Theorems · Definition · category theory

CategoryTheory.Functor.splitMonoEquiv

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        (F : CategoryTheory.Functor C D) →
          {X Y : C} →
            (f : Y ⟶ X) → [F.Full] → [F.Faithful] → CategoryTheory.SplitMono f ≃ CategoryTheory.SplitMono (F.map f)

If F is a fully faithful functor, split monomorphisms are preserved and reflected by F.

Defined in

Mathlib.CategoryTheory.Functor.EpiMono

Cited by

1 results in Mathlib

Foundations

Depth 15 from the axioms · uses propext, Classical.choice

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.Faithful