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### 2 Operations

#### 2.1 Functors and natural transformations

##### 2.1-1 ShiftFunctor
 ‣ ShiftFunctor( T ) ( operation )

Returns: a functor $$\mathcal{T}\to\mathcal{T}$$

The argument is a triangulated category $$\mathcal{T}$$. The output is the shift autoequivalence $$\Sigma:\mathcal{T}\to\mathcal{T}$$.

##### 2.1-2 InverseShiftFunctor
 ‣ InverseShiftFunctor( T ) ( attribute )

Returns: a functor $$\mathcal{T}\to\mathcal{T}$$

The argument is a triangulated category $$\mathcal{T}$$. The output is the auto-equivalence $$\Sigma^{-1}:\mathcal{T}\to\mathcal{T}$$.

 ‣ UnitOfShiftAdjunction( T ) ( attribute )

Returns: a natural transformation $$\mathrm{Id}_{\mathcal{T}}\Rightarrow\Sigma\circ\Sigma^{-1}$$

The argument is a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$\eta:\mathrm{Id}_{\mathcal{T}}\Rightarrow\Sigma\circ\Sigma^{-1}$$.

 ‣ InverseOfUnitOfShiftAdjunction( T ) ( attribute )

Returns: a natural isomorphism $$\Sigma\circ\Sigma^{-1}\Rightarrow\mathrm{Id}_{\mathcal{T}}$$

The argument is a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$\eta:\Sigma\circ\Sigma^{-1}\Rightarrow\mathrm{Id}_{\mathcal{T}}$$.

 ‣ CounitOfShiftAdjunction( T ) ( attribute )

Returns: a natural transformation $$\Sigma^{-1}\circ\Sigma\Rightarrow\mathrm{Id}_{\mathcal{T}}$$

The argument is a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$\eta:\Sigma^{-1}\circ\Sigma\Rightarrow\mathrm{Id}_{\mathcal{T}}$$.

 ‣ InverseOfCounitOfShiftAdjunction( T ) ( attribute )

Returns: a natural transformation $$\mathrm{Id}_{\mathcal{T}}\Rightarrow\Sigma^{-1}\circ\Sigma$$

The argument is a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$\eta:\mathrm{Id}_{\mathcal{T}}\Rightarrow\Sigma^{-1}\circ\Sigma$$.

##### 2.1-7 CommutativityNaturalTransformationWithShiftFunctor
 ‣ CommutativityNaturalTransformationWithShiftFunctor( F ) ( attribute )

Returns: a natural transformation $$F\circ \Sigma_{\mathcal{T}_1} \Rightarrow \Sigma_{\mathcal{T}_2} \circ F$$

The argument is an exact functor $$F:\mathcal{T}_1\to\mathcal{T}_2$$ between triangulated categories. The output is a natural isomorphism $$\eta:F\circ \Sigma_{\mathcal{T}_1} \Rightarrow \Sigma_{\mathcal{T}_2} \circ F$$.

##### 2.1-8 ExtendFunctorToCategoryOfTriangles
 ‣ ExtendFunctorToCategoryOfTriangles( F ) ( attribute )

Returns: a natural isomorphism $$F\circ \Sigma_{\mathcal{T}_1} \Rightarrow \Sigma_{\mathcal{T}_2} \circ F$$

The argument is an exact functor $$F:\mathcal{T}_1\to\mathcal{T}_2$$ between triangulated categories, for which the attribute CommutativityNaturalIsomorphismForExactFunctor has already been set. The output is the extension functor of $$F$$ to the categories of triangles over $$\mathcal{T}_1$$ and $$\mathcal{T}_2$$.

##### 2.1-9 RotationFunctor
 ‣ RotationFunctor( T, b ) ( operation )

Returns: an endofunctor $$T\to T$$

The arguments are a category of exact triangles $$T$$ of some triangulated category and a boolian $$b$$. The output is the rotation endofunctor on $$T$$. If $$b$$ = true, then the functor computes witnesses when applied on objects.

##### 2.1-10 InverseRotationFunctor
 ‣ InverseRotationFunctor( T, b ) ( operation )

Returns: an endofunctor $$T\to T$$

The arguments are a category of exact triangles $$T$$ of some triangulated category and a boolian $$b$$. The output is the inverse rotation endofunctor on $$T$$. If $$b$$ = true, then the functor computes witnesses when applied on objects.

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