algebraic_topology.dold_kan.equivalence_pseudoabelian ⟷ Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -67,21 +67,17 @@ def Γ : ChainComplex C ℕ ⥤ SimplicialObject C :=
 #align category_theory.idempotents.dold_kan.Γ CategoryTheory.Idempotents.DoldKan.Γ
 -/
 
-#print CategoryTheory.Idempotents.DoldKan.hN₁ /-
 theorem hN₁ :
     (toKaroubiEquivalence (SimplicialObject C)).Functor ⋙ Preadditive.DoldKan.equivalence.Functor =
       N₁ :=
   Functor.congr_obj (functorExtension₁_comp_whiskeringLeft_toKaroubi _ _) N₁
 #align category_theory.idempotents.dold_kan.hN₁ CategoryTheory.Idempotents.DoldKan.hN₁
--/
 
-#print CategoryTheory.Idempotents.DoldKan.hΓ₀ /-
 theorem hΓ₀ :
     (toKaroubiEquivalence (ChainComplex C ℕ)).Functor ⋙ Preadditive.DoldKan.equivalence.inverse =
       Γ ⋙ (toKaroubiEquivalence _).Functor :=
   Functor.congr_obj (functorExtension₂_comp_whiskeringLeft_toKaroubi _ _) Γ₀
 #align category_theory.idempotents.dold_kan.hΓ₀ CategoryTheory.Idempotents.DoldKan.hΓ₀
--/
 
 #print CategoryTheory.Idempotents.DoldKan.equivalence /-
 /-- The Dold-Kan equivalence for pseudoabelian categories given

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2022 Joël Riou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joël Riou
 -/
-import Mathbin.AlgebraicTopology.DoldKan.EquivalenceAdditive
-import Mathbin.AlgebraicTopology.DoldKan.Compatibility
-import Mathbin.CategoryTheory.Idempotents.SimplicialObject
+import AlgebraicTopology.DoldKan.EquivalenceAdditive
+import AlgebraicTopology.DoldKan.Compatibility
+import CategoryTheory.Idempotents.SimplicialObject
 
 #align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -55,7 +55,7 @@ open AlgebraicTopology.DoldKan
 of the equivalence `chain_complex C ℕ ≌ karoubi (chain_complex C ℕ)`. -/
 @[simps, nolint unused_arguments]
 def N : SimplicialObject C ⥤ ChainComplex C ℕ :=
-  N₁ ⋙ (toKaroubi_equivalence _).inverse
+  N₁ ⋙ (toKaroubiEquivalence _).inverse
 #align category_theory.idempotents.dold_kan.N CategoryTheory.Idempotents.DoldKan.N
 -/
 
@@ -69,7 +69,7 @@ def Γ : ChainComplex C ℕ ⥤ SimplicialObject C :=
 
 #print CategoryTheory.Idempotents.DoldKan.hN₁ /-
 theorem hN₁ :
-    (toKaroubi_equivalence (SimplicialObject C)).Functor ⋙ Preadditive.DoldKan.equivalence.Functor =
+    (toKaroubiEquivalence (SimplicialObject C)).Functor ⋙ Preadditive.DoldKan.equivalence.Functor =
       N₁ :=
   Functor.congr_obj (functorExtension₁_comp_whiskeringLeft_toKaroubi _ _) N₁
 #align category_theory.idempotents.dold_kan.hN₁ CategoryTheory.Idempotents.DoldKan.hN₁
@@ -77,8 +77,8 @@ theorem hN₁ :
 
