Theorems · Theorem · category theory

CategoryTheory.Adjunction.Quadruple.epi_leftTriple_rightToLeft_iff_mono_rightTriple_leftToRight

∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor D C} {G : CategoryTheory.Functor C D}
  {R : CategoryTheory.Functor D C} (q : CategoryTheory.Adjunction.Quadruple L F G R) [inst_2 : F.Full]
  [inst_3 : F.Faithful] [CategoryTheory.Limits.HasPullbacks C] [CategoryTheory.Limits.HasPushouts D],
  CategoryTheory.Epi q.leftTriple.rightToLeft ↔ CategoryTheory.Mono q.rightTriple.leftToRight

For an adjoint quadruple L ⊣ F ⊣ G ⊣ R where F (and hence also R) is fully faithful and its domain / codomain has all pushouts resp. pullbacks, the natural transformation G ⟶ L is an epimorphism iff the natural transformation F ⟶ R is a monomorphism.

Defined in

Mathlib.CategoryTheory.Adjunction.Quadruple

Cited by

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Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.Limits.HasPullbacksCategoryTheory.Limits.HasPushouts