Theorems · Theorem · category theory

CategoryTheory.LiftLeftAdjoint.constructLeftAdjointEquiv_symm_apply

∀ {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) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U)
  [inst_3 : CategoryTheory.Limits.HasReflexiveCoequalizers A]
  (h : (X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)) (Y : A) (X : B) (a : X ⟶ R.obj Y),
  (CategoryTheory.LiftLeftAdjoint.constructLeftAdjointEquiv R F' adj₁ adj₂ h Y X).symm a =
    (CategoryTheory.Limits.Cofork.IsColimit.homIso
          (CategoryTheory.Limits.colimit.isColimit
            (CategoryTheory.Limits.parallelPair (F'.map (U.map (adj₁.counit.app X)))
              (CategoryTheory.LiftLeftAdjoint.otherMap R F' adj₁ adj₂ X)))
          Y).symm
      ⟨(adj₂.homEquiv (U.obj X) Y).symm
          ((adj₁.homEquiv (U.obj X) (R.obj Y))
            ↑((CategoryTheory.Limits.Cofork.IsColimit.homIso (CategoryTheory.LiftLeftAdjoint.counitCoequalises adj₁ h X)
                  (R.obj Y))
                a)),
        ⋯⟩

Defined in

Mathlib.CategoryTheory.Adjunction.Lifting.Left

Cited by

0 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasReflexiveCoequalizers