category_theory.extensive ⟷ Mathlib.CategoryTheory.Limits.VanKampen

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

@@ -605,8 +605,8 @@ theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExten
 #print CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism /-
 theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]
     [HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]
-    [PreservesColimitsOfShape (Discrete WalkingPair) F] [ReflectsIsomorphisms F] :
-    FinitaryExtensive C :=
+    [PreservesColimitsOfShape (Discrete WalkingPair) F]
+    [CategoryTheory.Functor.ReflectsIsomorphisms F] : FinitaryExtensive C :=
   by
   haveI : reflects_limits_of_shape walking_cospan F :=
     reflects_limits_of_shape_of_reflects_isomorphisms

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

@@ -6,7 +6,7 @@ Authors: Andrew Yang
 import CategoryTheory.Limits.Shapes.CommSq
 import CategoryTheory.Limits.Shapes.StrictInitial
 import CategoryTheory.Limits.Shapes.Types
-import Topology.Category.Top.Limits.Pullbacks
+import Topology.Category.TopCat.Limits.Pullbacks
 import CategoryTheory.Limits.FunctorCategory
 
 #align_import category_theory.extensive from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"

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

@@ -193,7 +193,7 @@ theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
     let X' := F'.obj ⟨walking_pair.left⟩; let Y' := F'.obj ⟨walking_pair.right⟩
     have : F' = pair X' Y' := by apply functor.hext; · rintro ⟨⟨⟩⟩ <;> rfl;
       · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simpa
-    clear_value X' Y'; subst this; change binary_cofan X' Y' at c' 
+    clear_value X' Y'; subst this; change binary_cofan X' Y' at c'
     rw [H c' _ _ _ (nat_trans.congr_app hα ⟨walking_pair.left⟩)
         (nat_trans.congr_app hα ⟨walking_pair.right⟩)]
     constructor; · rintro H ⟨⟨⟩⟩; exacts [H.1, H.2]; · intro H; exact ⟨H _, H _⟩
@@ -365,9 +365,9 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
         intro x
         cases h : s.fst x
         · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
-        · apply_fun f at h 
+        · apply_fun f at h
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val : _).symm
-      delta ExistsUnique at this 
+      delta ExistsUnique at this
       choose l hl hl'
       exact
         ⟨l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h₁ h₂ =>
@@ -377,10 +377,10 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
-        · apply_fun f at h 
+        · apply_fun f at h
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val : _).symm
         · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
-      delta ExistsUnique at this 
+      delta ExistsUnique at this
       choose l hl hl'
       exact
         ⟨l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h₁ h₂ =>
@@ -486,11 +486,11 @@ instance : FinitaryExtensive TopCat.{u} :=
         intro x
         cases h : s.fst x
         · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
-        · apply_fun f at h 
+        · apply_fun f at h
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαY val : _).symm
-      delta ExistsUnique at this 
+      delta ExistsUnique at this
       choose l hl hl'
       refine'
         ⟨⟨l, _⟩, ContinuousMap.ext fun a => (hl a).symm, Top.is_terminal_punit.hom_ext _ _,
@@ -503,12 +503,12 @@ instance : FinitaryExtensive TopCat.{u} :=
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
-        · apply_fun f at h 
+        · apply_fun f at h
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαX val : _).symm
         · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
-      delta ExistsUnique at this 
+      delta ExistsUnique at this
       choose l hl hl'
       refine'
         ⟨⟨l, _⟩, ContinuousMap.ext fun a => (hl a).symm, Top.is_terminal_punit.hom_ext _ _,

mathlib commit https://github.com/leanprover-community/mathlib/commit/3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe

@@ -321,14 +321,14 @@ theorem hasStrictInitial_of_isUniversal [HasInitial C]
 #align category_theory.has_strict_initial_of_is_universal CategoryTheory.hasStrictInitial_of_isUniversal
 -/
 
-#print CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive /-
-instance (priority := 100) hasStrictInitialObjects_of_finitaryExtensive [FinitaryExtensive C] :
+#print CategoryTheory.hasStrictInitialObjects_of_finitaryPreExtensive /-
+instance (priority := 100) hasStrictInitialObjects_of_finitaryPreExtensive [FinitaryExtensive C] :
     HasStrictInitialObjects C :=
   hasStrictInitial_of_isUniversal
     (FinitaryExtensive.vanKampen _
         ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr
             (by dsimp; infer_instance)).some).isUniversal
-#align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive
+#align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryPreExtensive
 -/
 
 #print CategoryTheory.finitaryExtensive_iff_of_isTerminal /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe

@@ -548,9 +548,9 @@ theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKamp
 #align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso
 -/
 
-#print CategoryTheory.IsVanKampenColimit.of_map /-
-theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
-    [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
+#print CategoryTheory.IsVanKampenColimit.of_mapCocone /-
+theorem IsVanKampenColimit.of_mapCocone {D : Type _} [Category D] (G : C ⥤ D) {F : J ⥤ C}
+    {c : Cocone F} [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
     [PreservesColimitsOfShape J G] [ReflectsColimitsOfShape J G]
     (H : IsVanKampenColimit (G.mapCocone c)) : IsVanKampenColimit c :=
   by
@@ -563,7 +563,7 @@ theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J
       (forall_congr' fun j => _)
   · exact ⟨fun h => ⟨is_colimit_of_preserves G h.some⟩, fun h => ⟨is_colimit_of_reflects G h.some⟩⟩
   · exact is_pullback.map_iff G (nat_trans.congr_app h.symm j)
-#align category_theory.is_van_kampen_colimit.of_map CategoryTheory.IsVanKampenColimit.of_map
+#align category_theory.is_van_kampen_colimit.of_map CategoryTheory.IsVanKampenColimit.of_mapCocone
 -/
 
 #print CategoryTheory.isVanKampenColimit_of_evaluation /-

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

@@ -3,11 +3,11 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathbin.CategoryTheory.Limits.Shapes.CommSq
-import Mathbin.CategoryTheory.Limits.Shapes.StrictInitial
-import Mathbin.CategoryTheory.Limits.Shapes.Types
-import Mathbin.Topology.Category.Top.Limits.Pullbacks
-import Mathbin.CategoryTheory.Limits.FunctorCategory
+import CategoryTheory.Limits.Shapes.CommSq
+import CategoryTheory.Limits.Shapes.StrictInitial
+import CategoryTheory.Limits.Shapes.Types
+import Topology.Category.Top.Limits.Pullbacks
+import CategoryTheory.Limits.FunctorCategory
 
 #align_import category_theory.extensive from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.extensive
-! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Limits.Shapes.CommSq
 import Mathbin.CategoryTheory.Limits.Shapes.StrictInitial
@@ -14,6 +9,8 @@ import Mathbin.CategoryTheory.Limits.Shapes.Types
 import Mathbin.Topology.Category.Top.Limits.Pullbacks
 import Mathbin.CategoryTheory.Limits.FunctorCategory
 
+#align_import category_theory.extensive from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+
 /-!
 
