Goto Chapter: Top 1 2 3 4 Ind

### 4 Triangulated Categories

#### 4.1 Categorical operations

##### 4.1-1 StandardConeObject
 ‣ StandardConeObject( alpha ) ( attribute )

Returns: an object

The argument is a morphism $$\alpha:A\to B$$ in a triangulated category. The output is the standard cone object $$C(\alpha)$$ of $$\alpha$$.

##### 4.1-2 MorphismIntoStandardConeObjectWithGivenStandardConeObject
 ‣ MorphismIntoStandardConeObjectWithGivenStandardConeObject( alpha, C ) ( operation )

Returns: a morphism $$\iota(\alpha):B\to C(\alpha)$$

The arguments are a morphism $$\alpha: A \to B$$ in a triangulated category and an object $$C:=C(\alpha)$$. The output is the morphism $$\iota(\alpha):B\to C(\alpha)$$ into the standard cone object $$C(\alpha)$$.

##### 4.1-3 MorphismIntoStandardConeObject
 ‣ MorphismIntoStandardConeObject( alpha ) ( attribute )

Returns: a morphism $$\iota(\alpha):B\to C(\alpha)$$

The argument is a morphism $$\alpha: A \to B$$ in a triangulated category. The output is the morphism $$\iota(\alpha):B\to C(\alpha)$$ into the standard cone object $$C(\alpha)$$.

##### 4.1-4 MorphismFromStandardConeObjectWithGivenObjects
 ‣ MorphismFromStandardConeObjectWithGivenObjects( alpha, C ) ( operation )

Returns: a morphism $$\pi(\alpha):C(\alpha)\to\Sigma A$$

The arguments are a morphism $$\alpha: A \to B$$ in a triangulated category and an object $$C:=C(\alpha)$$. The output is the morphism $$\pi(\alpha):C(\alpha)\to\Sigma A$$ from the standard cone object $$C(\alpha)$$.

##### 4.1-5 MorphismFromStandardConeObject
 ‣ MorphismFromStandardConeObject( alpha, C ) ( attribute )

Returns: a morphism $$\pi(\alpha):C(\alpha)\to\Sigma A$$

The argument is a morphism $$\alpha: A \to B$$ in a triangulated category. The output is the morphism $$\pi(\alpha):C(\alpha)\to\Sigma A$$ from the standard cone object $$C(\alpha)$$.

##### 4.1-6 ShiftOfObject
 ‣ ShiftOfObject( A ) ( attribute )

Returns: $$\Sigma A$$

The argument is an object $$A$$ in a triangulated category $$\mathcal{T}$$. The output is $$\Sigma A$$.

##### 4.1-7 ShiftOfMorphismWithGivenObjects
 ‣ ShiftOfMorphismWithGivenObjects( sigma_A, alpha, sigma_B ) ( operation )

Returns: $$\Sigma \alpha:\Sigma A \to \Sigma B$$

The arguments are an object $$\Sigma A$$, a morphism $$\alpha:A\to B$$ and an object $$\Sigma B$$ in a triangulated category $$\mathcal{T}$$. The output is $$\Sigma \alpha:\Sigma A \to \Sigma B$$.

##### 4.1-8 ShiftOfMorphism
 ‣ ShiftOfMorphism( alpha ) ( attribute )

Returns: $$\Sigma \alpha:\Sigma A \to \Sigma B$$

The argument is a morphism $$\alpha:A\to B$$ in a triangulated category $$\mathcal{T}$$. The output is $$\Sigma \alpha:\Sigma A \to \Sigma B$$.

##### 4.1-9 Shift
 ‣ Shift( c ) ( operation )

Returns: $$\Sigma c$$

This is a convenience method to apply the shift functor on objects or morphisms. The operation delegates to either ShiftOfObject or ShiftOfMorphism.

##### 4.1-10 InverseShiftOfObject
 ‣ InverseShiftOfObject( A ) ( operation )

Returns: $$\Sigma^{-1} A$$

The argument is an object $$A$$ in a triangulated category $$\mathcal{T}$$. The output is $$\Sigma^{-1} A$$.

