refactor(Limits/Preserves/Finite): review API (#21890)

- Discard unneeded `Subsingleton` instances. Probably, these instances were added when the typeclasses were in `Type*`.
- Require a proof for `Fin _` in the definition, then prove an instance for `J : Type u`.
- Assume `Finite _` instead of `Fintype _` here and there.
- Drop some no longer needed `Nonempty` wrappers.

Author: urkud

Mathlib/CategoryTheory/Closed/Ideal.lean

@@ -302,7 +302,7 @@ lemma preservesBinaryProducts_of_exponentialIdeal :
 /--
 If a reflective subcategory is an exponential ideal, then the reflector preserves finite products.
 -/
-lemma preservesFiniteProducts_of_exponentialIdeal (J : Type) [Fintype J] :
+lemma preservesFiniteProducts_of_exponentialIdeal (J : Type) [Finite J] :
     PreservesLimitsOfShape (Discrete J) (reflector i) := by
   letI := preservesBinaryProducts_of_exponentialIdeal i
   letI : PreservesLimitsOfShape _ (reflector i) := leftAdjoint_preservesTerminal_of_reflective.{0} i

Mathlib/CategoryTheory/Galois/Action.lean

@@ -69,12 +69,12 @@ instance : Functor.ReflectsIsomorphisms (functorToAction F) where
     isIso_of_reflects_iso f F
 
 noncomputable instance : PreservesFiniteCoproducts (functorToAction F) :=
-  ⟨fun J _ ↦ Action.preservesColimitsOfShape_of_preserves (functorToAction F)
-    (inferInstanceAs <| PreservesColimitsOfShape (Discrete J) F)⟩
+  ⟨fun _ ↦ Action.preservesColimitsOfShape_of_preserves (functorToAction F)
+    (inferInstanceAs <| PreservesColimitsOfShape (Discrete _) F)⟩
 
 noncomputable instance : PreservesFiniteProducts (functorToAction F) :=
-  ⟨fun J _ ↦ Action.preservesLimitsOfShape_of_preserves (functorToAction F)
-    (inferInstanceAs <| PreservesLimitsOfShape (Discrete J) F)⟩
+  ⟨fun _ ↦ Action.preservesLimitsOfShape_of_preserves (functorToAction F)
+    (inferInstanceAs <| PreservesLimitsOfShape (Discrete _) F)⟩
 
 noncomputable instance (G : Type*) [Group G] [Finite G] :
     PreservesColimitsOfShape (SingleObj G) (functorToAction F) :=

Mathlib/CategoryTheory/Galois/Examples.lean

@@ -89,7 +89,7 @@ instance : PreGaloisCategory (Action FintypeCat (MonCat.of G)) where
 
 /-- The forgetful functor from finite `G`-sets to sets is a `FiberFunctor`. -/
 noncomputable instance : FiberFunctor (Action.forget FintypeCat (MonCat.of G)) where
-  preservesFiniteCoproducts := ⟨fun _ _ ↦ inferInstance⟩
+  preservesFiniteCoproducts := ⟨fun _ ↦ inferInstance⟩
   preservesQuotientsByFiniteGroups _ _ _ := inferInstance
   reflectsIsos := ⟨fun f (_ : IsIso f.hom) => inferInstance⟩
 

Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean

@@ -156,12 +156,11 @@ lemma preservesShape_fin_of_preserves_binary_and_terminal (n : ℕ) :
     apply preservesLimit_of_iso_diagram F that
 
 /-- If `F` preserves the terminal object and binary products then it preserves finite products. -/
-lemma preservesFiniteProducts_of_preserves_binary_and_terminal (J : Type*) [Fintype J] :
-    PreservesLimitsOfShape (Discrete J) F := by
-  classical
-    let e := Fintype.equivFin J
-    haveI := preservesShape_fin_of_preserves_binary_and_terminal F (Fintype.card J)
-    apply preservesLimitsOfShape_of_equiv (Discrete.equivalence e).symm
+lemma preservesFiniteProducts_of_preserves_binary_and_terminal (J : Type*) [Finite J] :
+    PreservesLimitsOfShape (Discrete J) F :=
+  let ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
+  have := preservesShape_fin_of_preserves_binary_and_terminal F n
+  preservesLimitsOfShape_of_equiv (Discrete.equivalence e).symm _
 
 end Preserves
 
@@ -287,12 +286,11 @@ lemma preservesShape_fin_of_preserves_binary_and_initial (n : ℕ) :
     apply preservesColimit_of_iso_diagram F that
 
 /-- If `F` preserves the initial object and binary coproducts then it preserves finite products. -/
-lemma preservesFiniteCoproductsOfPreservesBinaryAndInitial (J : Type*) [Fintype J] :
-    PreservesColimitsOfShape (Discrete J) F := by
-  classical
-    let e := Fintype.equivFin J
-    haveI := preservesShape_fin_of_preserves_binary_and_initial F (Fintype.card J)
-    apply preservesColimitsOfShape_of_equiv (Discrete.equivalence e).symm
+lemma preservesFiniteCoproductsOfPreservesBinaryAndInitial (J : Type*) [Finite J] :
+    PreservesColimitsOfShape (Discrete J) F :=
+  let ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
+  have := preservesShape_fin_of_preserves_binary_and_initial F n
+  preservesColimitsOfShape_of_equiv (Discrete.equivalence e).symm _
 
 end Preserves
 

Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean

@@ -204,14 +204,9 @@ lemma preservesFiniteLimits_of_preservesEqualizers_and_finiteProducts [HasEquali
     [HasFiniteProducts C] (G : C ⥤ D) [PreservesLimitsOfShape WalkingParallelPair G]
     [PreservesFiniteProducts G] : PreservesFiniteLimits G where
   preservesFiniteLimits := by
-    intro J sJ fJ
-    haveI : Fintype J := inferInstance
-    haveI : Fintype ((p : J × J) × (p.fst ⟶ p.snd)) := inferInstance
-    apply @preservesLimit_of_preservesEqualizers_and_product _ _ _ sJ _ _ ?_ ?_ _ G _ ?_ ?_
-    · apply hasLimitsOfShape_discrete _ _
-    · apply hasLimitsOfShape_discrete _
-    · apply PreservesFiniteProducts.preserves _
-    · apply PreservesFiniteProducts.preserves _
+    intros
+    apply preservesLimit_of_preservesEqualizers_and_product
+
 
 /-- If G preserves equalizers and products, it preserves all limits. -/
 lemma preservesLimits_of_preservesEqualizers_and_products [HasEqualizers C]
@@ -292,7 +287,7 @@ lemma preservesFiniteLimits_of_preservesTerminal_and_pullbacks [HasTerminal C]
       preservesEqualizers_of_preservesPullbacks_and_binaryProducts G
   apply
     @preservesFiniteLimits_of_preservesEqualizers_and_finiteProducts _ _ _ _ _ _ G _ ?_
-  apply PreservesFiniteProducts.mk
+  refine ⟨fun n ↦ ?_⟩
   apply preservesFiniteProducts_of_preserves_binary_and_terminal G
 
 attribute [local instance] preservesFiniteLimits_of_preservesTerminal_and_pullbacks in
@@ -493,13 +488,7 @@ lemma preservesFiniteColimits_of_preservesCoequalizers_and_finiteCoproducts
     [PreservesFiniteCoproducts G] : PreservesFiniteColimits G where
   preservesFiniteColimits := by
     intro J sJ fJ
-    haveI : Fintype J := inferInstance
-    haveI : Fintype ((p : J × J) × (p.fst ⟶ p.snd)) := inferInstance
-    apply @preservesColimit_of_preservesCoequalizers_and_coproduct _ _ _ sJ _ _ ?_ ?_ _ G _ ?_ ?_
-    · apply hasColimitsOfShape_discrete _ _
-    · apply hasColimitsOfShape_discrete _
-    · apply PreservesFiniteCoproducts.preserves _
-    · apply PreservesFiniteCoproducts.preserves _
+    apply preservesColimit_of_preservesCoequalizers_and_coproduct
 
 /-- If G preserves coequalizers and coproducts, it preserves all colimits. -/
 lemma preservesColimits_of_preservesCoequalizers_and_coproducts [HasCoequalizers C]
@@ -580,7 +569,7 @@ lemma preservesFiniteColimits_of_preservesInitial_and_pushouts [HasInitial C]
       (preservesCoequalizers_of_preservesPushouts_and_binaryCoproducts G)
   refine
     @preservesFiniteColimits_of_preservesCoequalizers_and_finiteCoproducts _ _ _ _ _ _ G _ ?_
-  apply PreservesFiniteCoproducts.mk
+  refine ⟨fun _ ↦ ?_⟩
   apply preservesFiniteCoproductsOfPreservesBinaryAndInitial G
 
 attribute [local instance] preservesFiniteColimits_of_preservesInitial_and_pushouts in

