Theorems · Theorem · category theory

CategoryTheory.Limits.pullback_map_diagonal_isPullback

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X Y : C} [CategoryTheory.Limits.HasPullbacks C]
  {S T : C} (f : X ⟶ T) (g : Y ⟶ T) (i : T ⟶ S) [inst_2 : CategoryTheory.Limits.HasPullback i i]
  [inst_3 : CategoryTheory.Limits.HasPullback f g]
  [inst_4 :
    CategoryTheory.Limits.HasPullback (CategoryTheory.CategoryStruct.comp f i) (CategoryTheory.CategoryStruct.comp g i)]
  [CategoryTheory.Limits.HasPullback (CategoryTheory.Limits.pullback.diagonal i)
      (CategoryTheory.Limits.pullback.map (CategoryTheory.CategoryStruct.comp f i)
        (CategoryTheory.CategoryStruct.comp g i) i i f g (CategoryTheory.CategoryStruct.id S) ⋯ ⋯)],
  CategoryTheory.IsPullback (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst f g) f)
    (CategoryTheory.Limits.pullback.map f g (CategoryTheory.CategoryStruct.comp f i)
      (CategoryTheory.CategoryStruct.comp g i) (CategoryTheory.CategoryStruct.id X) (CategoryTheory.CategoryStruct.id Y)
      i ⋯ ⋯)
    (CategoryTheory.Limits.pullback.diagonal i)
    (CategoryTheory.Limits.pullback.map (CategoryTheory.CategoryStruct.comp f i)
      (CategoryTheory.CategoryStruct.comp g i) i i f g (CategoryTheory.CategoryStruct.id S) ⋯ ⋯)

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Cited by

2 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasPullbacksCategoryTheory.Limits.HasPullbackCategoryTheory.Limits.HasPullbackCategoryTheory.Limits.HasPullbackCategoryTheory.Limits.HasPullback