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### 1 Category of triangles

#### 1.1 Gap categories

##### 1.1-1 IsCategoryOfExactTriangles
 ‣ IsCategoryOfExactTriangles( T ) ( filter )

Returns: true or false

The GAP category for the category of triangles over some triangulated category.

##### 1.1-2 IsCategoryOfExactTrianglesObject
 ‣ IsCategoryOfExactTrianglesObject( triangle ) ( filter )

Returns: true or false

The GAP category for exact triangles.

##### 1.1-3 IsCategoryOfExactTrianglesMorphism
 ‣ IsCategoryOfExactTrianglesMorphism( mu ) ( filter )

Returns: true or false

The GAP category for morphism of exact triangles

#### 1.2 Constructors

##### 1.2-1 CategoryOfExactTriangles
 ‣ 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}.

##### 1.2-2 UnderlyingCategory
 ‣ 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}.

##### 1.2-3 ExactTriangle
 ‣ 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.

##### 1.2-4 DomainMorphism
 ‣ 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.

##### 1.2-5 MorphismIntoConeObject
 ‣ 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.

##### 1.2-6 MorphismFromConeObject
 ‣ 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.

##### 1.2-7 ObjectAt
 ‣ 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.

##### 1.2-9 MorphismAt
 ‣ 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.

##### 1.2-11 StandardExactTriangle
 ‣ 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.

##### 1.2-12 StandardExactTriangle
 ‣ 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)).

##### 1.2-13 IsStandardExactTriangle
 ‣ 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).

##### 1.2-14 WitnessIsomorphismIntoStandardExactTriangle
 ‣ 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)).

##### 1.2-15 WitnessIsomorphismFromStandardExactTriangle
 ‣ 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.

##### 1.2-16 MorphismOfExactTriangles
 ‣ 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.

##### 1.2-17 MorphismAt
 ‣ 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.

##### 1.2-19 MorphismBetweenConeObjects
 ‣ 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.

##### 1.2-20 MorphismOfExactTriangles
 ‣ 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.

##### 1.2-21 ExactTriangleByOctahedralAxiom
 ‣ 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.

##### 1.2-22 ExactTriangleByOctahedralAxiom
 ‣ 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.

##### 1.2-23 ExactTriangleByOctahedralAxiom
 ‣ 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.

##### 1.2-24 ExactTriangleByOctahedralAxiom
 ‣ 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.

##### 1.2-25 Rotation
 ‣ 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).

##### 1.2-26 Rotation
 ‣ 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.

##### 1.2-27 InverseRotation
 ‣ 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).

##### 1.2-28 InverseRotation
 ‣ 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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