Mathlib/CategoryTheory/Limits/Preserves/Finite.lean

@@ -43,9 +43,6 @@ class PreservesFiniteLimits (F : C ⥤ D) : Prop where
 
 attribute [instance] PreservesFiniteLimits.preservesFiniteLimits
 
-instance (F : C ⥤ D) : Subsingleton (PreservesFiniteLimits F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
-
 /-- Preserving finite limits also implies preserving limits over finite shapes in higher universes,
 though through a noncomputable instance. -/
 instance (priority := 100) preservesLimitsOfShapeOfPreservesFiniteLimits (F : C ⥤ D)
@@ -93,29 +90,25 @@ lemma preservesFiniteLimits_of_natIso {F G : C ⥤ D} (h : F ≅ G) [PreservesFi
 
 /- Porting note: adding this class because quantified classes don't behave well
 https://github.com/leanprover-community/mathlib4/pull/2764 -/
-/-- A functor `F` preserves finite products if it preserves all from `Discrete J`
-for `Fintype J` -/
+/-- A functor `F` preserves finite products if it preserves all from `Discrete J` for `Finite J`.
+We require this for `J = Fin n` in the definition,
+then generalize to `J : Type u` in the instance. -/
 class PreservesFiniteProducts (F : C ⥤ D) : Prop where
-  preserves : ∀ (J : Type) [Fintype J], PreservesLimitsOfShape (Discrete J) F
-
-attribute [instance] PreservesFiniteProducts.preserves
-
-instance (F : C ⥤ D) : Subsingleton (PreservesFiniteProducts F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
+  preserves : ∀ n, PreservesLimitsOfShape (Discrete (Fin n)) F
 
 instance (priority := 100) (F : C ⥤ D) (J : Type u) [Finite J]
     [PreservesFiniteProducts F] : PreservesLimitsOfShape (Discrete J) F := by
-  apply Nonempty.some
   obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
-  exact ⟨preservesLimitsOfShape_of_equiv (Discrete.equivalence e.symm) F⟩
+  have := PreservesFiniteProducts.preserves (F := F) n
+  exact preservesLimitsOfShape_of_equiv (Discrete.equivalence e.symm) F
 
 instance comp_preservesFiniteProducts (F : C ⥤ D) (G : D ⥤ E)
     [PreservesFiniteProducts F] [PreservesFiniteProducts G] :
     PreservesFiniteProducts (F ⋙ G) where
-  preserves _ _ := inferInstance
+  preserves _ := inferInstance
 
 instance (F : C ⥤ D) [PreservesFiniteLimits F] : PreservesFiniteProducts F where
-  preserves _ _ := inferInstance
+  preserves _ := inferInstance
 
 /--
 A functor is said to reflect finite limits, if it reflects all limits of shape `J`,
@@ -127,20 +120,20 @@ class ReflectsFiniteLimits (F : C ⥤ D) : Prop where
 
 attribute [instance] ReflectsFiniteLimits.reflects
 
-instance (F : C ⥤ D) : Subsingleton (ReflectsFiniteLimits F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
-
 /- Similarly to preserving finite products, quantified classes don't behave well. -/
 /--
-A functor `F` preserves finite products if it reflects limits of shape `Discrete J` for finite `J`
+A functor `F` preserves finite products if it reflects limits of shape `Discrete J` for finite `J`.
+We require this for `J = Fin n` in the definition,
+then generalize to `J : Type u` in the instance.
 -/
 class ReflectsFiniteProducts (F : C ⥤ D) : Prop where
-  reflects : ∀ (J : Type) [Fintype J], ReflectsLimitsOfShape (Discrete J) F
+  reflects : ∀ n, ReflectsLimitsOfShape (Discrete (Fin n)) F
 
-attribute [instance] ReflectsFiniteProducts.reflects
-
-instance (F : C ⥤ D) : Subsingleton (ReflectsFiniteProducts F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
+instance (priority := 100) (F : C ⥤ D) [ReflectsFiniteProducts F] (J : Type u) [Finite J] :
+    ReflectsLimitsOfShape (Discrete J) F :=
+  let ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
+  have := ReflectsFiniteProducts.reflects (F := F) n
+  reflectsLimitsOfShape_of_equiv (Discrete.equivalence e.symm) _
 
