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2 Operations

2.1 Functors and natural transformations

2.1-1 ShiftFunctor

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

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}.

2.1-3 UnitOfShiftAdjunction

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}.

2.1-4 InverseOfUnitOfShiftAdjunction

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}}.

2.1-5 CounitOfShiftAdjunction

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}}.

2.1-6 InverseOfCounitOfShiftAdjunction

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

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

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

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

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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