##### 4.1-11 InverseShiftOfMorphismWithGivenObjects
 ‣ InverseShiftOfMorphismWithGivenObjects( rev_sigma_A, alpha, rev_sigma_B ) ( operation )

Returns: $$\Sigma^{-1} \alpha:\Sigma^{-1} A \to \Sigma^{-1} B$$

The arguments are an object $$\Sigma^{-1} A$$, a morphism $$\alpha:A\to B$$ and an object $$\Sigma^{-1} B$$ in a triangulated category $$\mathcal{T}$$. The output is $$\Sigma^{-1} \alpha:\Sigma^{-1} A \to \Sigma^{-1} B$$.

##### 4.1-12 InverseShiftOfMorphism
 ‣ InverseShiftOfMorphism( alpha ) ( attribute )

Returns: $$\Sigma^{-1} \alpha:\Sigma^{-1} A \to \Sigma^{-1} B$$

The argument is a morphism $$\alpha:A\to B$$ in a triangulated category $$\mathcal{T}$$. The output is $$\Sigma^{-1} \alpha:\Sigma^{-1} A \to \Sigma^{-1} B$$.

##### 4.1-13 InverseShift
 ‣ InverseShift( c ) ( operation )

Returns: $$\Sigma^{-1} c$$

This is a convenience method to apply the inverse shift functor on objects or morphisms. The operation delegates to either InverseShiftOfObject or InverseShiftOfMorphism.

 ‣ UnitOfShiftAdjunctionWithGivenObject( A, sigma_o_rev_sigma_A ) ( operation )

Returns: a morphism $$A \to (\Sigma \circ \Sigma^{-1}) A$$

The arguments are two objects $$A$$ and $$(\Sigma \circ \Sigma^{-1}) A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$A \to (\Sigma \circ \Sigma^{-1}) A$$

 ‣ UnitOfShiftAdjunction( A ) ( attribute )

Returns: a morphism $$A \to (\Sigma \circ \Sigma^{-1}) A$$

The argument is an object $$A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$A \to (\Sigma \circ \Sigma^{-1}) A$$

 ‣ InverseOfCounitOfShiftAdjunctionWithGivenObject( A, rev_sigma_o_sigma_A ) ( operation )

Returns: a morphism $$A \to (\Sigma^{-1} \circ \Sigma) A$$

The arguments are two objects $$A$$ and $$(\Sigma^{-1} \circ \Sigma) A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$A \to (\Sigma^{-1} \circ \Sigma) A$$

 ‣ InverseOfCounitOfShiftAdjunction( A ) ( attribute )

Returns: a morphism $$A \to (\Sigma^{-1} \circ \Sigma) A$$

The argument is an object $$A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$A \to (\Sigma^{-1} \circ \Sigma) A$$

 ‣ InverseOfUnitOfShiftAdjunctionWithGivenObject( A, sigma_o_rev_sigma_A ) ( operation )

Returns: a morphism $$(\Sigma \circ \Sigma^{-1}) A \to A$$

The arguments are two objects $$A$$ and $$(\Sigma \circ \Sigma^{-1}) A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$(\Sigma \circ \Sigma^{-1}) A \to A$$

 ‣ InverseOfUnitOfShiftAdjunction( A ) ( attribute )

Returns: a morphism $$(\Sigma \circ \Sigma^{-1}) A \to A$$

The argument in an objects $$A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$(\Sigma \circ \Sigma^{-1}) A \to A$$

 ‣ CounitOfShiftAdjunctionWithGivenObject( A, rev_sigma_o_sigma_A ) ( operation )

Returns: a morphism $$(\Sigma^{-1} \circ \Sigma) A \to A$$

The arguments are two objects $$A$$ and $$(\Sigma^{-1} \circ \Sigma) A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$(\Sigma^{-1} \circ \Sigma) A \to A$$

 ‣ CounitOfShiftAdjunction( A ) ( attribute )

Returns: a morphism $$(\Sigma^{-1} \circ \Sigma) A \to A$$

The argument is an object $$A$$ in a triangulated category $$\mathcal{T}$$. The output is the natural isomorphism $$(\Sigma^{-1} \circ \Sigma) A \to A$$

##### 4.1-22 MorphismBetweenStandardConeObjectsWithGivenObjects
 ‣ MorphismBetweenStandardConeObjectsWithGivenObjects( C_alpha_1, list, C_alpha_2 ) ( operation )

