Theorems · Definition · category theory

CategoryTheory.Abelian.PreservesCoimageImageComparison.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) →
              [inst_4 : F.PreservesZeroMorphisms] →
                {X Y : C} →
                  (f : X ⟶ Y) →
                    [inst_5 : CategoryTheory.Limits.HasKernel f] →
                      [inst_6 : CategoryTheory.Limits.HasCokernel f] →
                        [inst_7 : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π f)] →
                          [inst_8 : CategoryTheory.Limits.HasCokernel (CategoryTheory.Limits.kernel.ι f)] →
                            [inst_9 : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) F] →
                              [inst_10 :
                                  CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f 0) F] →
                                [CategoryTheory.Limits.PreservesLimit
                                      (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.cokernel.π f) 0) F] →
                                  [CategoryTheory.Limits.PreservesColimit
                                        (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.kernel.ι f) 0) F] →
                                    [inst_13 :
                                        CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π (F.map f))] →
                                      [inst_14 :
                                          CategoryTheory.Limits.HasCokernel
                                            (CategoryTheory.Limits.kernel.ι (F.map f))] →
                                        CategoryTheory.Arrow.mk
                                            (F.map (CategoryTheory.Abelian.coimageImageComparison f)) ≅
                                          CategoryTheory.Arrow.mk
                                            (CategoryTheory.Abelian.coimageImageComparison (F.map f))

If a functor preserves kernels and cokernels, it preserves coimage-image comparisons.

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.AbelianImages

Cited by

4 results in Mathlib

Foundations

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

Assumes

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