[Merged by Bors] - feat(Algebra/Homology): two descriptions of the derived category as a localized category#9660
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[Merged by Bors] - feat(Algebra/Homology): two descriptions of the derived category as a localized category#9660
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… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
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… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
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… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
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… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
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… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
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… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
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In this PR, it is shown that under certain conditions on the complex shape
c, the categoryHomologicalComplexUpToQuasiIso C cof homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian categoryCcan be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.