Documentation

Mathlib.CategoryTheory.Abelian.LeftDerived

Left-derived functors #

We define the left-derived functors F.leftDerived n : C ⥤ D for any additive functor F out of a category with projective resolutions.

We first define a functor F.leftDerivedToHomotopyCategory : C ⥤ HomotopyCategory D (ComplexShape.down ℕ) which is projectiveResolutions C ⋙ F.mapHomotopyCategory _. We show that if X : C and P : ProjectiveResolution X, then F.leftDerivedToHomotopyCategory.obj X identifies to the image in the homotopy category of the functor F applied objectwise to P.complex (this isomorphism is P.isoLeftDerivedToHomotopyCategoryObj F).

Then, the left-derived functors F.leftDerived n : C ⥤ D are obtained by composing F.leftDerivedToHomotopyCategory with the homology functors on the homotopy category.

Similarly we define natural transformations between left-derived functors coming from natural transformations between the original additive functors, and show how to compute the components.

Main results #

When F : C ⥤ D is an additive functor, this is the functor C ⥤ HomotopyCategory D (ComplexShape.down ℕ) which sends X : C to F applied to a projective resolution of X.

Equations
  • One or more equations did not get rendered due to their size.
Instances For

    If P : ProjectiveResolution Z and F : C ⥤ D is an additive functor, this is an isomorphism between F.leftDerivedToHomotopyCategory.obj X and the complex obtained by applying F to P.complex.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For

      We can compute a left derived functor using a chosen projective resolution.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For

        If P : ProjectiveResolution X and F is an additive functor, this is the canonical morphism from the opcycles in degree 0 of (F.mapHomologicalComplex _).obj P.complex to F.obj X.

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For

          The natural transformation F.leftDerived 0 ⟶ F.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For