‣ HomologyFunctor( Ch_\bullet(A), A, n ) | ( operation ) |
‣ CohomologyFunctor( Ch^\bullet(A), A, n ) | ( operation ) |
Returns: a functor
The first argument in the input must be the chain (resp. cochain) complex category of an abelian category \(A\), the second argument is an integer n. The output is the \(n\)'th homology (resp. cohomology) functor \(\mathrm{Ch}_\bullet(A) \rightarrow A\) ( resp. \(\mathrm{Ch}^\bullet(A) \rightarrow A\))
‣ ShiftFunctor( Comp(A), n ) | ( operation ) |
Returns: a functor
The inputs are complex category \(\mathrm{Comp}(A)\) and an integer. The output is a the endofunctor \(T[n]\) that sends any complex \(C\) to \(C[n]\) and any complex morphism \(\phi:C\rightarrow D\) to \(\phi[n]:C[n]\rightarrow D[n]\). The shift chain complex \(C[n]\) of a chain complex \(C\) is defined by \(C[n]_i=C_{n+i}, d_{i}^{C[n]}=(-1)^{n}d_{n+i}^{C}\) and the same for chain complex morphisms, i.e., \(\phi[n]_i=\phi_{n+i}\). The same holds for cochain complexes and morphisms.
‣ UnsignedShiftFunctor( Comp(A), n ) | ( operation ) |
Returns: a functor
The inputs are complex category \(\mathrm{Comp}(A)\) and an integer. The output is a the endofunctor \(T[n]\) that sends any complex \(C\) to \(C[n]\) and any complex morphism \(\phi:C\rightarrow D\) to \(\phi[n]:C[n]\rightarrow D[n]\). The shift chain complex \(C[n]\) of a chain complex \(C\) is defined by \(C[n]_i=C_{n+i}, d_{i}^{C[n]}=d_{n+i}^{C}\) and the same for chain complex morphisms, i.e., \(\phi[n]_i=\phi_{n+i}\). The same holds for cochain complexes and morphisms.
‣ ChainToCochainComplexFunctor( Ch(A)_\bullet, Ch(A)^\bullet ) | ( operation ) |
Returns: a functor
The arguments are \(\mathrm{Ch}_\bullet(A)\) and \(\mathrm{Ch}^\bullet(A)\) for some category \(A\). The output is the functor \(F:\mathrm{Ch}_\bullet(A)\rightarrow\mathrm{Ch}^\bullet(A)\) defined by \(C_{\bullet}\mapsto C^{\bullet}\) for any for any chain complex \(C_{\bullet}\in \mathrm{Ch}_\bullet(A)\) and by \(\phi_{\bullet}\mapsto \phi^{\bullet}\) for any morphism \(\phi_\bullet\) where \(C^\bullet_{i}=C_\bullet^{-i}\) and \(\phi^\bullet_{i}=\phi_\bullet^{-i}\) for any \(i\in\mathbb{Z}\).
‣ CochainToChainComplexFunctor( Ch(A)^\bullet, Ch(A)_\bullet ) | ( operation ) |
Returns: a functor
The arguments are \(\mathrm{Ch}^\bullet(A)\) and \(\mathrm{Ch}_\bullet(A)\) for some category \(A\). The output is the functor \(F:\mathrm{Ch}^\bullet(A)\rightarrow\mathrm{Ch}_\bullet(A)\) defined by \(C^\bullet\mapsto C_\bullet\) for any for any chain complex \(C^\bullet\in \mathrm{Ch}^\bullet(A)\) and by \(\phi^\bullet\mapsto \phi_\bullet\) for any morphism \(\phi^\bullet\) where \(C_\bullet^{i}=C^\bullet_{-i}\) and \(\phi_\bullet^{i}=\phi^\bullet_{-i}\) for any \(i\in\mathbb{Z}\).
‣ ExtendFunctorToChainComplexCategories( F ) | ( attribute ) |
Returns: a functor
The input is a functor \(F:A\rightarrow B\). The output is its extention functor
\[\mathrm{Ch}_\bullet F:\mathrm{Ch}_\bullet(A)\rightarrow \mathrm{Ch}_\bullet(B).\]
‣ ExtendFunctorToCochainComplexCategories( F ) | ( attribute ) |
Returns: a functor
The input is a functor \(F:A\rightarrow B\). The output is its extention functor
\[\mathrm{Ch}^\bullet F:\mathrm{Ch}^\bullet(A)\rightarrow \mathrm{Ch}^\bullet(B)\]
.
‣ BrutalTruncationAboveFunctor( Com(A), n ) | ( operation ) |
Returns: a endofunctor
The input is a complex category \(\mathrm{Com}(A)\) of some Cap category \(A\) and an integer \(n\). The output is an endofunctor from \(\mathrm{Com}(A) \rightarrow \mathrm{Com}(A)\).
‣ BrutalTruncationBelowFunctor( Com(A), n ) | ( operation ) |
Returns: a endofunctor
The input is a complex category \(\mathrm{Com}(A)\) of some Cap category \(A\) and an integer \(n\). The output is an endofunctor from \(\mathrm{Com}(A) \rightarrow \mathrm{Com}(A)\).
‣ GoodTruncationAboveFunctor( A ) | ( operation ) |
Returns: a functor
To do.
‣ ExtendNaturalTransformationToChainComplexCategories( eta ) | ( attribute ) |
Returns: a natural transformation
The input is a natural transformation \(\eta:F\to G\). The output is its extension to the chain complexes.
‣ ExtendNaturalTransformationToCochainComplexCategories( eta ) | ( attribute ) |
Returns: a natural transformation
The input is a natural transformation \(\eta:F\to G\). The output is its extension to the cochain complexes.
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