‣ DecompositionFunctorOfProjectiveQuiverRepresentations( cat ) | ( attribute ) |
Returns: a functor
The argument is a category of quiver representations over some finite dimensional quiver algebra. The output is an equivalence from full subcategory generated by the projective representations into the additive closure of the full subcategory generated by indecomposable projective representations.
‣ QuasiInverseOfDecompositionFunctorOfProjectiveQuiverRepresentations( cat ) | ( attribute ) |
Returns: a functor
The argument is a category of quiver representations over some finite dimensional quiver algebra. The output is an equivalence from the additive closure of the full subcategory generated by indecomposable projective representations into full subcategory generated by the projective representations.
‣ DecompositionFunctorOfInjectiveQuiverRepresentations( cat ) | ( attribute ) |
Returns: a full subcategory
The argument is a category of quiver representations over some finite dimensional quiver algebra. The output is an equivalence from full subcategory generated by the injective representations into the additive closure of the full subcategory generated by indecomposable injective representations.
‣ QuasiInverseOfDecompositionFunctorOfInjectiveQuiverRepresentations( cat ) | ( attribute ) |
Returns: a functor
The argument is a category of quiver representations over some finite dimensional quiver algebra. The output is an equivalence from the additive closure of the full subcategory generated by indecomposable injective representations into full subcategory generated by the injective representations.
‣ IsomorphismOntoAlgebroid( E ) | ( attribute ) |
Returns: a functor
The input is an exceptional collection \(E\) and the output is an isomorphism functor into algebroid defined over the endomorphism algebra of \(\bigoplus_i E_i\) for \(E_i\in\)E.
‣ IsomorphismFromAlgebroid( E ) | ( attribute ) |
Returns: a functor
The input is an exceptional collection E and the output is an isomorphism functor from algebroid defined over the endomorphism algebra of \(\bigoplus_i E_i\) for \(E_i \in\) E.
‣ YonedaIsomorphismOntoFullSubcategoryOfCategoryOfQuiverRepresentations( alg ) | ( attribute ) |
Returns: a functor
The input is an algebroid alg defined by a finite dimensional quiver algebra \(A\). The output is the isomorphism functor from alg into the full subcategory of category of \(A^{op}\)-quiver representations, which is generated by the indecomposable projective objects.
‣ YonedaIsomorphismOntoFullSubcategoryOfCategoryOfQuiverRepresentations( E ) | ( attribute ) |
Returns: a functor
The input is an exceptional collection E and the output is an isomorphism from the full subcategory generated by the exceptional collection into the full subcategory of category of \(A^{op}\)-quiver representations, which is generated by the indecomposable projective objects.
‣ InverseOfYonedaIsomorphismOntoFullSubcategoryOfCategoryOfQuiverRepresentations( E ) | ( attribute ) |
Returns: a functor
The input is an exceptional collection E and the output is an isomorphism into the full subcategory generated by the exceptional collection from the full subcategory of category of \(A^{op}\)-quiver representations, which is generated by the indecomposable projective objects.
‣ InverseOfYonedaIsomorphismOntoFullSubcategoryOfCategoryOfQuiverRepresentations( alg ) | ( attribute ) |
Returns: a functor
The input is an algebroid alg defined by a finite dimensional quiver algebra \(A\). The output is the isomorphism functor from the full subcategory of category of \(A^{op}\)-quiver representations, which is generated by the indecomposable projective objects into alg.
‣ LocalizationFunctor( H ) | ( attribute ) |
Returns: a functor
The input is a homotopy category H := HomotopyCategory(\(C\)) of some abelian category \(C\). The output is the localization functor \(L\) from H into DerivedCategory(\(C\)).
‣ UniversalFunctorFromDerivedCategory( H ) | ( attribute ) |
Returns: a functor
The input is a localization functor for some homotopy category H := HomotopyCategory(\(C\)) of some abelian category \(C\) in some category \(B\), i.e., send quasi-isomorphisms in H to isomorphisms in \(B\). The output is the universal functor from DerivedCategory(\(C\)) into \(B\).
‣ LeftDerivedFunctor( G ) | ( operation ) |
Returns: a functor
The input is a functor G:\(D\to C\) such that \(D\) is abelian category with computable enough projectives or extension of such a functor to homotopy categories. The output the left derived functor \(LG:D^b(D)\to D^b(C)\).
‣ RightDerivedFunctor( F ) | ( operation ) |
Returns: a functor
The input is a functor F:\(C\to D\) such that \(C\) is abelian category with computable enough injectives or extension of such a functor to homotopy categories. The output the right derived functor \(RF:D^b(C)\to D^b(D)\).
‣ TensorFunctorFromCategoryOfQuiverRepresentations( collection ) | ( attribute ) |
Returns: a functor
The argument is an exceptional collection E which is defined by some full subcategory generated by finite number of objects \((E_i)_i\) in some category \(C\) with homomorphism structure. The output is the functor \(- \otimes_{\mathrm{End}(\oplus_i E_i)} \oplus_i E_i:\mathrm{mod}\mbox{-}\mathrm{End}(\oplus_i E_i) \to C\).
‣ TensorFunctorFromCategoryOfQuiverRepresentationsOnIndecProjectiveObjects( collection ) | ( attribute ) |
Returns: a functor
The argument is an exceptional collection E which is defined by some full subcategory generated by finite number of objects \((E_i)_i\) in some category \(C\) with homomorphism structure. The output is the functor \(- \otimes_{\mathrm{End}(\oplus_i E_i)} \oplus_i E_i\): FullSubcategoryGeneratedByIndecProjectiveObjects\((\mathrm{mod}\mbox{-}\mathrm{End}(\oplus_i E_i)) \to C\).
‣ TensorFunctorFromCategoryOfQuiverRepresentationsOnProjectiveObjects( collection ) | ( attribute ) |
Returns: a functor
The argument is an exceptional collection E which is defined by some full subcategory generated by finite number of objects \((E_i)_i\) in some category \(C\) with homomorphism structure. The output is the functor \(- \otimes_{\mathrm{End}(\oplus_i E_i)} \oplus_i E_i\): FullSubcategoryGeneratedByProjectiveObjects\((\mathrm{mod}\mbox{-}\mathrm{End}(\oplus_i E_i)) \to C\).
‣ UnitOfTensorHomAdjunction( E ) | ( operation ) |
Returns: a natural transformation
The input is an exceptional collection E in some abelian category C. The output is the unit of the tensor-hom adjunction \(\lambda:\mathrm{Id}_{ \mathrm{mod}\mbox{-}\mathrm{End}(\oplus_i E_i)} \to \mathrm{Hom}(T,-\otimes_{\mathrm{End}(\oplus_i E_i)} \oplus_i E_i)\).
‣ CounitOfTensorHomAdjunction( E ) | ( operation ) |
Returns: a natural transformation
The input is an exceptional collection E. The output is the unit of the tensor-hom adjunction \(\eta: \mathrm{Hom}(\oplus_i E_i,-)\otimes_{\mathrm{End}(\oplus_i E_i)} \oplus_i E_i \to \mathrm{Id}_C\).
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