 -- This is a dangerous instance as it has unbound universe variables.
 /-- If we reflect limits of some arbitrary size, then we reflect all finite limits. -/
@@ -174,7 +167,7 @@ finite products.
 -/
 lemma preservesFiniteProducts_of_reflects_of_preserves (F : C ⥤ D) (G : D ⥤ E)
     [PreservesFiniteProducts (F ⋙ G)] [ReflectsFiniteProducts G] : PreservesFiniteProducts F where
-  preserves _ _ := preservesLimitsOfShape_of_reflects_of_preserves F G
+  preserves _ := preservesLimitsOfShape_of_reflects_of_preserves F G
 
 instance reflectsFiniteLimits_of_reflectsIsomorphisms (F : C ⥤ D)
     [F.ReflectsIsomorphisms] [HasFiniteLimits C] [PreservesFiniteLimits F] :
@@ -184,15 +177,15 @@ instance reflectsFiniteLimits_of_reflectsIsomorphisms (F : C ⥤ D)
 instance reflectsFiniteProducts_of_reflectsIsomorphisms (F : C ⥤ D)
     [F.ReflectsIsomorphisms] [HasFiniteProducts C] [PreservesFiniteProducts F] :
       ReflectsFiniteProducts F where
-  reflects _ _ := reflectsLimitsOfShape_of_reflectsIsomorphisms
+  reflects _ := reflectsLimitsOfShape_of_reflectsIsomorphisms
 
 instance comp_reflectsFiniteProducts (F : C ⥤ D) (G : D ⥤ E)
     [ReflectsFiniteProducts F] [ReflectsFiniteProducts G] :
     ReflectsFiniteProducts (F ⋙ G) where
-  reflects _ _ := inferInstance
+  reflects _ := inferInstance
 
 instance (F : C ⥤ D) [ReflectsFiniteLimits F] : ReflectsFiniteProducts F where
-  reflects _ _ := inferInstance
+  reflects _ := inferInstance
 
 /-- A functor is said to preserve finite colimits, if it preserves all colimits of
 shape `J`, where `J : Type` is a finite category.
@@ -204,9 +197,6 @@ class PreservesFiniteColimits (F : C ⥤ D) : Prop where
 
 attribute [instance] PreservesFiniteColimits.preservesFiniteColimits
 
-instance (F : C ⥤ D) : Subsingleton (PreservesFiniteColimits F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
-
 /--
 Preserving finite colimits also implies preserving colimits over finite shapes in higher
 universes.
@@ -257,44 +247,36 @@ lemma preservesFiniteColimits_of_natIso {F G : C ⥤ D} (h : F ≅ G) [Preserves
 
 /- Porting note: adding this class because quantified classes don't behave well
 https://github.com/leanprover-community/mathlib4/pull/2764 -/
-/-- A functor `F` preserves finite products if it preserves all from `Discrete J`
-for `Fintype J` -/
+/-- A functor `F` preserves finite products if it preserves all from `Discrete J` for `Fintype J`.
+We require this for `J = Fin n` in the definition,
+then generalize to `J : Type u` in the instance. -/
 class PreservesFiniteCoproducts (F : C ⥤ D) : Prop where
-  /-- preservation of colimits indexed by `Discrete J` when `[Fintype J]` -/
-  preserves : ∀ (J : Type) [Fintype J], PreservesColimitsOfShape (Discrete J) F
-
-attribute [instance] PreservesFiniteCoproducts.preserves
-
-instance (F : C ⥤ D) : Subsingleton (PreservesFiniteCoproducts F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
+  /-- preservation of colimits indexed by `Discrete (Fin n)`. -/
+  preserves : ∀ n, PreservesColimitsOfShape (Discrete (Fin n)) F
 
 instance (priority := 100) (F : C ⥤ D) (J : Type u) [Finite J]
-    [PreservesFiniteCoproducts F] : PreservesColimitsOfShape (Discrete J) F := by
-  apply Nonempty.some
-  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
-  exact ⟨preservesColimitsOfShape_of_equiv (Discrete.equivalence e.symm) F⟩
+    [PreservesFiniteCoproducts F] : PreservesColimitsOfShape (Discrete J) F :=
+  let ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
+  have := PreservesFiniteCoproducts.preserves (F := F) n
+  preservesColimitsOfShape_of_equiv (Discrete.equivalence e.symm) F
 
 instance comp_preservesFiniteCoproducts (F : C ⥤ D) (G : D ⥤ E)
     [PreservesFiniteCoproducts F] [PreservesFiniteCoproducts G] :
     PreservesFiniteCoproducts (F ⋙ G) where
-  preserves _ _ := inferInstance
+  preserves _ := inferInstance
 
 instance (F : C ⥤ D) [PreservesFiniteColimits F] : PreservesFiniteCoproducts F where
-  preserves _ _ := inferInstance
+  preserves _ := inferInstance
 
