Theorems · Theorem · category theory

CategoryTheory.TwoSquare.isIso_lanBaseChange_app_iff

∀ {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [inst : CategoryTheory.Category.{v₁, u₁} C₁]
  [inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] [inst_2 : CategoryTheory.Category.{v₃, u₃} C₃]
  [inst_3 : CategoryTheory.Category.{v₄, u₄} C₄] [inst_4 : CategoryTheory.Category.{v₅, u₅} D]
  {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄}
  {B : CategoryTheory.Functor C₃ C₄} (w : CategoryTheory.TwoSquare T L R B)
  [inst_5 : ∀ (F : CategoryTheory.Functor C₁ D), L.HasLeftKanExtension F]
  [inst_6 : ∀ (F : CategoryTheory.Functor C₂ D), R.HasLeftKanExtension F] (F : CategoryTheory.Functor C₂ D),
  CategoryTheory.IsIso (w.lanBaseChange.app F) ↔
    (CategoryTheory.StructuredArrow.right
          ((CategoryTheory.Functor.LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare
            w)).IsLeftKanExtension
      (CategoryTheory.StructuredArrow.hom
        ((CategoryTheory.Functor.LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w))

Defined in

Mathlib.CategoryTheory.GuitartExact.KanExtension

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.HasLeftKanExtensionCategoryTheory.Functor.HasLeftKanExtension