Theorems · Theorem · category theory

CategoryTheory.Limits.PreservesKernel.of_iso_comparison

∀ {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) [inst_4 : G.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y)
  [inst_5 : CategoryTheory.Limits.HasKernel f] [inst_6 : CategoryTheory.Limits.HasKernel (G.map f)]
  [i : CategoryTheory.IsIso (CategoryTheory.Limits.kernelComparison f G)],
  CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) G

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

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Cited by

0 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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