 #print CategoryTheory.Idempotents.DoldKan.hΓ₀ /-
 theorem hΓ₀ :
-    (toKaroubi_equivalence (ChainComplex C ℕ)).Functor ⋙ Preadditive.DoldKan.equivalence.inverse =
-      Γ ⋙ (toKaroubi_equivalence _).Functor :=
+    (toKaroubiEquivalence (ChainComplex C ℕ)).Functor ⋙ Preadditive.DoldKan.equivalence.inverse =
+      Γ ⋙ (toKaroubiEquivalence _).Functor :=
   Functor.congr_obj (functorExtension₂_comp_whiskeringLeft_toKaroubi _ _) Γ₀
 #align category_theory.idempotents.dold_kan.hΓ₀ CategoryTheory.Idempotents.DoldKan.hΓ₀
 -/
@@ -110,7 +110,7 @@ for the construction of our counit isomorphism `η` -/
 theorem hη :
     Compatibility.τ₀ =
       Compatibility.τ₁ (eqToIso hN₁) (eqToIso hΓ₀)
-        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubi_equivalence (ChainComplex C ℕ)).Functor) :=
+        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).Functor) :=
   by
   ext K : 3
   simpa only [compatibility.τ₀_hom_app, compatibility.τ₁_hom_app, eq_to_iso.hom,
@@ -124,7 +124,7 @@ theorem hη :
 @[simps]
 def η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ) :=
   Compatibility.equivalenceCounitIso
-    (N₁Γ₀ : (Γ : ChainComplex C ℕ ⥤ _) ⋙ N₁ ≅ (toKaroubi_equivalence _).Functor)
+    (N₁Γ₀ : (Γ : ChainComplex C ℕ ⥤ _) ⋙ N₁ ≅ (toKaroubiEquivalence _).Functor)
 #align category_theory.idempotents.dold_kan.η CategoryTheory.Idempotents.DoldKan.η
 -/
 
@@ -139,7 +139,7 @@ theorem equivalence_counitIso :
 theorem hε :
     Compatibility.υ (eqToIso hN₁) =
       (Γ₂N₁ :
-        (toKaroubi_equivalence _).Functor ≅
+        (toKaroubiEquivalence _).Functor ≅
           (N₁ : SimplicialObject C ⥤ _) ⋙ Preadditive.DoldKan.equivalence.inverse) :=
   by
   ext X : 4

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -49,47 +49,62 @@ namespace DoldKan
 
 open AlgebraicTopology.DoldKan
 
+#print CategoryTheory.Idempotents.DoldKan.N /-
 /-- The functor `N` for the equivalence is obtained by composing
 `N' : simplicial_object C ⥤ karoubi (chain_complex C ℕ)` and the inverse
 of the equivalence `chain_complex C ℕ ≌ karoubi (chain_complex C ℕ)`. -/
 @[simps, nolint unused_arguments]
-def n : SimplicialObject C ⥤ ChainComplex C ℕ :=
+def N : SimplicialObject C ⥤ ChainComplex C ℕ :=
   N₁ ⋙ (toKaroubi_equivalence _).inverse
-#align category_theory.idempotents.dold_kan.N CategoryTheory.Idempotents.DoldKan.n
+#align category_theory.idempotents.dold_kan.N CategoryTheory.Idempotents.DoldKan.N
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.Γ /-
 /-- The functor `Γ` for the equivalence is `Γ'`. -/
 @[simps, nolint unused_arguments]
 def Γ : ChainComplex C ℕ ⥤ SimplicialObject C :=
   Γ₀
 #align category_theory.idempotents.dold_kan.Γ CategoryTheory.Idempotents.DoldKan.Γ
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.hN₁ /-
 theorem hN₁ :
     (toKaroubi_equivalence (SimplicialObject C)).Functor ⋙ Preadditive.DoldKan.equivalence.Functor =
       N₁ :=
   Functor.congr_obj (functorExtension₁_comp_whiskeringLeft_toKaroubi _ _) N₁
 #align category_theory.idempotents.dold_kan.hN₁ CategoryTheory.Idempotents.DoldKan.hN₁
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.hΓ₀ /-
 theorem hΓ₀ :
     (toKaroubi_equivalence (ChainComplex C ℕ)).Functor ⋙ Preadditive.DoldKan.equivalence.inverse =
       Γ ⋙ (toKaroubi_equivalence _).Functor :=
   Functor.congr_obj (functorExtension₂_comp_whiskeringLeft_toKaroubi _ _) Γ₀
 #align category_theory.idempotents.dold_kan.hΓ₀ CategoryTheory.Idempotents.DoldKan.hΓ₀
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.equivalence /-
 /-- The Dold-Kan equivalence for pseudoabelian categories given
 by the functors `N` and `Γ`. It is obtained by applying the results in
 `compatibility.lean` to the equivalence `preadditive.dold_kan.equivalence`. -/
 def equivalence : SimplicialObject C ≌ ChainComplex C ℕ :=
   Compatibility.equivalence (eqToIso hN₁) (eqToIso hΓ₀)
 #align category_theory.idempotents.dold_kan.equivalence CategoryTheory.Idempotents.DoldKan.equivalence
+-/
 
-theorem equivalence_functor : (equivalence : SimplicialObject C ≌ _).Functor = n :=
+#print CategoryTheory.Idempotents.DoldKan.equivalence_functor /-
+theorem equivalence_functor : (equivalence : SimplicialObject C ≌ _).Functor = N :=
   rfl
 #align category_theory.idempotents.dold_kan.equivalence_functor CategoryTheory.Idempotents.DoldKan.equivalence_functor
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.equivalence_inverse /-
 theorem equivalence_inverse : (equivalence : SimplicialObject C ≌ _).inverse = Γ :=
   rfl
 #align category_theory.idempotents.dold_kan.equivalence_inverse CategoryTheory.Idempotents.DoldKan.equivalence_inverse
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.hη /-
 /-- The natural isomorphism `NΓ' satisfies the compatibility that is needed
 for the construction of our counit isomorphism `η` -/
 theorem hη :
@@ -102,19 +117,25 @@ theorem hη :
     preadditive.dold_kan.equivalence_counit_iso, N₂Γ₂_to_karoubi_iso_hom, eq_to_hom_map,
     eq_to_hom_trans_assoc, eq_to_hom_app] using N₂Γ₂_compatible_with_N₁Γ₀ K
 #align category_theory.idempotents.dold_kan.hη CategoryTheory.Idempotents.DoldKan.hη
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.η /-
 /-- The counit isomorphism induced by `N₁Γ₀` -/
 @[simps]
-def η : Γ ⋙ n ≅ 𝟭 (ChainComplex C ℕ) :=
+def η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ) :=
   Compatibility.equivalenceCounitIso
     (N₁Γ₀ : (Γ : ChainComplex C ℕ ⥤ _) ⋙ N₁ ≅ (toKaroubi_equivalence _).Functor)
 #align category_theory.idempotents.dold_kan.η CategoryTheory.Idempotents.DoldKan.η
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.equivalence_counitIso /-
 theorem equivalence_counitIso :
-    DoldKan.equivalence.counitIso = (η : Γ ⋙ n ≅ 𝟭 (ChainComplex C ℕ)) :=
+    DoldKan.equivalence.counitIso = (η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ)) :=
   Compatibility.equivalenceCounitIso_eq hη
 #align category_theory.idempotents.dold_kan.equivalence_counit_iso CategoryTheory.Idempotents.DoldKan.equivalence_counitIso
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.hε /-
 theorem hε :
     Compatibility.υ (eqToIso hN₁) =
       (Γ₂N₁ :
@@ -129,15 +150,20 @@ theorem hε :
   dsimp
   simpa only [id_comp, eq_to_hom_app, eq_to_hom_map, eq_to_hom_trans]
 #align category_theory.idempotents.dold_kan.hε CategoryTheory.Idempotents.DoldKan.hε
+-/
 
+#print CategoryTheory.Idempotents.DoldKan.ε /-
 /-- The unit isomorphism induced by `Γ₂N₁`. -/
-def ε : 𝟭 (SimplicialObject C) ≅ n ⋙ Γ :=
+def ε : 𝟭 (SimplicialObject C) ≅ N ⋙ Γ :=
   Compatibility.equivalenceUnitIso (eqToIso hΓ₀) Γ₂N₁
 #align category_theory.idempotents.dold_kan.ε CategoryTheory.Idempotents.DoldKan.ε
+-/
 
-theorem equivalence_unitIso : DoldKan.equivalence.unitIso = (ε : 𝟭 (SimplicialObject C) ≅ n ⋙ Γ) :=
+#print CategoryTheory.Idempotents.DoldKan.equivalence_unitIso /-
+theorem equivalence_unitIso : DoldKan.equivalence.unitIso = (ε : 𝟭 (SimplicialObject C) ≅ N ⋙ Γ) :=
   Compatibility.equivalenceUnitIso_eq hε
 #align category_theory.idempotents.dold_kan.equivalence_unit_iso CategoryTheory.Idempotents.DoldKan.equivalence_unitIso
+-/
 
 end DoldKan
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -7,7 +7,7 @@ import Mathbin.AlgebraicTopology.DoldKan.EquivalenceAdditive
 import Mathbin.AlgebraicTopology.DoldKan.Compatibility
 import Mathbin.CategoryTheory.Idempotents.SimplicialObject
 
-#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
+#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
 
 /-!
 
@@ -30,6 +30,8 @@ the composition of `N₁ : simplicial_object C ⥤ karoubi (chain_complex C ℕ)
 `idempotents.dold_kan.Γ` of the equivalence is by definition the functor
 `Γ₀` introduced in `functor_gamma.lean`.
 