 /--
 A functor is said to reflect finite colimits, if it reflects all colimits of shape `J`,
 where `J : Type` is a finite category.
 -/
 class ReflectsFiniteColimits (F : C ⥤ D) : Prop where
-  reflects : ∀ (J : Type) [SmallCategory J] [FinCategory J], ReflectsColimitsOfShape J F := by
-    infer_instance
+  [reflects : ∀ (J : Type) [SmallCategory J] [FinCategory J], ReflectsColimitsOfShape J F]
 
 attribute [instance] ReflectsFiniteColimits.reflects
 
-instance (F : C ⥤ D) : Subsingleton (ReflectsFiniteColimits F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
-
 -- This is a dangerous instance as it has unbound universe variables.
 /-- If we reflect colimits of some arbitrary size, then we reflect all finite colimits. -/
 lemma ReflectsColimitsOfSize.reflectsFiniteColimits
@@ -315,16 +297,20 @@ instance (priority := 120) (F : C ⥤ D)
 
 /- Similarly to preserving finite coproducts, quantified classes don't behave well. -/
 /--
-A functor `F` preserves finite coproducts if it reflects colimits of shape `Discrete J` for
-finite `J`
+A functor `F` preserves finite coproducts if it reflects colimits of shape `Discrete J`
+for finite `J`.
+
+We require this for `J = Fin n` in the definition,
+then generalize to `J : Type u` in the instance.
 -/
 class ReflectsFiniteCoproducts (F : C ⥤ D) : Prop where
-  reflects : ∀ (J : Type) [Fintype J], ReflectsColimitsOfShape (Discrete J) F
-
-attribute [instance] ReflectsFiniteCoproducts.reflects
+  reflects : ∀ n, ReflectsColimitsOfShape (Discrete (Fin n)) F
 
-instance (F : C ⥤ D) : Subsingleton (ReflectsFiniteCoproducts F) :=
-  ⟨fun ⟨a⟩ ⟨b⟩ => by congr⟩
+instance (priority := 100) (F : C ⥤ D) [ReflectsFiniteCoproducts F] (J : Type u) [Finite J] :
+    ReflectsColimitsOfShape (Discrete J) F :=
+  let ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
+  have := ReflectsFiniteCoproducts.reflects (F := F) n
+  reflectsColimitsOfShape_of_equiv (Discrete.equivalence e.symm) _
 
 /--
 If `F ⋙ G` preserves finite colimits and `G` reflects finite colimits, then `F` preserves finite
@@ -341,7 +327,7 @@ finite coproducts.
 lemma preservesFiniteCoproducts_of_reflects_of_preserves (F : C ⥤ D) (G : D ⥤ E)
     [PreservesFiniteCoproducts (F ⋙ G)] [ReflectsFiniteCoproducts G] :
     PreservesFiniteCoproducts F where
-  preserves _ _ := preservesColimitsOfShape_of_reflects_of_preserves F G
+  preserves _ := preservesColimitsOfShape_of_reflects_of_preserves F G
 
 instance reflectsFiniteColimitsOfReflectsIsomorphisms (F : C ⥤ D)
     [F.ReflectsIsomorphisms] [HasFiniteColimits C] [PreservesFiniteColimits F] :
@@ -351,14 +337,14 @@ instance reflectsFiniteColimitsOfReflectsIsomorphisms (F : C ⥤ D)
 instance reflectsFiniteCoproductsOfReflectsIsomorphisms (F : C ⥤ D)
     [F.ReflectsIsomorphisms] [HasFiniteCoproducts C] [PreservesFiniteCoproducts F] :
       ReflectsFiniteCoproducts F where
-  reflects _ _ := reflectsColimitsOfShape_of_reflectsIsomorphisms
+  reflects _ := reflectsColimitsOfShape_of_reflectsIsomorphisms
 
 instance comp_reflectsFiniteCoproducts (F : C ⥤ D) (G : D ⥤ E)
     [ReflectsFiniteCoproducts F] [ReflectsFiniteCoproducts G] :
     ReflectsFiniteCoproducts (F ⋙ G) where
-  reflects _ _ := inferInstance
+  reflects _ := inferInstance
 
 instance (F : C ⥤ D) [ReflectsFiniteColimits F] : ReflectsFiniteCoproducts F where
-  reflects _ _ := inferInstance
+  reflects _ := inferInstance
 
 end CategoryTheory.Limits