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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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.FiniteSumTransport
variable {C D E : Type*} [Category C] [Category D] [Category E]
[Preadditive C] [Preadditive D] [Preadditive E]
[HasFiniteBiproducts C] [HasFiniteBiproducts D]
variable (F : C ⥤ E) (G : D ⥤ E) [F.Additive] [G.Additive]
variable {I : Type} [Finite I] (X : I → C) (Y : I → D)
variable (α : ∀ i, F.obj (X i) ≅ G.obj (Y i))
def iso : F.obj (⨁ X) ≅ G.obj (⨁ Y) :=
F.mapBiproduct X ≪≫ biproduct.mapIso α ≪≫ (G.mapBiproduct Y).symm
@[reassoc (attr := simp)]
theorem injection (i : I) :
F.map (biproduct.ι X i) ≫ (iso F G X Y α).hom =
(α i).hom ≫ G.map (biproduct.ι Y i) := by
have hi : F.map (biproduct.ι X i) ≫ (F.mapBiproduct X).hom =
biproduct.ι (fun j => F.obj (X j)) i := by
apply (cancel_mono (F.mapBiproduct X).inv).1
rw [Category.assoc,Iso.hom_inv_id,Category.comp_id]
simp only [Functor.mapBiproduct_inv,biproduct.ι_desc]
simp only [iso,Iso.trans_hom,Iso.symm_hom,← Category.assoc,hi]
simp [biproduct.mapIso_hom]
erw [Category.assoc, biproduct.ι_desc]
rfl
/-- Coordinate isomorphisms commute with the supplied map from a finite sum. -/
theorem desc_square {U : C} {V : D} (β : F.obj U ≅ G.obj V)
(f : ∀ i, X i ⟶ U) (g : ∀ i, Y i ⟶ V)
(h : ∀ i, (α i).hom ≫ G.map (g i) = F.map (f i) ≫ β.hom) :
(iso F G X Y α).hom ≫ G.map (biproduct.desc g) =
F.map (biproduct.desc f) ≫ β.hom := by
apply (cancel_epi (F.mapBiproduct X).inv).1
apply biproduct.hom_ext'
intro i
simp only [Functor.mapBiproduct_inv,biproduct.ι_desc_assoc]
rw [← Category.assoc,injection,Category.assoc,← Functor.map_comp,
biproduct.ι_desc,h]
rw [← Category.assoc,← Functor.map_comp,biproduct.ι_desc]
@[reassoc (attr := simp)]
theorem projection (i : I) :
(iso F G X Y α).hom ≫ G.map (biproduct.π Y i) =
F.map (biproduct.π X i) ≫ (α i).hom := by
apply (cancel_epi (F.mapBiproduct X).inv).1
apply biproduct.hom_ext'
intro j
simp only [Functor.mapBiproduct_inv,biproduct.ι_desc_assoc]
rw [← Category.assoc,injection,Category.assoc,← Functor.map_comp,
← Category.assoc (F.map (biproduct.ι X j)),← Functor.map_comp]
classical
by_cases hji : j=i
· subst j; simp
· simp [hji]
/-- Entrywise coordinate squares identify the entire supplied finite matrix map. -/
theorem matrix_square {J : Type} [Finite J] (X' : J → C) (Y' : J → D)
(β : ∀ j, F.obj (X' j) ≅ G.obj (Y' j))
(f : (⨁ X) ⟶ (⨁ X')) (g : (⨁ Y) ⟶ (⨁ Y'))
(h : ∀ i j, (α i).hom ≫ G.map (biproduct.ι Y i ≫ g ≫ biproduct.π Y' j) =
F.map (biproduct.ι X i ≫ f ≫ biproduct.π X' j) ≫ (β j).hom) :
(iso F G X Y α).hom ≫ G.map g = F.map f ≫ (iso F G X' Y' β).hom := by
apply (cancel_epi (F.mapBiproduct X).inv).1
apply biproduct.hom_ext'
intro i
simp only [Functor.mapBiproduct_inv,biproduct.ι_desc_assoc]
apply (cancel_mono (G.mapBiproduct Y').hom).1
apply biproduct.hom_ext
intro j
have hp : (G.mapBiproduct Y').hom ≫ biproduct.π (G.obj ∘ Y') j =
G.map (biproduct.π Y' j) := by
change (biproduct.lift (fun j => G.map (biproduct.π Y' j))) ≫
biproduct.π (fun j => G.obj (Y' j)) j = _
exact biproduct.lift_π _ j
simp only [Category.assoc,hp,injection_assoc,projection]
simpa only [Functor.map_comp,Category.assoc] using h i j
end LittleFinitistic.FiniteSumTransport
end
end OAI
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