‣ DoubleChainComplex( A, rows, cols ) | ( operation ) |
Returns: a double chain complex
The input is a Cap category \mathcal{A} and two \mathbb{Z}-functions rows and cols. The entry in index j of rows should be an \mathbb{Z}-function that represents the j'th row of the double complex. I.e., h^D_{i,j}:= rows[j][i] for all i\in\mathbb{Z}. Again, the entry in index i of cols should be an \mathbb{Z}-function that represents the i'th column of the double complex. I.e., v^D_{i,j}:=cols[i][j].
‣ DoubleChainComplex( A, H, V ) | ( operation ) |
Returns: a double chain complex
The input is a Cap category \mathcal{A} and two functions R and V. The output is the double chain complex D defined by h^D_{i,j}=H(i,j) and v^D_{i,j}=V(i,j).
‣ DoubleChainComplex( C ) | ( operation ) |
Returns: a double chain complex
The input is chain complex of chain complexes C. The output is the double chain complex D defined using sign trick. I.e., h^D_{i,j}=(d^C_i)_j and v^D_{i,j}=(-1)^id^{C_i}_j.
‣ DoubleChainComplex( C ) | ( operation ) |
Returns: a double chain complex
The input is double cochain complex D. The output is the double chain complex E defined by h^E_{i,j}=h_D^{-i,-j} and v^E_{i,j}=v_D^{-i,-j}.
‣ DoubleCochainComplex( A, rows, cols ) | ( operation ) |
Returns: a double cochain complex
The input is a Cap category \mathcal{A} and two \mathbb{Z}-functions rows and cols. The entry in index j of rows should be an \mathbb{Z}-function that represents the j'th row of the double complex. I.e., h_D^{i,j}:= rows[j][i] for all i\in\mathbb{Z}. Again, the entry in index i of cols should be an \mathbb{Z}-function that represents the i'th column of the double complex. I.e., v_D^{i,j}:=cols[i][j].
‣ DoubleCochainComplex( A, H, V ) | ( operation ) |
Returns: a double cochain complex
The input is a Cap category \mathcal{A} and two functions R and V. The output is the double chain complex D defined by h_D^{i,j}=H(i,j) and v_D^{i,j}=V(i,j).
‣ DoubleCochainComplex( C ) | ( operation ) |
Returns: a double cochain complex
The input is cochain complex of cochain complexes C. The output is the double cochain complex D defined using sign trick. I.e., h_D^{i,j}=(d_C^i)^j and v_D^{i,j}=(-1)^id_{C^i}^j.
‣ DoubleCochainComplex( C ) | ( operation ) |
Returns: a double cochain complex
The input is double chain complex D. The output is the double cochain complex E defined by h_E^{i,j}=h^D_{-i,-j} and v_E^{i,j}=v^D_{-i,-j}.
‣ Rows( D ) | ( attribute ) |
Returns: an \mathbb{Z}-function of \mathbb{Z}-functions.
The input is double chain or cochain complex D. The output is the \mathbb{Z}-function of rows.
‣ Columns( D ) | ( attribute ) |
Returns: an \mathbb{Z}-function of \mathbb{Z}-functions.
The input is double chain or cochain complex D. The output is the \mathbb{Z}-function of columns.
‣ CertainRow( D, j ) | ( operation ) |
Returns: an \mathbb{Z}-function
The input is double chain or cochain complex D and integer j. The output is the \mathbb{Z}-function that represents the j'th row of D.
‣ CertainColumn( D, i ) | ( operation ) |
Returns: an \mathbb{Z}-function
The input is double chain or cochain complex D and integer i. The output is the \mathbb{Z}-function that represents the i'th column of D.
‣ ObjectAt( D, i, j ) | ( operation ) |
Returns: an \mathbb{Z}-function
The input is double chain or cochain complex D and integers i,j. The output is the object of D in position (i,j).
‣ HorizontalDifferentialAt( D, i, j ) | ( operation ) |
Returns: a morphism
The input is double chain (resp. cochain) complex D and integers i,j. The output is the horizontal differential h^D_{i,j} (resp. h_D^{i,j})
‣ VerticalDifferentialAt( D, i, j ) | ( operation ) |
Returns: a morphism
The input is double chain (resp. cochain) complex D and integers i,j. The output is the vertical differential v^D_{i,j} (resp. v_D^{i,j})
‣ SetAboveBound( D, i ) | ( operation ) |
‣ SetBelowBound( D, i ) | ( operation ) |
‣ SetRightBound( D, i ) | ( operation ) |
‣ SetLeftBound( D, i ) | ( operation ) |
Returns: a morphism
Here we can set bounds for the double complex.
‣ TotalComplex( D ) | ( attribute ) |
Returns: a morphism
To be able to compute the total complex the double complex we must have one of the following cases: 1. D has left and right bounds. 2. D has below and above bounds. 3. D has left and below bounds. 4. D has right and above bounds.
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