| 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 |
|
|
