Theorems · Definition · category theory

CategoryTheory.Abelian.PreservesImage.iso

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
      {D : Type u₂} →
        [inst_2 : CategoryTheory.Category.{v₂, u₂} D] →
          [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] →
            (F : CategoryTheory.Functor C D) →
              [F.PreservesZeroMorphisms] →
                {X Y : C} →
                  (f : X ⟶ Y) →
                    [inst_5 : CategoryTheory.Limits.HasCokernel f] →
                      [inst_6 : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π f)] →
                        [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f 0) F] →
                          [CategoryTheory.Limits.PreservesLimit
                                (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.cokernel.π f) 0) F] →
                            [inst_9 : CategoryTheory.Limits.HasCokernel (F.map f)] →
                              [inst_10 : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π (F.map f))] →
                                F.obj (CategoryTheory.Abelian.image f) ≅ CategoryTheory.Abelian.image (F.map f)

If a functor preserves kernels and cokernels, it preserves abelian images.

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.AbelianImages

Cited by

12 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.HasKernelCategoryTheory.Limits.PreservesColimitCategoryTheory.Limits.PreservesLimitCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.HasKernel