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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