 # Extensive categories

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -181,6 +181,7 @@ theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F')
 #align category_theory.map_pair_equifibered CategoryTheory.mapPair_equifibered
 -/
 
+#print CategoryTheory.BinaryCofan.isVanKampen_iff /-
 theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
     IsVanKampenColimit c ↔
       ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt)
@@ -200,7 +201,9 @@ theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
         (nat_trans.congr_app hα ⟨walking_pair.right⟩)]
     constructor; · rintro H ⟨⟨⟩⟩; exacts [H.1, H.2]; · intro H; exact ⟨H _, H _⟩
 #align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.isVanKampen_iff
+-/
 
+#print CategoryTheory.BinaryCofan.isVanKampen_mk /-
 theorem BinaryCofan.isVanKampen_mk {X Y : C} (c : BinaryCofan X Y)
     (cofans : ∀ X Y : C, BinaryCofan X Y) (colimits : ∀ X Y, IsColimit (cofans X Y))
     (cones : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), PullbackCone f g)
@@ -239,7 +242,9 @@ theorem BinaryCofan.isVanKampen_mk {X Y : C} (c : BinaryCofan X Y)
     apply binary_cofan.is_colimit_comp_left_iso (binary_cofan.mk _ _)
     exact h₂ f
 #align category_theory.binary_cofan.is_van_kampen_mk CategoryTheory.BinaryCofan.isVanKampen_mk
+-/
 
+#print CategoryTheory.BinaryCofan.mono_inr_of_isVanKampen /-
 theorem BinaryCofan.mono_inr_of_isVanKampen [HasInitial C] {X Y : C} {c : BinaryCofan X Y}
     (h : IsVanKampenColimit c) : Mono c.inr :=
   by
@@ -254,16 +259,21 @@ theorem BinaryCofan.mono_inr_of_isVanKampen [HasInitial C] {X Y : C} {c : Binary
       ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
           (by dsimp; infer_instance)).some
 #align category_theory.binary_cofan.mono_inr_of_is_van_kampen CategoryTheory.BinaryCofan.mono_inr_of_isVanKampen
+-/
 
+#print CategoryTheory.FinitaryExtensive.mono_inr_of_isColimit /-
 theorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
     (hc : IsColimit c) : Mono c.inr :=
   BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
 #align category_theory.finitary_extensive.mono_inr_of_is_colimit CategoryTheory.FinitaryExtensive.mono_inr_of_isColimit
+-/
 
+#print CategoryTheory.FinitaryExtensive.mono_inl_of_isColimit /-
 theorem FinitaryExtensive.mono_inl_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
     (hc : IsColimit c) : Mono c.inl :=
   FinitaryExtensive.mono_inr_of_isColimit (BinaryCofan.isColimitFlip hc)
 #align category_theory.finitary_extensive.mono_inl_of_is_colimit CategoryTheory.FinitaryExtensive.mono_inl_of_isColimit
+-/
 
 instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inl : X ⟶ X ⨿ Y) :=
   (FinitaryExtensive.mono_inl_of_isColimit (coprodIsCoprod X Y) : _)
@@ -271,6 +281,7 @@ instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inl : X ⟶ X ⨿ Y) :=
 instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inr : Y ⟶ X ⨿ Y) :=
   (FinitaryExtensive.mono_inr_of_isColimit (coprodIsCoprod X Y) : _)
 
+#print CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen /-
 theorem BinaryCofan.isPullback_initial_to_of_isVanKampen [HasInitial C] {c : BinaryCofan X Y}
     (h : IsVanKampenColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
   by
@@ -284,12 +295,15 @@ theorem BinaryCofan.isPullback_initial_to_of_isVanKampen [HasInitial C] {c : Bin
       ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
           (by dsimp; infer_instance)).some
 #align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen
+-/
 
+#print CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan /-
 theorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]
     {c : BinaryCofan X Y} (hc : IsColimit c) :
     IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
   BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
 #align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan
+-/
 
 #print CategoryTheory.hasStrictInitial_of_isUniversal /-
 theorem hasStrictInitial_of_isUniversal [HasInitial C]
@@ -398,6 +412,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
 
 section TopCat
 
+#print CategoryTheory.finitaryExtensiveTopCatAux /-
 /-- (Implementation) An auxiliary lemma for the proof that `Top` is finitary extensive. -/
 def finitaryExtensiveTopCatAux (Z : TopCat.{u})
     (f : Z ⟶ TopCat.of (Sum PUnit.{u + 1} PUnit.{u + 1})) :
@@ -457,6 +472,7 @@ def finitaryExtensiveTopCatAux (Z : TopCat.{u})
   · intro s m e₁ e₂; ext x; change m x = dite _ _ _
     split_ifs; · rw [← e₁]; rfl; · rw [← e₂]; rfl
 #align category_theory.finitary_extensive_Top_aux CategoryTheory.finitaryExtensiveTopCatAux
+-/
 
 instance : FinitaryExtensive TopCat.{u} :=
   by
@@ -535,6 +551,7 @@ theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKamp
 #align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso
 -/
 
+#print CategoryTheory.IsVanKampenColimit.of_map /-
 theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
     [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
     [PreservesColimitsOfShape J G] [ReflectsColimitsOfShape J G]
@@ -550,7 +567,9 @@ theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J
   · exact ⟨fun h => ⟨is_colimit_of_preserves G h.some⟩, fun h => ⟨is_colimit_of_reflects G h.some⟩⟩
   · exact is_pullback.map_iff G (nat_trans.congr_app h.symm j)
 #align category_theory.is_van_kampen_colimit.of_map CategoryTheory.IsVanKampenColimit.of_map
+-/
 
+#print CategoryTheory.isVanKampenColimit_of_evaluation /-
 theorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J D] (F : J ⥤ C ⥤ D)
     (c : Cocone F) (hc : ∀ x : C, IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)) :
     IsVanKampenColimit c := by
@@ -569,6 +588,7 @@ theorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J
       ⟨evaluation_jointly_reflects_colimits _ fun x =>
           ((this x).mpr fun j => (H j).map ((evaluation C D).obj x)).some⟩
 #align category_theory.is_van_kampen_colimit_of_evaluation CategoryTheory.isVanKampenColimit_of_evaluation
+-/
 
 instance [HasPullbacks C] [FinitaryExtensive C] : FinitaryExtensive (D ⥤ C) :=
   haveI : has_finite_coproducts (D ⥤ C) := ⟨fun n => limits.functor_category_has_colimits_of_shape⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.extensive
-! leanprover-community/mathlib commit 178a32653e369dce2da68dc6b2694e385d484ef1
+! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.CategoryTheory.Limits.FunctorCategory
 
 # Extensive categories
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Main definitions
 - `category_theory.is_van_kampen_colimit`: A (colimit) cocone over a diagram `F : J ⥤ C` is van
   Kampen if for every cocone `c'` over the pullback of the diagram `F' : J ⥤ C'`,
@@ -56,6 +59,7 @@ universe v' u' v u
 
 variable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]
 
+#print CategoryTheory.NatTrans.Equifibered /-
 /-- A natural transformation is equifibered if every commutative square of the following form is
 a pullback.
 ```
@@ -67,22 +71,30 @@ G(X) → G(Y)
 def NatTrans.Equifibered {F G : J ⥤ C} (α : F ⟶ G) : Prop :=
   ∀ ⦃i j : J⦄ (f : i ⟶ j), IsPullback (F.map f) (α.app i) (α.app j) (G.map f)
 #align category_theory.nat_trans.equifibered CategoryTheory.NatTrans.Equifibered
+-/
 
+#print CategoryTheory.NatTrans.equifibered_of_isIso /-
 theorem NatTrans.equifibered_of_isIso {F G : J ⥤ C} (α : F ⟶ G) [IsIso α] : α.Equifibered :=
   fun _ _ f => IsPullback.of_vert_isIso ⟨NatTrans.naturality _ f⟩
 #align category_theory.nat_trans.equifibered_of_is_iso CategoryTheory.NatTrans.equifibered_of_isIso
+-/
 
+#print CategoryTheory.NatTrans.Equifibered.comp /-
 theorem NatTrans.Equifibered.comp {F G H : J ⥤ C} {α : F ⟶ G} {β : G ⟶ H} (hα : α.Equifibered)
     (hβ : β.Equifibered) : (α ≫ β).Equifibered := fun i j f => (hα f).paste_vert (hβ f)
 #align category_theory.nat_trans.equifibered.comp CategoryTheory.NatTrans.Equifibered.comp
+-/
 
+#print CategoryTheory.IsUniversalColimit /-
 /-- A (colimit) cocone over a diagram `F : J ⥤ C` is universal if it is stable under pullbacks. -/
 def IsUniversalColimit {F : J ⥤ C} (c : Cocone F) : Prop :=
   ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.pt ⟶ c.pt)
     (h : α ≫ c.ι = c'.ι ≫ (Functor.const J).map f) (hα : α.Equifibered),
     (∀ j : J, IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)) → Nonempty (IsColimit c')
 #align category_theory.is_universal_colimit CategoryTheory.IsUniversalColimit
+-/
 
+#print CategoryTheory.IsVanKampenColimit /-
 /-- A (colimit) cocone over a diagram `F : J ⥤ C` is van Kampen if for every cocone `c'` over the
 pullback of the diagram `F' : J ⥤ C'`, `c'` is colimiting iff `c'` is the pullback of `c`.
 
@@ -94,11 +106,15 @@ def IsVanKampenColimit {F : J ⥤ C} (c : Cocone F) : Prop :=
     (h : α ≫ c.ι = c'.ι ≫ (Functor.const J).map f) (hα : α.Equifibered),
     Nonempty (IsColimit c') ↔ ∀ j : J, IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)
 #align category_theory.is_van_kampen_colimit CategoryTheory.IsVanKampenColimit
+-/
 
-theorem IsVanKampenColimit.is_universal {F : J ⥤ C} {c : Cocone F} (H : IsVanKampenColimit c) :
+#print CategoryTheory.IsVanKampenColimit.isUniversal /-
+theorem IsVanKampenColimit.isUniversal {F : J ⥤ C} {c : Cocone F} (H : IsVanKampenColimit c) :
     IsUniversalColimit c := fun _ c' α f h hα => (H c' α f h hα).mpr
-#align category_theory.is_van_kampen_colimit.is_universal CategoryTheory.IsVanKampenColimit.is_universal
+#align category_theory.is_van_kampen_colimit.is_universal CategoryTheory.IsVanKampenColimit.isUniversal
+-/
 
+#print CategoryTheory.IsVanKampenColimit.isColimit /-
 /-- A van Kampen colimit is a colimit. -/
 noncomputable def IsVanKampenColimit.isColimit {F : J ⥤ C} {c : Cocone F}
     (h : IsVanKampenColimit c) : IsColimit c :=
@@ -110,7 +126,9 @@ noncomputable def IsVanKampenColimit.isColimit {F : J ⥤ C} {c : Cocone F}
   haveI : is_iso (𝟙 c.X) := inferInstance
   exact is_pullback.of_vert_is_iso ⟨by erw [nat_trans.id_app, category.comp_id, category.id_comp]⟩
 #align category_theory.is_van_kampen_colimit.is_colimit CategoryTheory.IsVanKampenColimit.isColimit
+-/
 
+#print CategoryTheory.IsInitial.isVanKampenColimit /-
 theorem IsInitial.isVanKampenColimit [HasStrictInitialObjects C] {X : C} (h : IsInitial X) :
     IsVanKampenColimit (asEmptyCocone X) :=
   by
@@ -122,11 +140,13 @@ theorem IsInitial.isVanKampenColimit [HasStrictInitialObjects C] {X : C} (h : Is
     ⟨by rintro _ ⟨⟨⟩⟩, fun _ =>
       ⟨is_colimit.of_iso_colimit h (cocones.ext (as_iso f).symm <| by rintro ⟨⟨⟩⟩)⟩⟩
 #align category_theory.is_initial.is_van_kampen_colimit CategoryTheory.IsInitial.isVanKampenColimit
+-/
 
 section Extensive
 
 variable {X Y : C}
 
+#print CategoryTheory.FinitaryExtensive /-
 /-- A category is (finitary) extensive if it has finite coproducts,
 and binary coproducts are van Kampen.
 
@@ -135,10 +155,12 @@ class FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where
   [HasFiniteCoproducts : HasFiniteCoproducts C]
   van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c
 #align category_theory.finitary_extensive CategoryTheory.FinitaryExtensive
+-/
 
 attribute [instance] finitary_extensive.has_finite_coproducts
 
-theorem FinitaryExtensive.van_kampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
+#print CategoryTheory.FinitaryExtensive.vanKampen /-
+theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
     (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c :=
   by
   let X := F.obj ⟨walking_pair.left⟩; let Y := F.obj ⟨walking_pair.right⟩
@@ -146,17 +168,20 @@ theorem FinitaryExtensive.van_kampen [FinitaryExtensive C] {F : Discrete Walking
     · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simpa
   clear_value X Y; subst this
   exact finitary_extensive.van_kampen' c hc
-#align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.van_kampen
+#align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.vanKampen
+-/
 
-theorem map_pair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') : α.Equifibered :=
+#print CategoryTheory.mapPair_equifibered /-
+theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') : α.Equifibered :=
   by
   rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩
   all_goals
     dsimp; simp only [discrete.functor_map_id]
     exact is_pullback.of_horiz_is_iso ⟨by simp only [category.comp_id, category.id_comp]⟩
-#align category_theory.map_pair_equifibered CategoryTheory.map_pair_equifibered
+#align category_theory.map_pair_equifibered CategoryTheory.mapPair_equifibered
+-/
 
-theorem BinaryCofan.is_van_kampen_iff (c : BinaryCofan X Y) :
+theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
     IsVanKampenColimit c ↔
       ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt)
         (hαX : αX ≫ c.inl = c'.inl ≫ f) (hαY : αY ≫ c.inr = c'.inr ≫ f),
@@ -174,9 +199,9 @@ theorem BinaryCofan.is_van_kampen_iff (c : BinaryCofan X Y) :
     rw [H c' _ _ _ (nat_trans.congr_app hα ⟨walking_pair.left⟩)
         (nat_trans.congr_app hα ⟨walking_pair.right⟩)]
     constructor; · rintro H ⟨⟨⟩⟩; exacts [H.1, H.2]; · intro H; exact ⟨H _, H _⟩
-#align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.is_van_kampen_iff
+#align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.isVanKampen_iff
 
-theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
+theorem BinaryCofan.isVanKampen_mk {X Y : C} (c : BinaryCofan X Y)
     (cofans : ∀ X Y : C, BinaryCofan X Y) (colimits : ∀ X Y, IsColimit (cofans X Y))
     (cones : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), PullbackCone f g)
     (limits : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), IsLimit (cones f g))
@@ -213,9 +238,9 @@ theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
     apply binary_cofan.is_colimit_comp_right_iso (binary_cofan.mk _ _)
