TODO
‣ MorphismBetweenProjectiveCochainResolutions( alpha, bool ) | ( operation ) |
Returns: a cochain morphism
#TODO
‣ MorphismBetweenProjectiveChainResolutions( alpha ) | ( attribute ) |
Returns: a chain morphism
#TODO
‣ MorphismBetweenProjectiveChainResolutions( alpha, bool ) | ( operation ) |
Returns: a chain morphism
#TODO
‣ InjectiveResolution( arg ) | ( attribute ) |
Returns: a (co)chain complex
If the input is bounded above chain complex or bounded below cochain complex then the output is injective resolution in the sense of the above definition. If the input is an object \(M\) which is not a complex and its category has enough injectives, then the output is its injective resolution in the classical sense , i.e., complex \(I^\bullet\) which is exact everywhere but in index \(0\), where \(H^0(I^\bullet)\cong M\).
‣ InjectiveResolution( C, bool ) | ( operation ) |
Returns: a (co)chain complex
The arguments are an object C and a boolian bool. If bool = false then the ouput is InjectiveResolution(C), otherwise the output is InjectiveResolution(C) after trying to find its bounds.
‣ InjectiveCochainResolution( M ) | ( attribute ) |
Returns: a cochain complex
The input is an object \(M\) in an abelian category with enough injectives. The output is InjectiveResolution(M) as a cochain complex.
‣ InjectiveCochainResolution( M ) | ( operation ) |
Returns: a cochain complex
The input is an object \(M\) in an abelian category with enough injectives. The output is InjectiveResolution(M) as a cochain complex after trying to set its bounds.
‣ InjectiveChainResolution( M ) | ( attribute ) |
Returns: a chain complex
The input is an object \(M\) in an abelian category with enough injectives. The output is InjectiveResolution(M) as a chain complex after trying to set its bounds.
‣ InjectiveChainResolution( M ) | ( operation ) |
Returns: a chain complex
The input is an object \(M\) in an abelian category with enough injectives. The output is InjectiveResolution(M) as a chain complex after trying to set its bounds.
‣ MorphismBetweenInjectiveResolutions( alpha ) | ( attribute ) |
Returns: a (co)chain morphism
The input is a morphism \(\alpha\) whose category is abelian with enough injectives. The output is the induced cochain morphism between the injective resolutions of the source and range of \(\alpha\). This morphism is unique up to homotopy.
‣ MorphismBetweenInjectiveResolutions( alpha, bool ) | ( operation ) |
Returns: a (co)chain morphism
The arguments are a morphism \(\alpha\) and a boolian bool. If bool = false then the ouput is MorphismBetweenInjectiveResolutions(\(\alpha\)), otherwise the output is MorphismBetweenInjectiveResolutions(\(\alpha\)) after trying to find its bounds.
‣ MorphismBetweenInjectiveCochainResolutions( alpha ) | ( attribute ) |
Returns: a cochain morphism
#TODO
‣ MorphismBetweenInjectiveCochainResolutions( alpha, bool ) | ( operation ) |
Returns: a cochain morphism
#TODO
‣ MorphismBetweenInjectiveChainResolutions( alpha ) | ( attribute ) |
Returns: a chain morphism
#TODO
‣ MorphismBetweenInjectiveChainResolutions( alpha, bool ) | ( operation ) |
Returns: a chain morphism
#TODO
‣ QuasiIsomorphismFromProjectiveResolution( C ) | ( attribute ) |
‣ QuasiIsomorphismFromProjectiveResolution( C ) | ( attribute ) |
Returns: a (co)chain epimorphism
The input is an above bounded cochain complex \(C^\bullet\). The output is a quasi-isomorphism \(q:P^\bullet \rightarrow C^\bullet\) such that \(P^\bullet\) is upper bounded and all its objects are projective in the underlying abelian category. In the second command the input is a below bounded chain complex \(C_\bullet\). The output is a quasi-isomorphism \(q:P_\bullet \rightarrow C_\bullet\) such that \(P_\bullet\) is lower bounded and all its objects are projective in the underlying abelian category.
‣ QuasiIsomorphismFromProjectiveResolution( C ) | ( operation ) |
Returns: a bounded (co)chain epimorphism
The input is chain or cochain complex and the output is a quasi-isomorphism from its projective resolution, after trying to find its bounds.
‣ QuasiIsomorphismIntoInjectiveResolution( C ) | ( attribute ) |
‣ QuasiIsomorphismIntoInjectiveResolution( C ) | ( attribute ) |
Returns: a (co)chain epimorphism
The input is a below bounded cochain complex \(C^\bullet\). The output is a quasi-isomorphism \(q:C^\bullet \rightarrow I^\bullet\) such that \(I^\bullet\) is below bounded and all its objects are injective in the underlying abelian category. In the second command the input is an above bounded chain complex \(C_\bullet\). The output is a quasi-isomorphism \(q: C_\bullet\rightarrow I_\bullet\) such that \(I_\bullet\) is below bounded and all its objects are injective in the underlying abelian category.
‣ QuasiIsomorphismIntoInjectiveResolution( C ) | ( operation ) |
Returns: a bounded (co)chain epimorphism
The input is chain or cochain complex and the output is a quasi-isomorphism into its injective resolution, after trying to find its bounds.
‣ MorphismFromHorseshoeResolution( C ) | ( attribute ) |
Returns: chain morphism of chain complexes
The input is a short exact sequence defined as a chain complex and the output is a chain morphism from the Horseshoe resolution (which is a complex of complexes and each object in this complex is again a complex that consists of a short exact sequence of projective objects). The total complex of the resolution is quasi isomorphic to \(C\) and both are exact complexes.
‣ HorseshoeResolution( C ) | ( attribute ) |
Returns: chain complex of chain complexes
The input is a short exact sequence defined as a chain complex and the output is the source of the morphism from Horseshoe resolution.
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