Theorems · Theorem · category theory

CategoryTheory.Limits.pullback_diagonal_map_snd_fst_fst

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X Y : C} [inst_1 : CategoryTheory.Limits.HasPullbacks C]
  {U V₁ V₂ : C} (f : X ⟶ Y) (i : U ⟶ Y) (i₁ : V₁ ⟶ CategoryTheory.Limits.pullback f i)
  (i₂ : V₂ ⟶ CategoryTheory.Limits.pullback f i),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.pullback.snd (CategoryTheory.Limits.pullback.diagonal f)
        (CategoryTheory.Limits.pullback.map
          (CategoryTheory.CategoryStruct.comp i₁ (CategoryTheory.Limits.pullback.snd f i))
          (CategoryTheory.CategoryStruct.comp i₂ (CategoryTheory.Limits.pullback.snd f i)) f f
          (CategoryTheory.CategoryStruct.comp i₁ (CategoryTheory.Limits.pullback.fst f i))
          (CategoryTheory.CategoryStruct.comp i₂ (CategoryTheory.Limits.pullback.fst f i)) i ⋯ ⋯))
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Limits.pullback.fst
          (CategoryTheory.CategoryStruct.comp i₁ (CategoryTheory.Limits.pullback.snd f i))
          (CategoryTheory.CategoryStruct.comp i₂ (CategoryTheory.Limits.pullback.snd f i)))
        (CategoryTheory.CategoryStruct.comp i₁ (CategoryTheory.Limits.pullback.fst f i))) =
    CategoryTheory.Limits.pullback.fst (CategoryTheory.Limits.pullback.diagonal f)
      (CategoryTheory.Limits.pullback.map
        (CategoryTheory.CategoryStruct.comp i₁ (CategoryTheory.Limits.pullback.snd f i))
        (CategoryTheory.CategoryStruct.comp i₂ (CategoryTheory.Limits.pullback.snd f i)) f f
        (CategoryTheory.CategoryStruct.comp i₁ (CategoryTheory.Limits.pullback.fst f i))
        (CategoryTheory.CategoryStruct.comp i₂ (CategoryTheory.Limits.pullback.fst f i)) i ⋯ ⋯)

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Cited by

1 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasPullbacks