‣ 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 \(X=\Sigma^{-1} \bigoplus_i A_i\).
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