‣ 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.
‣ 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).
‣ 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).
‣ 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).
‣ 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).
‣ ShiftOfObject( A ) | ( attribute ) |
Returns: \Sigma A
The argument is an object A in a triangulated category \mathcal{T}. The output is \Sigma A.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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
‣ 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.
‣ 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.
‣ 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).
‣ 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}.
‣ 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).
‣ 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).
‣ 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:
‣ 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}.
‣ 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 ...
‣ 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 ...
‣ ConeObjectByRotationAxiom( alpha ) | ( attribute ) |
Returns: an object \Sigma A
The argument is a morphism \alpha:A\to B The output is \Sigma A.
‣ 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).
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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 ???
‣ 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 ???
‣ ConeObjectByInverseRotationAxiom( alpha ) | ( attribute ) |
Returns: an object \Sigma A
The argument is a morphism \alpha:A\to B The output is \Sigma A.
‣ 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).
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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 ???
‣ 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 ???
‣ 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.
‣ 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
‣ 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.
‣ 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.
‣ 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}.
‣ 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
‣ 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}.
‣ 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.
generated by GAPDoc2HTML