+(See `equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)
+
 -/
 
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -111,7 +111,7 @@ theorem hη :
         (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by
   ext K : 3
   simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app]
-  refine' (N₂Γ₂_compatible_with_N₁Γ₀ K).trans (by simp )
+  exact (N₂Γ₂_compatible_with_N₁Γ₀ K).trans (by simp )
 #align category_theory.idempotents.dold_kan.hη CategoryTheory.Idempotents.DoldKan.hη
 
 /-- The counit isomorphism induced by `N₁Γ₀` -/

In the context of idempotent completion of categories and the Dold-Kan equivalence, constructing some isomorphisms was very slow in Lean 3 but has been much faster in Lean 4, while using equalities of functors in Lean 3 was fast, but in Lean 4 it became very slow. In this PR, we switch to using mostly isomorphisms of functors: this also became necessary in order to make the refactor #8531 possible.

@@ -66,24 +66,33 @@ def Γ [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ ⥤
 
 variable [IsIdempotentComplete C] [HasFiniteCoproducts C]
 
-theorem hN₁ :
-    (toKaroubiEquivalence (SimplicialObject C)).functor ⋙ Preadditive.DoldKan.equivalence.functor =
-      N₁ :=
-  Functor.congr_obj (functorExtension₁_comp_whiskeringLeft_toKaroubi _ _) N₁
-set_option linter.uppercaseLean3 false in
-#align category_theory.idempotents.dold_kan.hN₁ CategoryTheory.Idempotents.DoldKan.hN₁
-
-theorem hΓ₀ :
-    (toKaroubiEquivalence (ChainComplex C ℕ)).functor ⋙ Preadditive.DoldKan.equivalence.inverse =
+/-- A reformulation of the isomorphism `toKaroubi (SimplicialObject C) ⋙ N₂ ≅ N₁` -/
+def isoN₁ :
+    (toKaroubiEquivalence (SimplicialObject C)).functor ⋙
+      Preadditive.DoldKan.equivalence.functor ≅ N₁ := toKaroubiCompN₂IsoN₁
+
+@[simp]
+lemma isoN₁_hom_app_f (X : SimplicialObject C) :
+    (isoN₁.hom.app X).f = PInfty := rfl
+
+/-- A reformulation of the canonical isomorphism
+`toKaroubi (ChainComplex C ℕ) ⋙ Γ₂ ≅ Γ ⋙ toKaroubi (SimplicialObject C)`. -/
+def isoΓ₀ :
+    (toKaroubiEquivalence (ChainComplex C ℕ)).functor ⋙ Preadditive.DoldKan.equivalence.inverse ≅
       Γ ⋙ (toKaroubiEquivalence _).functor :=
-  Functor.congr_obj (functorExtension₂_comp_whiskeringLeft_toKaroubi _ _) Γ₀
-#align category_theory.idempotents.dold_kan.hΓ₀ CategoryTheory.Idempotents.DoldKan.hΓ₀
+  (functorExtension₂CompWhiskeringLeftToKaroubiIso _ _).app Γ₀
+
+@[simp]
+lemma N₂_map_isoΓ₀_hom_app_f (X : ChainComplex C ℕ) :
+    (N₂.map (isoΓ₀.hom.app X)).f = PInfty := by
+  ext
+  apply comp_id
 
 /-- The Dold-Kan equivalence for pseudoabelian categories given
 by the functors `N` and `Γ`. It is obtained by applying the results in
 `Compatibility.lean` to the equivalence `Preadditive.DoldKan.Equivalence`. -/
 def equivalence : SimplicialObject C ≌ ChainComplex C ℕ :=
-  Compatibility.equivalence (eqToIso hN₁) (eqToIso hΓ₀)
+  Compatibility.equivalence isoN₁ isoΓ₀
 #align category_theory.idempotents.dold_kan.equivalence CategoryTheory.Idempotents.DoldKan.equivalence
 
 theorem equivalence_functor : (equivalence : SimplicialObject C ≌ _).functor = N :=
@@ -98,12 +107,11 @@ theorem equivalence_inverse : (equivalence : SimplicialObject C ≌ _).inverse =
 for the construction of our counit isomorphism `η` -/
 theorem hη :
     Compatibility.τ₀ =
-      Compatibility.τ₁ (eqToIso hN₁) (eqToIso hΓ₀)
+      Compatibility.τ₁ isoN₁ isoΓ₀
         (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by
   ext K : 3
-  simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app, eqToIso.hom]
-  refine' (N₂Γ₂_compatible_with_N₁Γ₀ K).trans _
-  simp only [N₂Γ₂ToKaroubiIso_hom, eqToHom_map, eqToHom_app, eqToHom_trans_assoc]
+  simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app]
+  refine' (N₂Γ₂_compatible_with_N₁Γ₀ K).trans (by simp )
 #align category_theory.idempotents.dold_kan.hη CategoryTheory.Idempotents.DoldKan.hη
 
