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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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
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