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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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