Theorems · Theorem · category theory

CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv_assoc

∀ {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)]
  [inst_7 : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) G] {X' Y' : C} (g : X' ⟶ Y')
  [inst_8 : CategoryTheory.Limits.HasKernel g] [inst_9 : CategoryTheory.Limits.HasKernel (G.map g)]
  [inst_10 : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
  (hpq : CategoryTheory.CategoryStruct.comp f q = CategoryTheory.CategoryStruct.comp p g) {Z : D}
  (h : G.obj (CategoryTheory.Limits.kernel g) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.map (G.map f) (G.map g) (G.map p) (G.map q) ⋯)
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesKernel.iso G g).inv h) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesKernel.iso G f).inv
      (CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.kernel.map f g p q hpq)) h)

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Cited by

0 results in Mathlib

Foundations

Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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