| import Mathlib.CategoryTheory.Limits.Preserves.Basic |
| import Mathlib.CategoryTheory.EssentialImage |
| import Mathlib.CategoryTheory.Discrete.Basic |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits |
| noncomputable section |
| namespace DerivedBar |
| universe u v u' v' |
| variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D] |
| (G : C ⥤ D) [G.EssSurj] (F : Discrete ℕ ⥤ C) (s : Cocone F) (hs : IsColimit s) |
| (hp : ∀ T : C, Function.Bijective (G.map : (s.pt ⟶ T) → (G.obj s.pt ⟶ G.obj T))) |
| (hF : ∀ (r : Discrete ℕ) (T : C), |
| Function.Bijective (G.map : (F.obj r ⟶ T) → (G.obj (F.obj r) ⟶ G.obj T))) |
|
|
| def homBijectionCoconeLift (t : Cocone (F ⋙ G)) : Cocone F where |
| pt := G.objPreimage t.pt |
| ι := Discrete.natTrans (fun r => |
| (Equiv.ofBijective _ (hF r (G.objPreimage t.pt))).symm |
| (t.ι.app r ≫ (G.objObjPreimageIso t.pt).inv)) |
|
|
| theorem homBijectionCoconeLift_map (t : Cocone (F ⋙ G)) (r : Discrete ℕ) : |
| G.map ((homBijectionCoconeLift G F hF t).ι.app r) = |
| t.ι.app r ≫ (G.objObjPreimageIso t.pt).inv := |
| (Equiv.ofBijective _ (hF r (G.objPreimage t.pt))).apply_symm_apply _ |
|
|
| /-- A coproduct's full Hom universal property transfers through an essentially |
| surjective functor when Hom is bijective on its summands and its point. -/ |
| def mapCoproductOfHomBijections : IsColimit (G.mapCocone s) where |
| desc t := G.map (hs.desc (homBijectionCoconeLift G F hF t)) ≫ (G.objObjPreimageIso t.pt).hom |
| fac t r := by |
| change G.map (s.ι.app r) ≫ G.map (hs.desc (homBijectionCoconeLift G F hF t)) ≫ |
| (G.objObjPreimageIso t.pt).hom = t.ι.app r |
| erw [← Category.assoc, ← Functor.map_comp, hs.fac, homBijectionCoconeLift_map] |
| change (t.ι.app r ≫ (G.objObjPreimageIso t.pt).inv) ≫ (G.objObjPreimageIso t.pt).hom = _ |
| simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id] |
| uniq t m hm := by |
| let e := G.objObjPreimageIso t.pt |
| let u := (Equiv.ofBijective _ (hp (G.objPreimage t.pt))).symm (m ≫ e.inv) |
| have hu : G.map u=m ≫ e.inv := |
| (Equiv.ofBijective _ (hp (G.objPreimage t.pt))).apply_symm_apply _ |
| have heq : u=hs.desc (homBijectionCoconeLift G F hF t) := by |
| apply hs.hom_ext |
| intro r |
| erw [hs.fac] |
| apply (hF r (G.objPreimage t.pt)).injective |
| have hmr := hm r |
| change G.map (s.ι.app r) ≫ m = t.ι.app r at hmr |
| have hleft : G.map (s.ι.app r ≫ u) = t.ι.app r ≫ e.inv := by |
| rw [Functor.map_comp, hu, ← Category.assoc, hmr] |
| exact hleft.trans (homBijectionCoconeLift_map G F hF t r).symm |
| calc |
| m = G.map u ≫ e.hom := by rw [hu]; simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id] |
| _ = _ := by rw [heq]; rfl |
| end DerivedBar |
| |
| end |
| |
| end OAI |
| |
