Theorems · Theorem · category theory
CategoryTheory.isIso_of_reflects_iso
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] {A B : C} (f : A ⟶ B) (F : CategoryTheory.Functor C D)
[CategoryTheory.IsIso (F.map f)] [F.ReflectsIsomorphisms], CategoryTheory.IsIso fIf F reflects isos and F.map f is an iso, then f is an iso.
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- Quiver.Homstatement and proof · cited by 32,603
- CategoryTheory.Functor.objstatement · cited by 19,642
- CategoryTheory.Functorstatement and proof · cited by 16,252
- CategoryTheory.Functor.mapstatement and proof · cited by 8,698
- CategoryTheory.IsIsostatement and proof · cited by 1,156
- CategoryTheory.Functor.ReflectsIsomorphismsstatement and proof · cited by 82
- CategoryTheory.Functor.ReflectsIsomorphisms.reflectsproof · cited by 3
Cited by19
Results whose statement or proof uses this declaration.
- CategoryTheory.isIso_iff_of_reflects_isoproof · cited by 20
- CategoryTheory.ConcreteCategory.isIso_iff_bijectiveproof · cited by 12
- SimplexCategory.isIso_of_bijectiveproof · cited by 2
- TopCat.Presheaf.app_isIso_of_stalkFunctor_map_isoproof · cited by 2
- CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesLeftHomologyproof · cited by 1
- CategoryTheory.Sheaf.isConstant_of_forgetproof · cited by 1
- CategoryTheory.PreGaloisCategory.isIso_of_mono_of_eq_card_fiberproof · cited by 1
- CategoryTheory.GrothendieckTopology.Plus.isSheaf_of_sepproof · cited by 1
- CategoryTheory.Limits.reflectsColimit_of_reflectsIsomorphismsproof · cited by 1
- CategoryTheory.FintypeCat.Action.isConnected_of_transitiveproof · cited by 1
- CategoryTheory.Limits.reflectsLimit_of_reflectsIsomorphismsproof · cited by 1