Theorems · Theorem · category theory

CategoryTheory.ConcreteCategory.isIso_iff_bijective

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {FC : C → C → Type u_1} {CC : C → Type w}
  [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC]
  [(CategoryTheory.forget C).ReflectsIsomorphisms] {X Y : C} (f : X ⟶ Y),
  CategoryTheory.IsIso f ↔ Function.Bijective ⇑(CategoryTheory.ConcreteCategory.hom f)

If the forgetful functor of a concrete category reflects isomorphisms, being an isomorphism is equivalent to being bijective.

Defined in

Mathlib.CategoryTheory.ConcreteCategory.EpiMono

Cited by

12 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Functor.ReflectsIsomorphisms