[Merged by Bors] - feat(algebra/homology): projective resolutions #7486
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…ntially surjective
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I think my biggest question is whether in practical situations we won't get annoyed by the fact that Z is stuck into this single\0 complex.
Do you think it makes sense to write some machinery to go back and forth between "map of complexes to single\0" and "map from obj in degree 0 to Z"?
We already have this machinery, see Moreover, I think later we may be happy to have this formulation, as it's closer to the "(co)fibrant replacement" picture. |
Co-authored-by: Johan Commelin <johan@commelin.net>
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# Projective resolutions A projective resolution `P : ProjectiveResolution Z` of an object `Z : C` consists of a `ℕ`-indexed chain complex `P.complex` of projective objects, along with a chain map `P.π` from `C` to the chain complex consisting just of `Z` in degree zero, so that the augmented chain complex is exact. When `C` is abelian, this exactness condition is equivalent to `π` being a quasi-isomorphism. It turns out that this formulation allows us to set up the basic theory derived functors without even assuming `C` is abelian. (Typically, however, to show `has_projective_resolutions C` one will assume `enough_projectives C` and `abelian C`. This construction appears in `category_theory.abelian.projectives`.) We show that give `P : ProjectiveResolution X` and `Q : ProjectiveResolution Y`, any morphism `X ⟶ Y` admits a lift to a chain map `P.complex ⟶ Q.complex`. (It is a lift in the sense that the projection maps `P.π` and `Q.π` intertwine the lift and the original morphism.) Moreover, we show that any two such lifts are homotopic. As a consequence, if every object admits a projective resolution, we can construct a functor `projective_resolutions C : C ⥤ homotopy_category C`. Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>
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# Projective resolutions A projective resolution `P : ProjectiveResolution Z` of an object `Z : C` consists of a `ℕ`-indexed chain complex `P.complex` of projective objects, along with a chain map `P.π` from `C` to the chain complex consisting just of `Z` in degree zero, so that the augmented chain complex is exact. When `C` is abelian, this exactness condition is equivalent to `π` being a quasi-isomorphism. It turns out that this formulation allows us to set up the basic theory derived functors without even assuming `C` is abelian. (Typically, however, to show `has_projective_resolutions C` one will assume `enough_projectives C` and `abelian C`. This construction appears in `category_theory.abelian.projectives`.) We show that give `P : ProjectiveResolution X` and `Q : ProjectiveResolution Y`, any morphism `X ⟶ Y` admits a lift to a chain map `P.complex ⟶ Q.complex`. (It is a lift in the sense that the projection maps `P.π` and `Q.π` intertwine the lift and the original morphism.) Moreover, we show that any two such lifts are homotopic. As a consequence, if every object admits a projective resolution, we can construct a functor `projective_resolutions C : C ⥤ homotopy_category C`. Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>
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Projective resolutions
A projective resolution
P : ProjectiveResolution Zof an objectZ : Cconsists ofa
ℕ-indexed chain complexP.complexof projective objects,along with a chain map
P.πfromCto the chain complex consisting just ofZin degree zero,so that the augmented chain complex is exact.
When
Cis abelian, this exactness condition is equivalent toπbeing a quasi-isomorphism.It turns out that this formulation allows us to set up the basic theory derived functors
without even assuming
Cis abelian.(Typically, however, to show
has_projective_resolutions Cone will assume
enough_projectives Candabelian C.This construction appears in
category_theory.abelian.projectives.)We show that give
P : ProjectiveResolution XandQ : ProjectiveResolution Y,any morphism
X ⟶ Yadmits a lift to a chain mapP.complex ⟶ Q.complex.(It is a lift in the sense that
the projection maps
P.πandQ.πintertwine the lift and the original morphism.)Moreover, we show that any two such lifts are homotopic.
As a consequence, if every object admits a projective resolution,
we can construct a functor
projective_resolutions C : C ⥤ homotopy_category C.