 /-- The counit isomorphism induced by `N₁Γ₀` -/
@@ -119,9 +127,10 @@ theorem equivalence_counitIso :
 #align category_theory.idempotents.dold_kan.equivalence_counit_iso CategoryTheory.Idempotents.DoldKan.equivalence_counitIso
 
 theorem hε :
-    Compatibility.υ (eqToIso hN₁) =
+    Compatibility.υ (isoN₁) =
       (Γ₂N₁ : (toKaroubiEquivalence _).functor ≅
           (N₁ : SimplicialObject C ⥤ _) ⋙ Preadditive.DoldKan.equivalence.inverse) := by
+  dsimp only [isoN₁]
   ext1
   rw [← cancel_epi Γ₂N₁.inv, Iso.inv_hom_id]
   ext X : 2
@@ -129,13 +138,15 @@ theorem hε :
   erw [compatibility_Γ₂N₁_Γ₂N₂_natTrans X]
   rw [Compatibility.υ_hom_app, Preadditive.DoldKan.equivalence_unitIso, Iso.app_inv, assoc]
   erw [← NatTrans.comp_app_assoc, IsIso.hom_inv_id]
-  rw [NatTrans.id_app, id_comp, NatTrans.id_app, eqToIso.hom, eqToHom_app, eqToHom_map]
-  rw [compatibility_Γ₂N₁_Γ₂N₂_inv_app, eqToHom_trans, eqToHom_refl]
+  rw [NatTrans.id_app, id_comp, NatTrans.id_app, Γ₂N₂ToKaroubiIso_inv_app]
+  dsimp only [Preadditive.DoldKan.equivalence_inverse, Preadditive.DoldKan.Γ]
+  rw [← Γ₂.map_comp, Iso.inv_hom_id_app, Γ₂.map_id]
+  rfl
 #align category_theory.idempotents.dold_kan.hε CategoryTheory.Idempotents.DoldKan.hε
 
 /-- The unit isomorphism induced by `Γ₂N₁`. -/
 def ε : 𝟭 (SimplicialObject C) ≅ N ⋙ Γ :=
-  Compatibility.equivalenceUnitIso (eqToIso hΓ₀) Γ₂N₁
+  Compatibility.equivalenceUnitIso isoΓ₀ Γ₂N₁
 #align category_theory.idempotents.dold_kan.ε CategoryTheory.Idempotents.DoldKan.ε
 
 theorem equivalence_unitIso :

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -53,14 +53,14 @@ open AlgebraicTopology.DoldKan
 `N' : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` and the inverse
 of the equivalence `ChainComplex C ℕ ≌ Karoubi (ChainComplex C ℕ)`. -/
 @[simps!, nolint unusedArguments]
-def N  [IsIdempotentComplete C] [HasFiniteCoproducts C] : SimplicialObject C ⥤ ChainComplex C ℕ :=
+def N [IsIdempotentComplete C] [HasFiniteCoproducts C] : SimplicialObject C ⥤ ChainComplex C ℕ :=
   N₁ ⋙ (toKaroubiEquivalence _).inverse
 set_option linter.uppercaseLean3 false in
 #align category_theory.idempotents.dold_kan.N CategoryTheory.Idempotents.DoldKan.N
 
 /-- The functor `Γ` for the equivalence is `Γ'`. -/
 @[simps!, nolint unusedArguments]
-def Γ  [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ ⥤ SimplicialObject C :=
+def Γ [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ ⥤ SimplicialObject C :=
   Γ₀
 #align category_theory.idempotents.dold_kan.Γ CategoryTheory.Idempotents.DoldKan.Γ
 

@@ -7,7 +7,7 @@ import Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive
 import Mathlib.AlgebraicTopology.DoldKan.Compatibility
 import Mathlib.CategoryTheory.Idempotents.SimplicialObject
 
-#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
+#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
 
 /-!
 

