| import Mathlib | |
| open CategoryTheory | |
| namespace CAT_statement_S_0037 | |
| universe u v w | |
| variable {X : Type uX} [Category.{vX} X] | |
| 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 IsConcreteFunc {A B : ConcreteCat (X := X)} (F : A.C ⥤ B.C) : Prop := | |
| Nonempty ((F ⋙ B.U) ≅ A.U) | |
| def SetConcrete : ConcreteCat (X := Type u) := | |
| { C := Type u | |
| U := 𝟭 (Type u) } | |
| def TopConcrete : ConcreteCat (X := Type u) := | |
| { C := TopCat.{u} | |
| U := (forget TopCat) } | |
| def ConcreteFuncs (A B : ConcreteCat (X := Type u)) : Type _ := | |
| { F : A.C ⥤ B.C // IsConcreteFunc (A := A) (B := B) F } | |
| def ConcreteFuncsSetoid (A B : ConcreteCat (X := Type u)) : | |
| Setoid (ConcreteFuncs A B) where | |
| r F G := Nonempty (F.1 ≅ G.1) | |
| iseqv := by | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro F | |
| exact ⟨Iso.refl F.1⟩ | |
| · intro F G h | |
| rcases h with ⟨e⟩ | |
| exact ⟨e.symm⟩ | |
| · intro F G H hFG hGH | |
| rcases hFG with ⟨eFG⟩ | |
| rcases hGH with ⟨eGH⟩ | |
| exact ⟨eFG.trans eGH⟩ | |
| def ConcreteFuncClasses (A B : ConcreteCat (X := Type u)) : Type _ := | |
| Quotient (ConcreteFuncsSetoid A B) | |
| theorem only_two_concrete_functors_from_Set_to_Top_iso : | |
| Nat.card (ConcreteFuncClasses SetConcrete TopConcrete) = 2 := by | |
| sorry | |
| end CAT_statement_S_0037 | |