     apply binary_cofan.is_colimit_comp_left_iso (binary_cofan.mk _ _)
     exact h₂ f
-#align category_theory.binary_cofan.is_van_kampen_mk CategoryTheory.BinaryCofan.is_van_kampen_mk
+#align category_theory.binary_cofan.is_van_kampen_mk CategoryTheory.BinaryCofan.isVanKampen_mk
 
-theorem BinaryCofan.mono_inr_of_is_van_kampen [HasInitial C] {X Y : C} {c : BinaryCofan X Y}
+theorem BinaryCofan.mono_inr_of_isVanKampen [HasInitial C] {X Y : C} {c : BinaryCofan X Y}
     (h : IsVanKampenColimit c) : Mono c.inr :=
   by
   refine' pullback_cone.mono_of_is_limit_mk_id_id _ (is_pullback.is_limit _)
@@ -228,11 +253,11 @@ theorem BinaryCofan.mono_inr_of_is_van_kampen [HasInitial C] {X Y : C} {c : Bina
     exact
       ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
           (by dsimp; infer_instance)).some
-#align category_theory.binary_cofan.mono_inr_of_is_van_kampen CategoryTheory.BinaryCofan.mono_inr_of_is_van_kampen
+#align category_theory.binary_cofan.mono_inr_of_is_van_kampen CategoryTheory.BinaryCofan.mono_inr_of_isVanKampen
 
 theorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
     (hc : IsColimit c) : Mono c.inr :=
-  BinaryCofan.mono_inr_of_is_van_kampen (FinitaryExtensive.van_kampen c hc)
+  BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
 #align category_theory.finitary_extensive.mono_inr_of_is_colimit CategoryTheory.FinitaryExtensive.mono_inr_of_isColimit
 
 theorem FinitaryExtensive.mono_inl_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
@@ -246,7 +271,7 @@ instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inl : X ⟶ X ⨿ Y) :=
 instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inr : Y ⟶ X ⨿ Y) :=
   (FinitaryExtensive.mono_inr_of_isColimit (coprodIsCoprod X Y) : _)
 
-theorem BinaryCofan.isPullback_initial_to_of_is_van_kampen [HasInitial C] {c : BinaryCofan X Y}
+theorem BinaryCofan.isPullback_initial_to_of_isVanKampen [HasInitial C] {c : BinaryCofan X Y}
     (h : IsVanKampenColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
   by
   refine'
@@ -258,15 +283,16 @@ theorem BinaryCofan.isPullback_initial_to_of_is_van_kampen [HasInitial C] {c : B
     exact
       ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
           (by dsimp; infer_instance)).some
-#align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_is_van_kampen
+#align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen
 
 theorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]
     {c : BinaryCofan X Y} (hc : IsColimit c) :
     IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
-  BinaryCofan.isPullback_initial_to_of_is_van_kampen (FinitaryExtensive.van_kampen c hc)
+  BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
 #align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan
 
-theorem has_strict_initial_of_is_universal [HasInitial C]
+#print CategoryTheory.hasStrictInitial_of_isUniversal /-
+theorem hasStrictInitial_of_isUniversal [HasInitial C]
     (H : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))) : HasStrictInitialObjects C :=
   hasStrictInitialObjects_of_initial_is_strict
     (by
@@ -281,16 +307,20 @@ theorem has_strict_initial_of_is_universal [HasInitial C]
             (map_pair_equifibered _) _).some
       rintro ⟨⟨⟩⟩ <;> dsimp <;>
         exact is_pullback.of_horiz_is_iso ⟨(category.id_comp _).trans (category.comp_id _).symm⟩)
-#align category_theory.has_strict_initial_of_is_universal CategoryTheory.has_strict_initial_of_is_universal
+#align category_theory.has_strict_initial_of_is_universal CategoryTheory.hasStrictInitial_of_isUniversal
+-/
 
+#print CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive /-
 instance (priority := 100) hasStrictInitialObjects_of_finitaryExtensive [FinitaryExtensive C] :
     HasStrictInitialObjects C :=
-  has_strict_initial_of_is_universal
-    (FinitaryExtensive.van_kampen _
+  hasStrictInitial_of_isUniversal
+    (FinitaryExtensive.vanKampen _
         ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr
-            (by dsimp; infer_instance)).some).is_universal
+            (by dsimp; infer_instance)).some).isUniversal
 #align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive
+-/
 
+#print CategoryTheory.finitaryExtensive_iff_of_isTerminal /-
 theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]
     (T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :
     FinitaryExtensive C ↔ IsVanKampenColimit c₀ :=
@@ -306,7 +336,9 @@ theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFi
   obtain ⟨hl, hr⟩ := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp ⟨hc⟩
   rw [hl.paste_vert_iff hX.symm, hr.paste_vert_iff hY.symm]
 #align category_theory.finitary_extensive_iff_of_is_terminal CategoryTheory.finitaryExtensive_iff_of_isTerminal
+-/
 
+#print CategoryTheory.types.finitaryExtensive /-
 instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
   by
   rw [finitary_extensive_iff_of_is_terminal (Type u) PUnit types.is_terminal_punit _
@@ -322,7 +354,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
         intro x
         cases h : s.fst x
         · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
-        · apply_fun f  at h 
+        · apply_fun f at h 
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val : _).symm
       delta ExistsUnique at this 
       choose l hl hl'
@@ -334,7 +366,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
-        · apply_fun f  at h 
+        · apply_fun f at h 
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val : _).symm
         · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
       delta ExistsUnique at this 
@@ -362,11 +394,13 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
     · intro s; ext ⟨⟨x, ⟨⟩⟩, hx⟩; dsimp; split_ifs; · cases h.symm.trans hx; · rfl
     · intro s m e₁ e₂; ext x; split_ifs; · rw [← e₁]; rfl; · rw [← e₂]; rfl
 #align category_theory.types.finitary_extensive CategoryTheory.types.finitaryExtensive
+-/
 
 section TopCat
 
 /-- (Implementation) An auxiliary lemma for the proof that `Top` is finitary extensive. -/
-def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u + 1} PUnit.{u + 1})) :
+def finitaryExtensiveTopCatAux (Z : TopCat.{u})
+    (f : Z ⟶ TopCat.of (Sum PUnit.{u + 1} PUnit.{u + 1})) :
     IsColimit
       (BinaryCofan.mk
         (TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inl)
@@ -392,8 +426,7 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
     · revert h x
       apply (IsOpen.continuousOn_iff _).mp
       · rw [continuousOn_iff_continuous_restrict]
-        convert_to Continuous fun x : { x | f x = Sum.inl PUnit.unit } =>
-            s.inl ⟨(x, PUnit.unit), x.2⟩
+        convert_to Continuous fun x : {x | f x = Sum.inl PUnit.unit} => s.inl ⟨(x, PUnit.unit), x.2⟩
         · ext ⟨x, hx⟩; exact dif_pos hx
         continuity
       · convert f.2.1 _ openEmbedding_inl.open_range; ext x;
@@ -403,7 +436,8 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
     · revert h x