Returns: a morphism $$C(\alpha_1) \to C(\alpha_2)$$

The arguments are an object $$C_{\alpha_1}$$, a list of four morphisms $$\alpha_1:A_1\to B_1$$, $$u:A_1\to A_2$$, $$v:B_1\to B_2$$, $$\alpha_2:A_2\to B_2$$ and an object $$C_{\alpha_2}$$ such that $$C_{\alpha_1}:=C(\alpha_1)$$, $$C_{\alpha_2}:=C(\alpha_2)$$ and $$v\circ \alpha_1=\alpha_2\circ u$$. The output is a morphism $$w:C(\alpha_1) \to C(\alpha_2)$$ such that $$w\circ \iota(\alpha_1)=\iota(\alpha_2)\circ v$$ and $$\Sigma u\circ\pi(\alpha_1)=\pi(\alpha_2)\circ w$$.

##### 4.1-23 MorphismBetweenStandardConeObjects
 ‣ MorphismBetweenStandardConeObjects( alpha_1, u, v, alpha_2 ) ( operation )

Returns: a morphism $$C(\alpha_1) \to C(\alpha_2)$$

The arguments are morphisms $$\alpha_1:A_1\to B_1$$, $$u:A_1\to A_2$$, $$v:B_1\to B_2$$, $$\alpha_2:A_2\to B_2$$ such that $$v\circ \alpha_1=\alpha_2\circ u$$. The output is a morphism $$w:C(\alpha_1) \to C(\alpha_2)$$ such that $$w\circ \iota(\alpha_1)=\iota(\alpha_2)\circ v$$ and $$\Sigma u\circ\pi(\alpha_1)=\pi(\alpha_2)\circ w$$.

##### 4.1-24 ConeObjectByOctahedralAxiom
 ‣ ConeObjectByOctahedralAxiom( alpha, beta ) ( operation )

Returns: $$C(\beta)$$

The arguments are two morphisms $$\alpha:A\to B$$, $$\beta:B\to C$$. The output is the standard cone object $$C(\beta)$$.

##### 4.1-25 DomainMorphismByOctahedralAxiomWithGivenObjects
 ‣ DomainMorphismByOctahedralAxiomWithGivenObjects( alpha, beta ) ( operation )

Returns: a morphism $$u_{\alpha,\beta}$$:$$C(\alpha)\to C(\beta\circ\alpha)$$

The arguments are two morphisms $$\alpha:A\to B$$, $$\beta:B\to C$$. The output is a morphism $$u_{\alpha,\beta}$$:$$C(\alpha)\to C(\beta\circ\alpha)$$ such that $$u_{\alpha,\beta}\circ\iota(\alpha)=\iota(\beta\circ\alpha)\circ\beta$$ and $$\pi(\alpha)=\pi(\beta\circ\alpha)\circ u_{\alpha,\beta}$$.

##### 4.1-26 MorphismIntoConeObjectByOctahedralAxiomWithGivenObjects
 ‣ MorphismIntoConeObjectByOctahedralAxiomWithGivenObjects( alpha, beta ) ( operation )

Returns: a morphism $$C(\beta\circ\alpha) \to C(\beta)$$

The arguments are two morphisms $$\alpha:A\to B$$, $$\beta:B\to C$$. The output is a morphism $$\iota_{\alpha,\beta}$$:$$C(\beta\circ\alpha) \to C(\beta)$$ such that $$\iota_{\alpha,\beta}\circ\iota(\beta\circ\alpha)=\iota(\beta)$$ and $$\Sigma\alpha\circ\pi(\beta\circ\alpha)=\pi(\beta)$$.

##### 4.1-27 MorphismFromConeObjectByOctahedralAxiomWithGivenObjects
 ‣ MorphismFromConeObjectByOctahedralAxiomWithGivenObjects( alpha, beta ) ( operation )

Returns: a morphism $$C(\beta) \to \Sigma C(\alpha)$$

The arguments are two morphisms $$\alpha:A\to B$$, $$\beta:B\to C$$. The output is a morphism $$\pi_{\alpha,\beta}$$:$$C(\beta) \to \Sigma C(\alpha)$$ such that $$\pi_{\alpha,\beta}=\Sigma \iota(\alpha) \circ\pi(\beta)$$.