In this PR (which is a forward port of https://github.com/leanprover-community/mathlib/pull/17926), the Dold-Kan equivalence between simplicial objects and chain complexes in abelian categories is constructed.

Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>

@@ -30,6 +30,8 @@ the composition of `N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)`
 `Idempotents.DoldKan.Γ` of the equivalence is by definition the functor
 `Γ₀` introduced in `FunctorGamma.lean`.
 
+(See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)
+
 -/
 
 
@@ -52,7 +54,7 @@ open AlgebraicTopology.DoldKan
 of the equivalence `ChainComplex C ℕ ≌ Karoubi (ChainComplex C ℕ)`. -/
 @[simps!, nolint unusedArguments]
 def N  [IsIdempotentComplete C] [HasFiniteCoproducts C] : SimplicialObject C ⥤ ChainComplex C ℕ :=
-  N₁ ⋙ (toKaroubi_equivalence _).inverse
+  N₁ ⋙ (toKaroubiEquivalence _).inverse
 set_option linter.uppercaseLean3 false in
 #align category_theory.idempotents.dold_kan.N CategoryTheory.Idempotents.DoldKan.N
 
@@ -65,15 +67,15 @@ def Γ  [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ 
 variable [IsIdempotentComplete C] [HasFiniteCoproducts C]
 
 theorem hN₁ :
-    (toKaroubi_equivalence (SimplicialObject C)).functor ⋙ Preadditive.DoldKan.equivalence.functor =
+    (toKaroubiEquivalence (SimplicialObject C)).functor ⋙ Preadditive.DoldKan.equivalence.functor =
       N₁ :=
   Functor.congr_obj (functorExtension₁_comp_whiskeringLeft_toKaroubi _ _) N₁
 set_option linter.uppercaseLean3 false in
 #align category_theory.idempotents.dold_kan.hN₁ CategoryTheory.Idempotents.DoldKan.hN₁
 
 theorem hΓ₀ :
-    (toKaroubi_equivalence (ChainComplex C ℕ)).functor ⋙ Preadditive.DoldKan.equivalence.inverse =
-      Γ ⋙ (toKaroubi_equivalence _).functor :=
+    (toKaroubiEquivalence (ChainComplex C ℕ)).functor ⋙ Preadditive.DoldKan.equivalence.inverse =
+      Γ ⋙ (toKaroubiEquivalence _).functor :=
   Functor.congr_obj (functorExtension₂_comp_whiskeringLeft_toKaroubi _ _) Γ₀
 #align category_theory.idempotents.dold_kan.hΓ₀ CategoryTheory.Idempotents.DoldKan.hΓ₀
 
@@ -97,7 +99,7 @@ for the construction of our counit isomorphism `η` -/
 theorem hη :
     Compatibility.τ₀ =
       Compatibility.τ₁ (eqToIso hN₁) (eqToIso hΓ₀)
-        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubi_equivalence (ChainComplex C ℕ)).functor) := by
+        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by
   ext K : 3
   simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app, eqToIso.hom]
   refine' (N₂Γ₂_compatible_with_N₁Γ₀ K).trans _
@@ -108,7 +110,7 @@ theorem hη :
 @[simps!]
 def η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ) :=
   Compatibility.equivalenceCounitIso
-    (N₁Γ₀ : (Γ : ChainComplex C ℕ ⥤ _) ⋙ N₁ ≅ (toKaroubi_equivalence _).functor)
+    (N₁Γ₀ : (Γ : ChainComplex C ℕ ⥤ _) ⋙ N₁ ≅ (toKaroubiEquivalence _).functor)
 #align category_theory.idempotents.dold_kan.η CategoryTheory.Idempotents.DoldKan.η
 
 theorem equivalence_counitIso :
@@ -118,7 +120,7 @@ theorem equivalence_counitIso :
 
 theorem hε :
     Compatibility.υ (eqToIso hN₁) =
-      (Γ₂N₁ : (toKaroubi_equivalence _).functor ≅
+      (Γ₂N₁ : (toKaroubiEquivalence _).functor ≅
           (N₁ : SimplicialObject C ⥤ _) ⋙ Preadditive.DoldKan.equivalence.inverse) := by
   ext1
   rw [← cancel_epi Γ₂N₁.inv, Iso.inv_hom_id]

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -37,7 +37,7 @@ noncomputable section
 
 open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Idempotents
 
-variable {C : Type _} [Category C] [Preadditive C]
+variable {C : Type*} [Category C] [Preadditive C]
 
 namespace CategoryTheory