| import Mathlib | |
| open CategoryTheory | |
| namespace CAT_statement_S_0014 | |
| universe u uX | |
| variable {X : Type uX} [Category.{vX} X] | |
| namespace AHS2 | |
| 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 AHS2 | |
| namespace Poset | |
| def PosetConcrete : AHS2.ConcreteCat (Type u) where | |
| C := PartOrd.{u} | |
| cat := inferInstance | |
| U := forget PartOrd | |
| U_Faithful := inferInstance | |
| theorem injective_iff_suplattice (P : PartOrd.{u}) : | |
| AHS2.IsInjectiveObj (C := PosetConcrete) P ↔ ∀ (s : Set P), ∃ x, IsLUB s x := by | |
| sorry | |
| end Poset | |
| end CAT_statement_S_0014 | |