##### 4.1-28 WitnessIsomorphismIntoStandardConeObjectByOctahedralAxiomWithGivenObjects
 ‣ WitnessIsomorphismIntoStandardConeObjectByOctahedralAxiomWithGivenObjects( s, alpha, beta, r ) ( operation )

Returns: a morphism $$C(\beta)\to C(u_{\alpha,\beta})$$

The arguments are an object $$s=C(\beta)$$, a morphism $$\alpha:A\to B$$, a morphism $$\beta:B\to C$$ and an object $$r=C(u_{\alpha,\beta})$$. The output is an isomorphism $$w_{\alpha,\beta}:C(\beta)\to C(u_{\alpha,\beta})$$ such that $$w_{\alpha,\beta}\circ \iota_{\alpha,\beta}=\iota(u_{\alpha,\beta})$$ and $$\pi_{\alpha,\beta}=\pi(u_{\alpha,\beta})\circ w_{\alpha,\beta}$$. I.e., the following diagram is commutative:

##### 4.1-29 WitnessIsomorphismIntoStandardConeObjectByOctahedralAxiom
 ‣ WitnessIsomorphismIntoStandardConeObjectByOctahedralAxiom( alpha, beta ) ( operation )

Returns: a morphism $$C(\beta)\to C(u_{\alpha,\beta})$$.

The arguments are two morphisms $$\alpha:A\to B$$ and $$\beta:B\to C$$. The output is an isomorphism $$C(\beta)\to C(u_{\alpha,\beta})$$ such that $$w_{\alpha,\beta}\circ \iota_{\alpha,\beta}=\iota(u_{\alpha,\beta})$$ and $$\pi_{\alpha,\beta}=\pi(u_{\alpha,\beta})\circ w_{\alpha,\beta}$$.

##### 4.1-30 WitnessIsomorphismFromStandardConeObjectByOctahedralAxiomWithGivenObjects
 ‣ WitnessIsomorphismFromStandardConeObjectByOctahedralAxiomWithGivenObjects( alpha, beta ) ( operation )

Returns: a morphism $$C(u_{\alpha,\beta})\to C(\beta)$$

The arguments are two morphisms $$\alpha:A\to B$$ and $$\beta:B\to C$$. The output is ...

##### 4.1-31 WitnessIsomorphismFromStandardConeObjectByOctahedralAxiom
 ‣ WitnessIsomorphismFromStandardConeObjectByOctahedralAxiom( alpha, beta ) ( operation )

Returns: a morphism $$C(u_{\alpha,\beta})\to C(\beta)$$

The arguments are two morphisms $$\alpha:A\to B$$ and $$\beta:B\to C$$. The output is ...

##### 4.1-32 ConeObjectByRotationAxiom
 ‣ ConeObjectByRotationAxiom( alpha ) ( attribute )

Returns: an object $$\Sigma A$$

The argument is a morphism $$\alpha:A\to B$$ The output is $$\Sigma A$$.

##### 4.1-33 DomainMorphismByRotationAxiom
 ‣ DomainMorphismByRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$B\to C(\alpha)$$.

The argument is a morphism $$\alpha:A\to B$$. The output is $$\iota(\alpha):B\to C(\alpha)$$.

##### 4.1-34 MorphismIntoConeObjectByRotationAxiom
 ‣ MorphismIntoConeObjectByRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$C(\alpha)\to \Sigma A$$

The argument is a morphism $$\alpha:A\to B$$. The output is a morphism $$\pi(\alpha):C(\alpha)\to \Sigma A$$.

##### 4.1-35 MorphismFromConeObjectByRotationAxiom
 ‣ MorphismFromConeObjectByRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$\Sigma A\to\Sigma B$$

The argument is a morphism $$\alpha:A\to B$$. The output is a morphism $$-\Sigma \alpha:\Sigma A\to\Sigma B$$.

##### 4.1-36 WitnessIsomorphismIntoStandardConeObjectByRotationAxiom
 ‣ WitnessIsomorphismIntoStandardConeObjectByRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$\Sigma A \to C(\iota(\alpha))$$

The argument is a morphism $$\alpha:A\to B$$. The output is an isomorphism $$\Sigma A \to C(\iota(\alpha))$$ such that $$?\circ\pi(\alpha)=\iota(\iota(\alpha))$$ and $$\pi(\iota(\alpha))\circ ?=-\Sigma \alpha$$.

