Theorems · Definition · category theory

CategoryTheory.LiftLeftAdjoint.otherMap

{A : Type u₁} →
  {B : Type u₂} →
    {C : Type u₃} →
      [inst : CategoryTheory.Category.{v₁, u₁} A] →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} B] →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} C] →
            {U : CategoryTheory.Functor B C} →
              {F : CategoryTheory.Functor C B} →
                (R : CategoryTheory.Functor A B) →
                  (F' : CategoryTheory.Functor C A) →
                    (F ⊣ U) → (F' ⊣ R.comp U) → (X : B) → F'.obj (U.obj (F.obj (U.obj X))) ⟶ F'.obj (U.obj X)

(Implementation) To construct the left adjoint, we use the coequalizer of F' U ε_Y with the composite F' U F U X ⟶ F' U F U R F' U X ⟶ F' U R F' U X ⟶ F' U X where the first morphism is F' U F ι_UX, the second is F' U ε_RF'UX, and the third is δ_F'UX. We will show that this coequalizer exists and that it forms the object map for a left adjoint to R.

Defined in

Mathlib.CategoryTheory.Adjunction.Lifting.Left

Cited by

2 results in Mathlib

Foundations

Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category