| import Mathlib | |
| open CategoryTheory Limits TopologicalSpace | |
| namespace CAT_statement_S_0017 | |
| universe u uX | |
| variable {X : Type uX} [Category.{vX} X] | |
| namespace AHS | |
| structure ConcreteCat (X : Type v) [Category X] where | |
| C : Type u | |
| [cat : Category C] | |
| U : C ⥤ X | |
| [U_Faithful : U.Faithful] | |
| attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful | |
| def IsInitialHom {C : ConcreteCat (X:= X)} {A B : C.C} (f : A ⟶ B) : Prop := | |
| ∀ ⦃Z : C.C⦄ (g : C.U.obj Z ⟶ C.U.obj A), | |
| (∃ h : Z ⟶ B, C.U.map h = g ≫ C.U.map f) → | |
| (∃ k : Z ⟶ A, C.U.map k = g) | |
| def IsEmbedding {C : ConcreteCat (X:= X)} {A B : C.C} (f : A ⟶ B) : Prop := | |
| IsInitialHom f ∧ Mono (C.U.map f) | |
| def IsInjectiveObj {C : ConcreteCat (X:= X)} (I : C.C) : Prop := | |
| ∀ ⦃A B : C.C⦄ (m : A ⟶ B), | |
| IsEmbedding m → | |
| ∀ (f : A ⟶ I), ∃ g : B ⟶ I, m ≫ g = f | |
| end AHS | |
| def S : TopCat.{u} := | |
| letI : TopologicalSpace (Fin 3) := generateFrom {({0, 1} : Set (Fin 3))} | |
| TopCat.of (ULift.{u} (Fin 3)) | |
| def TopCatConcrete : AHS.ConcreteCat (X := Type u) := | |
| { C := TopCat.{u} | |
| U := forget TopCat} | |
| theorem Inj_in_TopCat {Y : TopCat.{u}} : | |
| AHS.IsInjectiveObj (C:= TopCatConcrete) Y ↔∃ (I : Type u), Nonempty (Retract Y (piObj (fun (_ : I) => S))) := by | |
| sorry | |
| end CAT_statement_S_0017 | |