##### 4.1-37 WitnessIsomorphismIntoStandardConeObjectByRotationAxiomWithGivenObjects
 ‣ WitnessIsomorphismIntoStandardConeObjectByRotationAxiomWithGivenObjects( s, alpha, r ) ( operation )

Returns: a morphism $$\Sigma A \to C(\iota(\alpha))$$

The arguments are an object $$s=\Sigma A$$, morphism $$\alpha:A\to B$$ and an object $$r=C(\iota A )$$. The output is an isomorphism $$\Sigma A \to C(\iota(\alpha))$$ such that $$?\circ\pi(\alpha)=\iota(\iota(\alpha))$$ and $$\pi(\iota(\alpha))\circ ?=-\Sigma \alpha$$.

##### 4.1-38 WitnessIsomorphismFromStandardConeObjectByRotationAxiom
 ‣ WitnessIsomorphismFromStandardConeObjectByRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$C(\iota(\alpha))\to\Sigma A$$

The argument is a morphism $$\alpha:A\to B$$. The output is an isomorphism $$C(\iota(\alpha))\to\Sigma A$$ such that ???

##### 4.1-39 WitnessIsomorphismFromStandardConeObjectByRotationAxiomWithGivenObjects
 ‣ WitnessIsomorphismFromStandardConeObjectByRotationAxiomWithGivenObjects( s, alpha, r ) ( operation )

Returns: a morphism $$C(\iota(\alpha))\to\Sigma A$$

The arguments are an object $$s=C(\iota(\alpha))$$, morphism $$\alpha:A\to B$$ and an object $$r=\Sigma A$$. The output is an isomorphism $$C(\iota(\alpha))\to\Sigma A$$ such that ???

##### 4.1-40 ConeObjectByInverseRotationAxiom
 ‣ ConeObjectByInverseRotationAxiom( alpha ) ( attribute )

Returns: an object $$\Sigma A$$

The argument is a morphism $$\alpha:A\to B$$ The output is $$\Sigma A$$.

##### 4.1-41 DomainMorphismByInverseRotationAxiom
 ‣ DomainMorphismByInverseRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$B\to C(\alpha)$$.

The argument is a morphism $$\alpha:A\to B$$. The output is $$\iota(\alpha):B\to C(\alpha)$$.

##### 4.1-42 MorphismIntoConeObjectByInverseRotationAxiom
 ‣ MorphismIntoConeObjectByInverseRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$C(\alpha)\to \Sigma A$$

The argument is a morphism $$\alpha:A\to B$$. The output is a morphism $$\pi(\alpha):C(\alpha)\to \Sigma A$$.

##### 4.1-43 MorphismFromConeObjectByInverseRotationAxiom
 ‣ MorphismFromConeObjectByInverseRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$\Sigma A\to\Sigma B$$

The argument is a morphism $$\alpha:A\to B$$. The output is a morphism $$-\Sigma \alpha:\Sigma A\to\Sigma B$$.

##### 4.1-44 WitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiom
 ‣ WitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$\Sigma A \to C(\iota(\alpha))$$

The argument is a morphism $$\alpha:A\to B$$. The output is an isomorphism $$\Sigma A \to C(\iota(\alpha))$$ such that $$?\circ\pi(\alpha)=\iota(\iota(\alpha))$$ and $$\pi(\iota(\alpha))\circ ?=-\Sigma \alpha$$.

##### 4.1-45 WitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiomWithGivenObjects
 ‣ WitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiomWithGivenObjects( s, alpha, r ) ( operation )

Returns: a morphism $$\Sigma A \to C(\iota(\alpha))$$

The arguments are an object $$s=\Sigma A$$, morphism $$\alpha:A\to B$$ and an object $$r=C(\iota A )$$. The output is an isomorphism $$\Sigma A \to C(\iota(\alpha))$$ such that $$?\circ\pi(\alpha)=\iota(\iota(\alpha))$$ and $$\pi(\iota(\alpha))\circ ?=-\Sigma \alpha$$.

##### 4.1-46 WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiom
 ‣ WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiom( alpha ) ( attribute )

Returns: a morphism $$C(\iota(\alpha))\to\Sigma A$$

The argument is a morphism $$\alpha:A\to B$$. The output is an isomorphism $$C(\iota(\alpha))\to\Sigma A$$ such that ???

