Theorems · Definition · category theory

CategoryTheory.Limits.PreservesKernel.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] →
            (G : CategoryTheory.Functor C D) →
              [G.PreservesZeroMorphisms] →
                {X Y : C} →
                  (f : X ⟶ Y) →
                    [inst_5 : CategoryTheory.Limits.HasKernel f] →
                      [inst_6 : CategoryTheory.Limits.HasKernel (G.map f)] →
                        [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) G] →
                          G.obj (CategoryTheory.Limits.kernel f) ≅ CategoryTheory.Limits.kernel (G.map f)

If G preserves the kernel of f, then the kernel comparison map for G at f is an isomorphism.

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Cited by

13 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Limits.HasKernelCategoryTheory.Limits.HasKernelCategoryTheory.Limits.PreservesLimit