1.2-8 \[\]
1.2-10 \^
1.2-18 \[\]
‣ IsCategoryOfExactTriangles( T ) | ( filter ) |
Returns: true or false
The GAP category for the category of triangles over some triangulated category.
‣ IsCategoryOfExactTrianglesObject( triangle ) | ( filter ) |
Returns: true or false
The GAP category for exact triangles.
‣ IsCategoryOfExactTrianglesMorphism( mu ) | ( filter ) |
Returns: true or false
The GAP category for morphism of exact triangles
‣ CategoryOfExactTriangles( T ) | ( attribute ) |
Returns: a CAP category
The argument is some triangulated category \(\mathcal{T}\). The output is the category of exact triangles over \(\mathcal{T}\).
‣ UnderlyingCategory( C ) | ( attribute ) |
Returns: a CAP category
The argument is a category of triangles over some triangulated category \(\mathcal{T}\). The output is \(\mathcal{T}\).
‣ ExactTriangle( alpha, iota, pi ) | ( operation ) |
Returns: an exact triangle
The arguments are three morphisms \(\alpha:A\to B\), \(\iota:B\to C\) and \(\pi:C\to\Sigma A\) in some triangulated category \(\mathcal{T}\). The output the exact triangle \(A\to B \to C \to \Sigma A\) defined by them.
‣ DomainMorphism( t ) | ( attribute ) |
Returns: a morphism
The arguments is an exact triangle defined by three morphisms \(\alpha:A\to B\), \(\iota:B\to C\) and \(\pi:C\to\Sigma A\). The output is \(\alpha:A\to B\).
‣ MorphismIntoConeObject( t ) | ( attribute ) |
Returns: a morphism
The arguments is an exact triangle defined by three morphisms \(\alpha:A\to B\), \(\iota:B\to C\) and \(\pi:C\to\Sigma A\). The output is \(\iota:B\to C\).
‣ MorphismFromConeObject( t ) | ( attribute ) |
Returns: a morphism
The arguments is an exact triangle defined by three morphisms \(\alpha:A\to B\), \(\iota:B\to C\) and \(\pi:C\to\Sigma A\). The output is \(\pi:C\to\Sigma A\).
‣ ObjectAt( t, i ) | ( operation ) |
Returns: an object
The arguments is an exact triangle defined by three morphisms \(\alpha:A\to B\), \(\iota:B\to C\) and \(\pi:C\to\Sigma A\) and an integer \(i\in\{0,1,2,3\}\). The output is \(A\) if \(i=0\), \(B\) if \(i=1\), \(C\) if \(i=2\) and \(\Sigma A\) if \(i=3\).
1.2-8 \[\]‣ \[\]( t, i ) | ( operation ) |
Returns: an object
Delegates to the operation ObjectAt.
‣ MorphismAt( t, i ) | ( operation ) |
Returns: a morphism
The arguments is an exact triangle defined by three morphisms \(\alpha:A\to B\), \(\iota:B\to C\) and \(\pi:C\to\Sigma A\) and an integer \(i\in\{0,1,2\}\). The output is \(\alpha\) if \(i=0\), \(\iota\) if \(i=1\), \(\pi\) if \(i=2\).
1.2-10 \^‣ \^( t, i ) | ( operation ) |
Returns: a morphism
Delegates to the operation MorphismAt.
‣ StandardExactTriangle( alpha ) | ( attribute ) |
Returns: an standard exact triangle
The arguments is a morphism \(\alpha:A\to B\) in some triangulated category \(\mathcal{T}\). The output the standard exact triangle \(A\to B \to C(\alpha) \to \Sigma A\) defined by \(\alpha\).
‣ StandardExactTriangle( t ) | ( attribute ) |
Returns: an standard exact triangle
The arguments is an exact triangle \(t=(\alpha,\iota,\pi)\). The output the standard exact triangle \((\alpha,\iota(\alpha),\pi(\alpha))\).
‣ IsStandardExactTriangle( t ) | ( property ) |
Returns: true or false
The argument is an exact triangle \(t=(\alpha,\iota,\pi)\). The operation checks whether \(t\) is standard exact triangle or not. I.e., it checks whether \(\iota=\iota(\alpha)\) and \(\pi=\pi(\alpha)\).
‣ WitnessIsomorphismIntoStandardExactTriangle( t ) | ( attribute ) |
Returns: a morphism of triangles
The argument is an exact triangle \(t=(\alpha,\iota,\pi)\). The output is an isomorphism of triangles from \(t\) into the standard exact triangle \((\alpha,\iota(\alpha),\pi(\alpha))\).
‣ WitnessIsomorphismFromStandardExactTriangle( t ) | ( attribute ) |
Returns: a morphism of triangles
The argument is an exact triangle \(t=(\alpha,\iota,\pi)\). The output is an isomorphism of triangles from the standard exact triangle \((\alpha,\iota(\alpha),\pi(\alpha))\) into t. This isomorphism is equal to the inverse of the witness isomorphism into the standard exact triangle.
