feat(category_theory/preadditive): additive functors preserve biprodu…

…cts (#6882)

An additive functor between preadditive categories creates and preserves biproducts.

Also, renames `src/category_theory/abelian/additive_functor.lean` to `src/category_theory/preadditive/additive_functor.lean` as it so far doesn't actually rely on anything being abelian. 

Also, moves the `.map_X` lemmas about additive functors into the `functor` namespace, so we can use dot notation `F.map_add` etc, when `[functor.additive F]` is available.



Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

src/algebra/homology/chain_complex.lean

@@ -6,7 +6,7 @@ Authors: Scott Morrison
 import data.int.basic
 import category_theory.graded_object
 import category_theory.differential_object
-import category_theory.abelian.additive_functor
+import category_theory.preadditive.additive_functor
 
 /-!
 # Chain complexes

src/category_theory/functor.lean

@@ -90,6 +90,11 @@ infixr ` ⋙ `:80 := comp
 protected lemma comp_id (F : C ⥤ D) : F ⋙ (𝟭 D) = F := by cases F; refl
 protected lemma id_comp (F : C ⥤ D) : (𝟭 C) ⋙ F = F := by cases F; refl
 
+@[simp] lemma map_dite (F : C ⥤ D) {X Y : C} {P : Prop} [decidable P]
+  (f : P → (X ⟶ Y)) (g : ¬P → (X ⟶ Y)) :
+  F.map (if h : P then f h else g h) = if h : P then F.map (f h) else F.map (g h) :=
+by { split_ifs; refl, }
+
 end
 
 @[mono] lemma monotone {α β : Type*} [preorder α] [preorder β] (F : α ⥤ β) :

