Theorems · Definition · category theory

CategoryTheory.Functor.leibnizPullback

{C₁ : Type u₁} →
  {C₂ : Type u₂} →
    {C₃ : Type u₃} →
      [inst : CategoryTheory.Category.{v₁, u₁} C₁] →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} C₃] →
            CategoryTheory.Functor C₁ᵒᵖ (CategoryTheory.Functor C₃ C₂) →
              [CategoryTheory.Limits.HasPullbacks C₂] →
                CategoryTheory.Functor (CategoryTheory.Arrow C₁)ᵒᵖ
                  (CategoryTheory.Functor (CategoryTheory.Arrow C₃) (CategoryTheory.Arrow C₂))

Given a bifunctor G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂ to a category C₂ which has pullbacks, the Leibniz pullback (pullback-power) of f₁ : X₁ ⟶ Y₁ in C₁ and f₃ : X₃ ⟶ Y₃ in C₃ is the map (G.obj (op Y₁)).obj X₃ ⟶ pullback ((G.obj (op X₁)).map f₃) ((G.map f₁.op).app Y₃) induced by the diagram `` (G.obj (op Y₁)).obj X₃ -> (G.obj (op X₁)).obj X₃ | | | | v v (G.obj (op Y₁)).obj Y₃ -> (G.obj (op X₁)).obj Y₃ ``

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Cited by

8 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasPullbacks