Theorems · Definition · category theory

CategoryTheory.ShortComplex.cyclesFunctorIso

{C : Type u_1} →
  {D : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C] →
          [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] →
            (F : CategoryTheory.Functor C D) →
              [inst_4 : F.PreservesZeroMorphisms] →
                [inst_5 : CategoryTheory.Limits.HasKernels C] →
                  [inst_6 : CategoryTheory.Limits.HasCokernels C] →
                    [inst_7 : CategoryTheory.Limits.HasKernels D] →
                      [inst_8 : CategoryTheory.Limits.HasCokernels D] →
                        [F.PreservesHomology] →
                          F.mapShortComplex.comp (CategoryTheory.ShortComplex.cyclesFunctor D) ≅
                            (CategoryTheory.ShortComplex.cyclesFunctor C).comp F

The natural isomorphism F.mapShortComplex ⋙ cyclesFunctor D ≅ cyclesFunctor C ⋙ F for a functor F : C ⥤ D which preserves homology.

Defined in

Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Cited by

0 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Limits.HasKernelsCategoryTheory.Limits.HasCokernelsCategoryTheory.Limits.HasKernelsCategoryTheory.Limits.HasCokernelsCategoryTheory.Functor.PreservesHomology