Type u is well-powered #
By building a categorical equivalence MonoOver α ≌ Set α for any α : Type u,
we deduce that Subobject α ≃o Set α and that Type u is well-powered.
One would hope that for a particular concrete category C (AddCommGroup, etc)
it's viable to prove [WellPowered C] without explicitly aligning Subobject X
with the "hand-rolled" definition of subobjects.
This may be possible using Lawvere theories, but it remains to be seen whether this just pushes lumps around in the carpet.
theorem
subtype_val_mono
{α : Type u}
(s : Set α)
:
CategoryTheory.Mono (CategoryTheory.asHom Subtype.val)
@[simp]
theorem
Types.monoOverEquivalenceSet_inverse_obj
(α : Type u)
(s : Set α)
:
(Types.monoOverEquivalenceSet α).inverse.obj s = CategoryTheory.MonoOver.mk' Subtype.val
@[simp]
theorem
Types.monoOverEquivalenceSet_counitIso
(α : Type u)
:
(Types.monoOverEquivalenceSet α).counitIso = CategoryTheory.NatIso.ofComponents (fun (s : Set α) => CategoryTheory.eqToIso ⋯) ⋯
@[simp]
theorem
Types.monoOverEquivalenceSet_inverse_map
(α : Type u)
{s : Set α}
{t : Set α}
(b : s ⟶ t)
:
(Types.monoOverEquivalenceSet α).inverse.map b = CategoryTheory.MonoOver.homMk
(fun (w : ((fun (s : Set α) => CategoryTheory.MonoOver.mk' Subtype.val) s).obj.left) =>
{ val := ↑w, property := ⋯ })
⋯
@[simp]
theorem
Types.monoOverEquivalenceSet_functor_obj
(α : Type u)
(f : CategoryTheory.MonoOver α)
:
(Types.monoOverEquivalenceSet α).functor.obj f = Set.range f.obj.hom
@[simp]
theorem
Types.monoOverEquivalenceSet_functor_map
(α : Type u)
{f : CategoryTheory.MonoOver α}
{g : CategoryTheory.MonoOver α}
(t : f ⟶ g)
:
(Types.monoOverEquivalenceSet α).functor.map t = CategoryTheory.homOfLE ⋯
@[simp]
theorem
Types.monoOverEquivalenceSet_unitIso
(α : Type u)
:
(Types.monoOverEquivalenceSet α).unitIso = CategoryTheory.NatIso.ofComponents
(fun (f : CategoryTheory.MonoOver α) =>
CategoryTheory.MonoOver.isoMk (Equiv.toIso (Equiv.ofInjective f.obj.hom ⋯)) ⋯)
⋯
The category of MonoOver α, for α : Type u, is equivalent to the partial order Set α.
Equations
- One or more equations did not get rendered due to their size.