Theorems · Theorem · category theory

CategoryTheory.Limits.kernelComparison_comp_kernel_map_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C}
  (f : X ⟶ Y) {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} [inst_5 : CategoryTheory.Limits.HasKernel f]
  [inst_6 : CategoryTheory.Limits.HasKernel (G.map f)] (g : X' ⟶ Y') [inst_7 : CategoryTheory.Limits.HasKernel g]
  [inst_8 : CategoryTheory.Limits.HasKernel (G.map g)] (p : X ⟶ X') (q : Y ⟶ Y')
  (hpq : CategoryTheory.CategoryStruct.comp f q = CategoryTheory.CategoryStruct.comp p g) {Z : D}
  (h : CategoryTheory.Limits.kernel (G.map g) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernelComparison f G)
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.map (G.map f) (G.map g) (G.map p) (G.map q) ⋯)
        h) =
    CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.kernel.map f g p q hpq))
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernelComparison g G) h)

Defined in

Mathlib.CategoryTheory.Limits.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.Limits.HasKernelCategoryTheory.Limits.HasKernel