‣ MorphismOfExactTriangles( t_1, mu_0, mu_1, mu_2, t_2 ) | ( operation ) |
Returns: a morphism \(t_1\to t_2\)
The arguments are an exact triangle \(t_1\), three morphisms \(\mu_0:t_1[0]\to t_2[0]\), \(\mu_1:t_1[1]\to t_2[1]\), \(\mu_2:t_1[2]\to t_2[2]\) and an exact triangle \(t_2\). The output is the morphism of exact triangles from \(t_1\to t_2\) defined by these morphisms.
‣ MorphismAt( phi, i ) | ( operation ) |
Returns: a morphism
The arguments is a morphism \(\mu:t_1\to t_2\) of exact triangles defined by three morphisms \(\mu_0:t_1[0]\to t_2[0]\), \(\mu_1:t_1[1]\to t_2[1]\) and \(\mu_2:t_1[2]\to t_2[2]\); and an integer \(i\in\{0,1,2\}\). The output is \(\mu_0\) if \(i=0\), \(\mu_1\) if \(i=1\), \(\mu_2\) if \(i=2\).
1.2-18 \[\]‣ \[\]( phi, i ) | ( operation ) |
Returns: a morphism
Delegates to the operation MorphismAt.
‣ MorphismBetweenConeObjects( t_1, mu_0, mu_1, t_2 ) | ( operation ) |
Returns: a morphism
The arguments are an exact triangle \(t_1\), two morphisms \(\mu_0:t_1[0]\to t_2[0]\), \(\mu_1:t_1[1]\to t_2[1]\), and an exact triangle \(t_2\). The output is some morphism \(\mu_2:t_1[2]\to t_2[2]\) such that \((t_1,\mu_0,\mu_1,\mu_2,t_2)\) is a morphism of exact triangles.
‣ MorphismOfExactTriangles( t_1, mu_0, mu_1, t_2 ) | ( operation ) |
Returns: a morphism
The arguments are an exact triangle \(t_1\), two morphisms \(\mu_0:t_1[0]\to t_2[0]\), \(\mu_1:t_1[1]\to t_2[1]\), and an exact triangle \(t_2\). The output is some morphism of exact triangles \((t_1,\mu_0,\mu_1,\mu_2,t_2)\). The morphism \(\mu_2\) will be computed by using the operation MorphismBetweenConeObjects.
‣ ExactTriangleByOctahedralAxiom( alpha, beta, gamma ) | ( operation ) |
Returns: a triangle
The arguments are two morphisms \(\alpha:A\to B\), \(\beta:B\to C\). The output is the exact triangle defined by the Octhedral axiom.
‣ ExactTriangleByOctahedralAxiom( alpha, beta, gamma, b ) | ( operation ) |
Returns: a triangle
The arguments are two morphisms \(\alpha:A\to B\), \(\beta:B\to C\) and a boolian \(b\). The output is the exact triangle defined by the Octhedral axiom. If \(b\)=true, then the operation will compute a witness isomorphism into the standard exact triangle.
‣ ExactTriangleByOctahedralAxiom( t_1, t_2, b ) | ( operation ) |
Returns: a triangle
The arguments are two exact triangles \(t_1,t_2\) such that \(t_1[1]=t_2[0]\) and a boolian \(b\). The output is ExactTriangleByOctahedralAxiom\((t_1,t_2)\). If \(b\) = true then the operation will compute a witness isomorphism into the standard exact triangle.
‣ ExactTriangleByOctahedralAxiom( t_1, t_2, t_3 ) | ( operation ) |
Returns: a triangle
The arguments are three exact triangles \(t_1,t_2,t_3\) such that \(t_1[1]=t_2[0]\), \(t_1[0]=t_3[0]\), \(t_2[1]=t_3[1]\) and \(t_2^0\circ t_1^0=t_3^0\). The output is the exact triangle defined by the Octahedral axiom.
‣ Rotation( t ) | ( attribute ) |
Returns: a triangle
The argument is an exact triangle \(t=(\alpha,\iota,\pi)\). The output is the exact triangle defined by the rotation axiom, i.e., the exact triangle \((\iota,\pi,-\Sigma \alpha)\).
‣ Rotation( t, b ) | ( operation ) |
Returns: a triangle
The arguments are an exact triangle \(t=(\alpha,\iota,\pi)\) and a boolian \(b\). The output is Rotation\((t)\). If \(b\)=true, then the operation will compute a witness isomorphism into the standard exact triangle.
‣ InverseRotation( t ) | ( attribute ) |
Returns: a triangle
The argument is an exact triangle \(t=(\alpha,\iota,\pi)\). The output is the exact triangle defined by the inverse rotation axiom, i.e., the exact triangle \((-\eta(A)\circ\Sigma^{-1}\pi,\alpha,\mu(C)\circ\iota)\), such that \(\eta\) := CounitIsomorphism, \(\mu\) := UnitIsomorphism, \(A\) := Source\((\alpha)\) and \(C\) := Range\((\iota)\).
‣ InverseRotation( t, bool ) | ( operation ) |
Returns: a triangle
The arguments are an exact triangle \(t=(\alpha,\iota,\pi)\) and a boolian \(b\). The output is InverseRotation\((t)\). If \(b\)=true, then the operation will compute a witness isomorphism into the standard exact triangle.
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