‣ 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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