src/category_theory/preadditive/additive_functor.lean

@@ -0,0 +1,135 @@
+/-
+Copyright (c) 2021 Adam Topaz. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Adam Topaz, Scott Morrison
+-/
+import category_theory.preadditive
+import category_theory.limits.shapes.biproducts
+
+/-!
+# Additive Functors
+
+A functor between two preadditive categories is called *additive*
+provided that the induced map on hom types is a morphism of abelian
+groups.
+
+An additive functor between preadditive categories creates and preserves biproducts.
+
+# Implementation details
+
+`functor.additive` is a `Prop`-valued class, defined by saying that
+for every two objects `X` and `Y`, the map
+`F.map : (X ⟶ Y) → (F.obj X ⟶ F.obj Y)` is a morphism of abelian
+groups.
+
+# Project:
+
+- Prove that a functor is additive if it preserves finite biproducts
+  (See https://stacks.math.columbia.edu/tag/010M.)
+-/
+
+namespace category_theory
+
+/-- A functor `F` is additive provided `F.map` is an additive homomorphism. -/
+class functor.additive {C D : Type*} [category C] [category D]
+  [preadditive C] [preadditive D] (F : C ⥤ D) : Prop :=
+(map_zero' : Π {X Y : C}, F.map (0 : X ⟶ Y) = 0 . obviously)
+(map_add' : Π {X Y : C} {f g : X ⟶ Y}, F.map (f + g) = F.map f + F.map g . obviously)
+
+section preadditive
+
+namespace functor
+
+section
+variables {C D : Type*} [category C] [category D] [preadditive C]
+  [preadditive D] (F : C ⥤ D) [functor.additive F]
+
+@[simp]
+lemma map_zero {X Y : C} : F.map (0 : X ⟶ Y) = 0 :=
+functor.additive.map_zero'
+
+@[simp]
+lemma map_add {X Y : C} {f g : X ⟶ Y} : F.map (f + g) = F.map f + F.map g :=
+functor.additive.map_add'
+
+instance : additive (𝟭 C) :=
+{}
+
+instance {E : Type*} [category E] [preadditive E] (G : D ⥤ E) [functor.additive G] :
+  additive (F ⋙ G) :=
+{}
+
+/-- `F.map_add_hom` is an additive homomorphism whose underlying function is `F.map`. -/
+@[simps]
+def map_add_hom {X Y : C} : (X ⟶ Y) →+ (F.obj X ⟶ F.obj Y) :=
+{ to_fun := λ f, F.map f,
+  map_zero' := F.map_zero,
+  map_add' := λ _ _, F.map_add }
+
+lemma coe_map_add_hom {X Y : C} : ⇑(F.map_add_hom : (X ⟶ Y) →+ _) = @map C _ D _ F X Y := rfl
+
+@[simp]
+lemma map_neg {X Y : C} {f : X ⟶ Y} : F.map (-f) = - F.map f :=
+F.map_add_hom.map_neg _
+
+@[simp]
+lemma map_sub {X Y : C} {f g : X ⟶ Y} : F.map (f - g) = F.map f - F.map g :=
+F.map_add_hom.map_sub _ _
+
+open_locale big_operators
+
+@[simp]
+lemma map_sum {X Y : C} {α : Type*} (f : α → (X ⟶ Y)) (s : finset α) :
+  F.map (∑ a in s, f a) = ∑ a in s, F.map (f a) :=
+(F.map_add_hom : (X ⟶ Y) →+ _).map_sum f s
+
+end
+
+section
+-- To talk about preservation of biproducts we need to specify universes explicitly.
+
+noncomputable theory
+universes v u₁ u₂
+
+variables {C : Type u₁} {D : Type u₂} [category.{v} C] [category.{v} D]
+  [preadditive C] [preadditive D] (F : C ⥤ D) [functor.additive F]
+
+open category_theory.limits
+
+/--
+An additive functor between preadditive categories creates finite biproducts.
+-/
+instance map_has_biproduct {J : Type v} [fintype J] [decidable_eq J] (f : J → C) [has_biproduct f] :
+  has_biproduct (λ j, F.obj (f j)) :=
+has_biproduct_of_total
+{ X := F.obj (⨁ f),
+  π := λ j, F.map (biproduct.π f j),
+  ι := λ j, F.map (biproduct.ι f j),
+  ι_π := λ j j', by { simp only [←F.map_comp], split_ifs, { subst h, simp, }, { simp [h], }, }, }
+(by simp_rw [←F.map_comp, ←F.map_sum, biproduct.total, functor.map_id])
+
+/--
+An additive functor between preadditive categories preserves finite biproducts.
+-/
+-- This essentially repeats the work of the previous instance,
+-- but gives good definitional reduction to `biproduct.lift` and `biproduct.desc`.
+@[simps]
+def map_biproduct {J : Type v} [fintype J] [decidable_eq J] (f : J → C) [has_biproduct f] :
+  F.obj (⨁ f) ≅ ⨁ (λ j, F.obj (f j)) :=
+{ hom := biproduct.lift (λ j, F.map (biproduct.π f j)),
+  inv := biproduct.desc (λ j, F.map (biproduct.ι f j)),
+  hom_inv_id' :=
+  by simp only [biproduct.lift_desc, ←F.map_comp, ←F.map_sum, biproduct.total, F.map_id],
+  inv_hom_id' :=
+  begin
+    ext j j',
+    simp only [category.comp_id,  category.assoc, biproduct.lift_π, biproduct.ι_desc_assoc,
+      ←F.map_comp, biproduct.ι_π, F.map_dite, dif_ctx_congr, eq_to_hom_map, F.map_zero],
+  end }
+
+end
+
+end functor
+end preadditive
+
+end category_theory

src/category_theory/triangulated/basic.lean

@@ -5,7 +5,7 @@ Authors: Luke Kershaw
 -/
 import category_theory.additive.basic
 import category_theory.shift
-import category_theory.abelian.additive_functor
+import category_theory.preadditive.additive_functor
 
 /-!
 # Triangles

src/category_theory/triangulated/rotate.lean

@@ -5,7 +5,7 @@ Authors: Luke Kershaw
 -/
 import category_theory.additive.basic
 import category_theory.shift
-import category_theory.abelian.additive_functor
+import category_theory.preadditive.additive_functor
 import category_theory.natural_isomorphism
 import category_theory.triangulated.basic