##### 4.1-47 WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiomWithGivenObjects
 ‣ WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiomWithGivenObjects( s, alpha, r ) ( operation )

Returns: a morphism $$C(\iota(\alpha))\to\Sigma A$$

The arguments are an object $$s=C(\iota(\alpha))$$, morphism $$\alpha:A\to B$$ and an object $$r=\Sigma A$$. The output is an isomorphism $$C(\iota(\alpha))\to\Sigma A$$ such that ???

##### 4.1-48 ShiftExpandingIsomorphismWithGivenObjects
 ‣ ShiftExpandingIsomorphismWithGivenObjects( X, L, Y ) ( operation )

Returns: a morphism

The arguments are list $$L=[A_1,\dots,A_n]$$ and two objects $$X=\Sigma \bigoplus_i A_i, Y=\bigoplus_i \Sigma A_i$$. The output is the isomorphism $$X \rightarrow Y$$ associated to $$\Sigma$$.

##### 4.1-49 ShiftExpandingIsomorphism
 ‣ ShiftExpandingIsomorphism( L ) ( operation )

Returns: a morphism

The argument is a list $$L=[A_1,\dots,A_n]$$. The output is the isomorphism $$X \rightarrow Y$$ associated to $$\Sigma$$, where $$X=\Sigma \bigoplus_i A_i$$ and $$Y=\bigoplus_i \Sigma A_i$$

##### 4.1-50 ShiftFactoringIsomorphismWithGivenObjects
 ‣ ShiftFactoringIsomorphismWithGivenObjects( Y, L, X ) ( operation )

Returns: a morphism

The arguments are list $$L=[A_1,\dots,A_n]$$ and two objects $$Y=\bigoplus_i \Sigma A_i, X=\Sigma \bigoplus_i A_i$$. The output is the isomorphism $$Y \rightarrow X$$ associated to $$\Sigma$$.

##### 4.1-51 ShiftFactoringIsomorphism
 ‣ ShiftFactoringIsomorphism( L ) ( operation )

Returns: a morphism

The argument is a list $$L=[A_1,\dots,A_n]$$. The output is the isomorphism $$Y \rightarrow X$$ associated to $$\Sigma$$, where $$Y=\bigoplus_i \Sigma A_i$$ and $$X=\Sigma \bigoplus_i A_i$$.

##### 4.1-52 InverseShiftExpandingIsomorphismWithGivenObjects
 ‣ InverseShiftExpandingIsomorphismWithGivenObjects( X, L, Y ) ( operation )

Returns: a morphism

The arguments are list $$L=[A_1,\dots,A_n]$$ and two objects $$X=\Sigma^{-1} \bigoplus_i A_i, Y=\bigoplus_i \Sigma^{-1} A_i$$. The output is the isomorphism $$X \rightarrow Y$$ associated to $$\Sigma^{-1}$$.

##### 4.1-53 InverseShiftExpandingIsomorphism
 ‣ InverseShiftExpandingIsomorphism( L ) ( operation )

Returns: a morphism

The argument is a list $$L=[A_1,\dots,A_n]$$. The output is the isomorphism $$X \rightarrow Y$$ associated to $$\Sigma$$, where $$X=\Sigma \bigoplus_i A_i$$ and $$Y=\bigoplus_i \Sigma A_i$$

##### 4.1-54 InverseShiftFactoringIsomorphismWithGivenObjects
 ‣ InverseShiftFactoringIsomorphismWithGivenObjects( Y, L, X ) ( operation )

Returns: a morphism

The arguments are list $$L=[A_1,\dots,A_n]$$ and two objects $$Y=\bigoplus_i \Sigma^{-1} A_i, X=\Sigma^{-1} \bigoplus_i A_i$$. The output is the isomorphism $$Y \rightarrow X$$ associated to $$\Sigma^{-1}$$.

##### 4.1-55 InverseShiftFactoringIsomorphism
 ‣ InverseShiftFactoringIsomorphism( L ) ( operation )

Returns: a morphism

The argument is a list $$L=[A_1,\dots,A_n]$$. The output is the isomorphism $$Y \rightarrow X$$ associated to $$\Sigma^{-1}$$, where $$Y=\bigoplus_i \Sigma^{-1} A_i$$ and $$X=\Sigma^{-1} \bigoplus_i A_i$$.

Goto Chapter: Top 1 2 3 4 Ind

generated by GAPDoc2HTML