       apply (IsOpen.continuousOn_iff _).mp
       · rw [continuousOn_iff_continuous_restrict]
-        convert_to Continuous fun x : { x | f x ≠ Sum.inl PUnit.unit } =>
+        convert_to
+          Continuous fun x : {x | f x ≠ Sum.inl PUnit.unit} =>
             s.inr ⟨(x, PUnit.unit), (this _).resolve_left x.2⟩
         · ext ⟨x, hx⟩; exact dif_neg hx
         continuity
@@ -422,7 +456,7 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
     split_ifs; · cases h.symm.trans hx; · rfl
   · intro s m e₁ e₂; ext x; change m x = dite _ _ _
     split_ifs; · rw [← e₁]; rfl; · rw [← e₂]; rfl
-#align category_theory.finitary_extensive_Top_aux CategoryTheory.finitaryExtensiveTopAux
+#align category_theory.finitary_extensive_Top_aux CategoryTheory.finitaryExtensiveTopCatAux
 
 instance : FinitaryExtensive TopCat.{u} :=
   by
@@ -439,7 +473,7 @@ instance : FinitaryExtensive TopCat.{u} :=
         intro x
         cases h : s.fst x
         · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
-        · apply_fun f  at h 
+        · apply_fun f at h 
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαY val : _).symm
@@ -456,7 +490,7 @@ instance : FinitaryExtensive TopCat.{u} :=
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
-        · apply_fun f  at h 
+        · apply_fun f at h 
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαX val : _).symm
@@ -479,11 +513,14 @@ universe v'' u''
 
 variable {D : Type u''} [Category.{v''} D]
 
+#print CategoryTheory.NatTrans.Equifibered.whiskerRight /-
 theorem NatTrans.Equifibered.whiskerRight {F G : J ⥤ C} {α : F ⟶ G} (hα : α.Equifibered) (H : C ⥤ D)
     [PreservesLimitsOfShape WalkingCospan H] : (whiskerRight α H).Equifibered := fun i j f =>
   (hα f).map H
 #align category_theory.nat_trans.equifibered.whisker_right CategoryTheory.NatTrans.Equifibered.whiskerRight
+-/
 
+#print CategoryTheory.IsVanKampenColimit.of_iso /-
 theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKampenColimit c)
     (e : c ≅ c') : IsVanKampenColimit c' :=
   by
@@ -496,6 +533,7 @@ theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKamp
   haveI : is_iso e.inv.hom := functor.map_is_iso (cocones.forget _) e.inv
   exact (is_pullback.of_vert_is_iso ⟨by simp⟩).paste_vert_iff (nat_trans.congr_app h j).symm
 #align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso
+-/
 
 theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
     [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
@@ -538,13 +576,16 @@ instance [HasPullbacks C] [FinitaryExtensive C] : FinitaryExtensive (D ⥤ C) :=
     is_van_kampen_colimit_of_evaluation _ c fun x =>
       finitary_extensive.van_kampen _ <| preserves_colimit.preserves hc⟩
 
+#print CategoryTheory.finitaryExtensive_of_preserves_and_reflects /-
 theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExtensive D]
     [HasFiniteCoproducts C] [PreservesLimitsOfShape WalkingCospan F]
     [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
     [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C :=
-  ⟨fun X Y c hc => (FinitaryExtensive.van_kampen _ (isColimitOfPreserves F hc)).of_map F⟩
+  ⟨fun X Y c hc => (FinitaryExtensive.vanKampen _ (isColimitOfPreserves F hc)).of_map F⟩
 #align category_theory.finitary_extensive_of_preserves_and_reflects CategoryTheory.finitaryExtensive_of_preserves_and_reflects
+-/
 
+#print CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism /-
 theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]
     [HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]
     [PreservesColimitsOfShape (Discrete WalkingPair) F] [ReflectsIsomorphisms F] :
@@ -556,6 +597,7 @@ theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [F
     reflects_colimits_of_shape_of_reflects_isomorphisms
   exact finitary_extensive_of_preserves_and_reflects F
 #align category_theory.finitary_extensive_of_preserves_and_reflects_isomorphism CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism
+-/
 
 end Functor
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -165,15 +165,15 @@ theorem BinaryCofan.is_van_kampen_iff (c : BinaryCofan X Y) :
   constructor
   · introv H hαX hαY
     rw [H c' (map_pair αX αY) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> assumption) (map_pair_equifibered _)]
-    constructor; · intro H; exact ⟨H _, H _⟩; · rintro H ⟨⟨⟩⟩; exacts[H.1, H.2]
+    constructor; · intro H; exact ⟨H _, H _⟩; · rintro H ⟨⟨⟩⟩; exacts [H.1, H.2]
   · introv H F' hα h
     let X' := F'.obj ⟨walking_pair.left⟩; let Y' := F'.obj ⟨walking_pair.right⟩
     have : F' = pair X' Y' := by apply functor.hext; · rintro ⟨⟨⟩⟩ <;> rfl;
       · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simpa
-    clear_value X' Y'; subst this; change binary_cofan X' Y' at c'
+    clear_value X' Y'; subst this; change binary_cofan X' Y' at c' 
     rw [H c' _ _ _ (nat_trans.congr_app hα ⟨walking_pair.left⟩)
         (nat_trans.congr_app hα ⟨walking_pair.right⟩)]
-    constructor; · rintro H ⟨⟨⟩⟩; exacts[H.1, H.2]; · intro H; exact ⟨H _, H _⟩
+    constructor; · rintro H ⟨⟨⟩⟩; exacts [H.1, H.2]; · intro H; exact ⟨H _, H _⟩
 #align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.is_van_kampen_iff
 
 theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
@@ -297,7 +297,7 @@ theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFi
   by
   refine' ⟨fun H => H.2 c₀ hc₀, fun H => _⟩
   constructor
-  simp_rw [binary_cofan.is_van_kampen_iff] at H⊢
+  simp_rw [binary_cofan.is_van_kampen_iff] at H ⊢
   intro X Y c hc X' Y' c' αX αY f hX hY
   obtain ⟨d, hd, hd'⟩ :=
     limits.binary_cofan.is_colimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)
@@ -322,9 +322,9 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
         intro x
         cases h : s.fst x
         · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
-        · apply_fun f  at h
+        · apply_fun f  at h 
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val : _).symm
-      delta ExistsUnique at this
+      delta ExistsUnique at this 
       choose l hl hl'
       exact
         ⟨l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h₁ h₂ =>
@@ -334,10 +334,10 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
-        · apply_fun f  at h
+        · apply_fun f  at h 
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val : _).symm
         · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
-      delta ExistsUnique at this
+      delta ExistsUnique at this 
       choose l hl hl'
       exact
         ⟨l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h₁ h₂ =>
@@ -346,7 +346,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
     dsimp [limits.types.binary_coproduct_cocone]
     delta types.pullback_obj
     have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by intro x;
-      rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩); exacts[Or.inl rfl, Or.inr rfl]
+      rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩); exacts [Or.inl rfl, Or.inr rfl]
     let eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
       ⟨fun p => ⟨p.1.1, by convert p.2⟩, fun x => ⟨⟨_, _⟩, x.2⟩, fun _ => by ext <;> rfl, fun _ =>
         by ext <;> rfl⟩
@@ -373,7 +373,7 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
         (TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inr)) :=
   by
   have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by intro x;
-    rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩); exacts[Or.inl rfl, Or.inr rfl]
+    rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩); exacts [Or.inl rfl, Or.inr rfl]
   let eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
     ⟨fun p => ⟨p.1.1, p.2.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)⟩, fun x =>
       ⟨⟨_, _⟩, x.2⟩, fun _ => by ext <;> rfl, fun _ => by ext <;> rfl⟩
@@ -439,11 +439,11 @@ instance : FinitaryExtensive TopCat.{u} :=
         intro x
         cases h : s.fst x
         · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
-        · apply_fun f  at h
+        · apply_fun f  at h 
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαY val : _).symm
-      delta ExistsUnique at this
+      delta ExistsUnique at this 
       choose l hl hl'
       refine'
         ⟨⟨l, _⟩, ContinuousMap.ext fun a => (hl a).symm, Top.is_terminal_punit.hom_ext _ _,
@@ -456,12 +456,12 @@ instance : FinitaryExtensive TopCat.{u} :=
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
-        · apply_fun f  at h
+        · apply_fun f  at h 
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαX val : _).symm
         · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
-      delta ExistsUnique at this
+      delta ExistsUnique at this 
       choose l hl hl'
       refine'
         ⟨⟨l, _⟩, ContinuousMap.ext fun a => (hl a).symm, Top.is_terminal_punit.hom_ext _ _,

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -141,14 +141,10 @@ attribute [instance] finitary_extensive.has_finite_coproducts
 theorem FinitaryExtensive.van_kampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
     (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c :=
   by
-  let X := F.obj ⟨walking_pair.left⟩
-  let Y := F.obj ⟨walking_pair.right⟩
-  have : F = pair X Y := by
-    apply functor.hext
-    · rintro ⟨⟨⟩⟩ <;> rfl
+  let X := F.obj ⟨walking_pair.left⟩; let Y := F.obj ⟨walking_pair.right⟩
+  have : F = pair X Y := by apply functor.hext; · rintro ⟨⟨⟩⟩ <;> rfl;
     · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simpa
-  clear_value X Y
-  subst this
+  clear_value X Y; subst this
   exact finitary_extensive.van_kampen' c hc
 #align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.van_kampen
 
@@ -169,28 +165,15 @@ theorem BinaryCofan.is_van_kampen_iff (c : BinaryCofan X Y) :
   constructor
   · introv H hαX hαY
     rw [H c' (map_pair αX αY) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> assumption) (map_pair_equifibered _)]
-    constructor
-    · intro H
-      exact ⟨H _, H _⟩
-    · rintro H ⟨⟨⟩⟩
-      exacts[H.1, H.2]
+    constructor; · intro H; exact ⟨H _, H _⟩; · rintro H ⟨⟨⟩⟩; exacts[H.1, H.2]
   · introv H F' hα h
-    let X' := F'.obj ⟨walking_pair.left⟩
-    let Y' := F'.obj ⟨walking_pair.right⟩
-    have : F' = pair X' Y' := by
-      apply functor.hext
-      · rintro ⟨⟨⟩⟩ <;> rfl
+    let X' := F'.obj ⟨walking_pair.left⟩; let Y' := F'.obj ⟨walking_pair.right⟩
+    have : F' = pair X' Y' := by apply functor.hext; · rintro ⟨⟨⟩⟩ <;> rfl;
       · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simpa
-    clear_value X' Y'
-    subst this
-    change binary_cofan X' Y' at c'
+    clear_value X' Y'; subst this; change binary_cofan X' Y' at c'
     rw [H c' _ _ _ (nat_trans.congr_app hα ⟨walking_pair.left⟩)
         (nat_trans.congr_app hα ⟨walking_pair.right⟩)]
-    constructor
-    · rintro H ⟨⟨⟩⟩
-      exacts[H.1, H.2]
-    · intro H
-      exact ⟨H _, H _⟩
+    constructor; · rintro H ⟨⟨⟩⟩; exacts[H.1, H.2]; · intro H; exact ⟨H _, H _⟩
 #align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.is_van_kampen_iff
 
 theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
@@ -213,17 +196,11 @@ theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
     obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [hX]) (by simp [hY])
     constructor
     · rw [← category.id_comp αX, ← iso.hom_inv_id_assoc e f]
-      have : c'.inl ≫ e.hom = 𝟙 X' ≫ (cofans X' Y').inl :=
-        by
-        dsimp
-        simp
+      have : c'.inl ≫ e.hom = 𝟙 X' ≫ (cofans X' Y').inl := by dsimp; simp
       haveI : is_iso (𝟙 X') := inferInstance
       exact (is_pullback.of_vert_is_iso ⟨this⟩).paste_vert hl
     · rw [← category.id_comp αY, ← iso.hom_inv_id_assoc e f]
-      have : c'.inr ≫ e.hom = 𝟙 Y' ≫ (cofans X' Y').inr :=
-        by
-        dsimp
-        simp
+      have : c'.inr ≫ e.hom = 𝟙 Y' ≫ (cofans X' Y').inr := by dsimp; simp
       haveI : is_iso (𝟙 Y') := inferInstance
       exact (is_pullback.of_vert_is_iso ⟨this⟩).paste_vert hr
   · rintro ⟨H₁, H₂⟩
@@ -250,9 +227,7 @@ theorem BinaryCofan.mono_inr_of_is_van_kampen [HasInitial C] {X Y : C} {c : Bina
   ·
     exact
       ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
-          (by
-            dsimp
-            infer_instance)).some
+          (by dsimp; infer_instance)).some
 #align category_theory.binary_cofan.mono_inr_of_is_van_kampen CategoryTheory.BinaryCofan.mono_inr_of_is_van_kampen
 
 theorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
@@ -282,9 +257,7 @@ theorem BinaryCofan.isPullback_initial_to_of_is_van_kampen [HasInitial C] {c : B
   ·
     exact
       ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
-          (by
-            dsimp
-            infer_instance)).some
+          (by dsimp; infer_instance)).some
 #align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_is_van_kampen
 
 theorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]
@@ -315,9 +288,7 @@ instance (priority := 100) hasStrictInitialObjects_of_finitaryExtensive [Finitar
   has_strict_initial_of_is_universal
     (FinitaryExtensive.van_kampen _
         ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr
-            (by
-              dsimp
-              infer_instance)).some).is_universal
+            (by dsimp; infer_instance)).some).is_universal
 #align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive
 
 theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]
@@ -350,8 +321,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
       have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
         intro x
         cases h : s.fst x
-        · simp_rw [sum.inl_injective.eq_iff]
-          exact exists_unique_eq'
+        · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
         · apply_fun f  at h
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val : _).symm
       delta ExistsUnique at this
@@ -360,15 +330,13 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
         ⟨l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h₁ h₂ =>
           funext fun x => hl' x (l' x) (congr_fun h₁ x).symm⟩
     · refine' ⟨⟨hαY.symm⟩, ⟨pullback_cone.is_limit_aux' _ _⟩⟩
-      intro s
-      dsimp
+      intro s; dsimp
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
         · apply_fun f  at h
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val : _).symm
-        · simp_rw [sum.inr_injective.eq_iff]
-          exact exists_unique_eq'
+        · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
       delta ExistsUnique at this
       choose l hl hl'
       exact
@@ -377,11 +345,8 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
   · intro Z f
     dsimp [limits.types.binary_coproduct_cocone]
     delta types.pullback_obj
-    have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit :=
-      by
-      intro x
-      rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
-      exacts[Or.inl rfl, Or.inr rfl]
+    have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by intro x;
+      rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩); exacts[Or.inl rfl, Or.inr rfl]
     let eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
       ⟨fun p => ⟨p.1.1, by convert p.2⟩, fun x => ⟨⟨_, _⟩, x.2⟩, fun _ => by ext <;> rfl, fun _ =>
         by ext <;> rfl⟩
@@ -393,23 +358,9 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) :=
       exact fun s x =>
         dite _ (fun h => s.inl <| eX.symm ⟨x, h⟩) fun h =>
           s.inr <| eY.symm ⟨x, (this x).resolve_left h⟩
-    · intro s
-      ext ⟨⟨x, ⟨⟩⟩, _⟩
-      dsimp
-      split_ifs <;> rfl
-    · intro s
-      ext ⟨⟨x, ⟨⟩⟩, hx⟩
-      dsimp
-      split_ifs
-      · cases h.symm.trans hx
-      · rfl
-    · intro s m e₁ e₂
-      ext x
-      split_ifs
-      · rw [← e₁]
-        rfl
-      · rw [← e₂]
-        rfl
+    · intro s; ext ⟨⟨x, ⟨⟩⟩, _⟩; dsimp; split_ifs <;> rfl
+    · intro s; ext ⟨⟨x, ⟨⟩⟩, hx⟩; dsimp; split_ifs; · cases h.symm.trans hx; · rfl
+    · intro s m e₁ e₂; ext x; split_ifs; · rw [← e₁]; rfl; · rw [← e₂]; rfl
 #align category_theory.types.finitary_extensive CategoryTheory.types.finitaryExtensive
 
 section TopCat
@@ -421,11 +372,8 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
         (TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inl)
         (TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inr)) :=
   by
-  have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit :=
-    by
-    intro x
-    rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
-    exacts[Or.inl rfl, Or.inr rfl]
+  have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by intro x;
+    rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩); exacts[Or.inl rfl, Or.inr rfl]
   let eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
     ⟨fun p => ⟨p.1.1, p.2.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)⟩, fun x =>
       ⟨⟨_, _⟩, x.2⟩, fun _ => by ext <;> rfl, fun _ => by ext <;> rfl⟩
@@ -446,11 +394,9 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
       · rw [continuousOn_iff_continuous_restrict]
         convert_to Continuous fun x : { x | f x = Sum.inl PUnit.unit } =>
             s.inl ⟨(x, PUnit.unit), x.2⟩
-        · ext ⟨x, hx⟩
-          exact dif_pos hx
+        · ext ⟨x, hx⟩; exact dif_pos hx
         continuity
-      · convert f.2.1 _ openEmbedding_inl.open_range
-        ext x
+      · convert f.2.1 _ openEmbedding_inl.open_range; ext x;
         exact
           ⟨fun h => ⟨_, h.symm⟩, fun ⟨e, h⟩ =>
             h.symm.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)⟩
@@ -459,11 +405,9 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
       · rw [continuousOn_iff_continuous_restrict]
         convert_to Continuous fun x : { x | f x ≠ Sum.inl PUnit.unit } =>
             s.inr ⟨(x, PUnit.unit), (this _).resolve_left x.2⟩
-        · ext ⟨x, hx⟩
-          exact dif_neg hx
+        · ext ⟨x, hx⟩; exact dif_neg hx
         continuity
-      · convert f.2.1 _ openEmbedding_inr.open_range
-        ext x
+      · convert f.2.1 _ openEmbedding_inr.open_range; ext x
         change f x ≠ Sum.inl PUnit.unit ↔ f x ∈ Set.range Sum.inr
         trans f x = Sum.inr PUnit.unit
         ·
@@ -473,24 +417,11 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
           exact
             ⟨fun h => ⟨_, h.symm⟩, fun ⟨e, h⟩ =>
               h.symm.trans (congr_arg Sum.inr <| Subsingleton.elim _ _)⟩
-  · intro s
-    ext ⟨⟨x, ⟨⟩⟩, _⟩
-    change dite _ _ _ = _
-    split_ifs <;> rfl
-  · intro s
-    ext ⟨⟨x, ⟨⟩⟩, hx⟩
-    change dite _ _ _ = _
-    split_ifs
-    · cases h.symm.trans hx
-    · rfl
-  · intro s m e₁ e₂
-    ext x
-    change m x = dite _ _ _
-    split_ifs
-    · rw [← e₁]
-      rfl
-    · rw [← e₂]
-      rfl
+  · intro s; ext ⟨⟨x, ⟨⟩⟩, _⟩; change dite _ _ _ = _; split_ifs <;> rfl
+  · intro s; ext ⟨⟨x, ⟨⟩⟩, hx⟩; change dite _ _ _ = _
+    split_ifs; · cases h.symm.trans hx; · rfl
+  · intro s m e₁ e₂; ext x; change m x = dite _ _ _
+    split_ifs; · rw [← e₁]; rfl; · rw [← e₂]; rfl
 #align category_theory.finitary_extensive_Top_aux CategoryTheory.finitaryExtensiveTopAux
 
 instance : FinitaryExtensive TopCat.{u} :=
@@ -507,8 +438,7 @@ instance : FinitaryExtensive TopCat.{u} :=
       have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
         intro x
         cases h : s.fst x
-        · simp_rw [sum.inl_injective.eq_iff]
-          exact exists_unique_eq'
+        · simp_rw [sum.inl_injective.eq_iff]; exact exists_unique_eq'
         · apply_fun f  at h
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
@@ -520,11 +450,9 @@ instance : FinitaryExtensive TopCat.{u} :=
           fun l' h₁ h₂ =>
           ContinuousMap.ext fun x => hl' x (l' x) (concrete_category.congr_hom h₁ x).symm⟩
       apply embedding_inl.to_inducing.continuous_iff.mpr
-      convert s.fst.2 using 1
-      exact (funext hl).symm
+      convert s.fst.2 using 1; exact (funext hl).symm
     · refine' ⟨⟨hαY.symm⟩, ⟨pullback_cone.is_limit_aux' _ _⟩⟩
-      intro s
-      dsimp
+      intro s; dsimp
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases h : s.fst x
@@ -532,8 +460,7 @@ instance : FinitaryExtensive TopCat.{u} :=
           cases
             ((concrete_category.congr_hom s.condition x).symm.trans h).trans
               (concrete_category.congr_hom hαX val : _).symm
-        · simp_rw [sum.inr_injective.eq_iff]
-          exact exists_unique_eq'
+        · simp_rw [sum.inr_injective.eq_iff]; exact exists_unique_eq'
       delta ExistsUnique at this
       choose l hl hl'
       refine'
@@ -541,10 +468,8 @@ instance : FinitaryExtensive TopCat.{u} :=
           fun l' h₁ h₂ =>
           ContinuousMap.ext fun x => hl' x (l' x) (concrete_category.congr_hom h₁ x).symm⟩
       apply embedding_inr.to_inducing.continuous_iff.mpr
-      convert s.fst.2 using 1
-      exact (funext hl).symm
-  · intro Z f
-    exact finitary_extensive_Top_aux Z f
+      convert s.fst.2 using 1; exact (funext hl).symm
+  · intro Z f; exact finitary_extensive_Top_aux Z f
 
 end TopCat
 
@@ -563,10 +488,7 @@ theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKamp
     (e : c ≅ c') : IsVanKampenColimit c' :=
   by
   intro F' c'' α f h hα
-  have : c'.ι ≫ (Functor.Const J).map e.inv.hom = c.ι :=
-    by
-    ext j
-    exact e.inv.2 j
+  have : c'.ι ≫ (Functor.Const J).map e.inv.hom = c.ι := by ext j; exact e.inv.2 j
   rw [H c'' α (f ≫ e.inv.1) (by rw [functor.map_comp, ← reassoc_of h, this]) hα]
   apply forall_congr'
   intro j
@@ -584,9 +506,7 @@ theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J
   refine'
     (Iff.trans _
           (H (G.map_cocone c') (whisker_right α G) (G.map f)
-            (by
-              ext j
-              simpa using G.congr_map (nat_trans.congr_app h j))
+            (by ext j; simpa using G.congr_map (nat_trans.congr_app h j))
             (hα.whisker_right G))).trans
       (forall_congr' fun j => _)
   · exact ⟨fun h => ⟨is_colimit_of_preserves G h.some⟩, fun h => ⟨is_colimit_of_reflects G h.some⟩⟩
@@ -600,11 +520,7 @@ theorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J
   have := fun x =>
     hc x (((evaluation C D).obj x).mapCocone c') (whisker_right α _)
       (((evaluation C D).obj x).map f)
-      (by
-        ext y
-        dsimp
-        exact nat_trans.congr_app (nat_trans.congr_app e y) x)
-      (hα.whisker_right _)
+      (by ext y; dsimp; exact nat_trans.congr_app (nat_trans.congr_app e y) x) (hα.whisker_right _)
   constructor
   · rintro ⟨hc'⟩ j
     refine' ⟨⟨(nat_trans.congr_app e j).symm⟩, ⟨evaluation_jointly_reflects_limits _ _⟩⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -495,7 +495,7 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
 
 instance : FinitaryExtensive TopCat.{u} :=
   by
-  rw [finitary_extensive_iff_of_is_terminal TopCat.{u} _ TopCat.isTerminalPunit _
+  rw [finitary_extensive_iff_of_is_terminal TopCat.{u} _ TopCat.isTerminalPUnit _
       (TopCat.binaryCofanIsColimit _ _)]
   apply
     binary_cofan.is_van_kampen_mk _ _ (fun X Y => TopCat.binaryCofanIsColimit X Y) _

mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd

@@ -4,14 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.extensive
-! leanprover-community/mathlib commit ac3ae212f394f508df43e37aa093722fa9b65d31
+! leanprover-community/mathlib commit 178a32653e369dce2da68dc6b2694e385d484ef1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Limits.Shapes.CommSq
 import Mathbin.CategoryTheory.Limits.Shapes.StrictInitial
 import Mathbin.CategoryTheory.Limits.Shapes.Types
-import Mathbin.Topology.Category.Top.Limits
+import Mathbin.Topology.Category.Top.Limits.Pullbacks
 import Mathbin.CategoryTheory.Limits.FunctorCategory
 
 /-!

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -444,8 +444,8 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
     · revert h x
       apply (IsOpen.continuousOn_iff _).mp
       · rw [continuousOn_iff_continuous_restrict]
-        convert_to
-          Continuous fun x : { x | f x = Sum.inl PUnit.unit } => s.inl ⟨(x, PUnit.unit), x.2⟩
+        convert_to Continuous fun x : { x | f x = Sum.inl PUnit.unit } =>
+            s.inl ⟨(x, PUnit.unit), x.2⟩
         · ext ⟨x, hx⟩
           exact dif_pos hx
         continuity
@@ -457,8 +457,7 @@ def finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z ⟶ TopCat.of (Sum PUnit.{u
     · revert h x
       apply (IsOpen.continuousOn_iff _).mp
       · rw [continuousOn_iff_continuous_restrict]
-        convert_to
-          Continuous fun x : { x | f x ≠ Sum.inl PUnit.unit } =>
+        convert_to Continuous fun x : { x | f x ≠ Sum.inl PUnit.unit } =>
             s.inr ⟨(x, PUnit.unit), (this _).resolve_left x.2⟩
         · ext ⟨x, hx⟩
           exact dif_neg hx

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -78,7 +78,7 @@ theorem NatTrans.Equifibered.comp {F G H : J ⥤ C} {α : F ⟶ G} {β : G ⟶ H
 
 /-- A (colimit) cocone over a diagram `F : J ⥤ C` is universal if it is stable under pullbacks. -/
 def IsUniversalColimit {F : J ⥤ C} (c : Cocone F) : Prop :=
-  ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.x ⟶ c.x)
+  ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.pt ⟶ c.pt)
     (h : α ≫ c.ι = c'.ι ≫ (Functor.const J).map f) (hα : α.Equifibered),
     (∀ j : J, IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)) → Nonempty (IsColimit c')
 #align category_theory.is_universal_colimit CategoryTheory.IsUniversalColimit
@@ -90,7 +90,7 @@ TODO: Show that this is iff the functor `C ⥤ Catᵒᵖ` sending `x` to `C/x` p
 TODO: Show that this is iff the inclusion functor `C ⥤ Span(C)` preserves it.
 -/
 def IsVanKampenColimit {F : J ⥤ C} (c : Cocone F) : Prop :=
-  ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.x ⟶ c.x)
+  ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.pt ⟶ c.pt)
     (h : α ≫ c.ι = c'.ι ≫ (Functor.const J).map f) (hα : α.Equifibered),
     Nonempty (IsColimit c') ↔ ∀ j : J, IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)
 #align category_theory.is_van_kampen_colimit CategoryTheory.IsVanKampenColimit
@@ -162,7 +162,7 @@ theorem map_pair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F')
 
 theorem BinaryCofan.is_van_kampen_iff (c : BinaryCofan X Y) :
     IsVanKampenColimit c ↔
-      ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.x ⟶ c.x)
+      ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt)
         (hαX : αX ≫ c.inl = c'.inl ≫ f) (hαY : αY ≫ c.inr = c'.inr ≫ f),
         Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr :=
   by
@@ -198,11 +198,12 @@ theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
     (cones : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), PullbackCone f g)
     (limits : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), IsLimit (cones f g))
     (h₁ :
-      ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').x ⟶ c.x)
+      ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt)
         (hαX : αX ≫ c.inl = (cofans X' Y').inl ≫ f) (hαY : αY ≫ c.inr = (cofans X' Y').inr ≫ f),
         IsPullback (cofans X' Y').inl αX f c.inl ∧ IsPullback (cofans X' Y').inr αY f c.inr)
     (h₂ :
-      ∀ {Z : C} (f : Z ⟶ c.x), IsColimit (BinaryCofan.mk (cones f c.inl).fst (cones f c.inr).fst)) :
+      ∀ {Z : C} (f : Z ⟶ c.pt),
+        IsColimit (BinaryCofan.mk (cones f c.inl).fst (cones f c.inr).fst)) :
     IsVanKampenColimit c := by
   rw [binary_cofan.is_van_kampen_iff]
   introv hX hY

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -69,11 +69,11 @@ def NatTrans.Equifibered {F G : J ⥤ C} (α : F ⟶ G) : Prop :=
 #align category_theory.nat_trans.equifibered CategoryTheory.NatTrans.Equifibered
 
 theorem NatTrans.equifibered_of_isIso {F G : J ⥤ C} (α : F ⟶ G) [IsIso α] : α.Equifibered :=
-  fun _ _ f => IsPullback.ofVertIsIso ⟨NatTrans.naturality _ f⟩
+  fun _ _ f => IsPullback.of_vert_isIso ⟨NatTrans.naturality _ f⟩
 #align category_theory.nat_trans.equifibered_of_is_iso CategoryTheory.NatTrans.equifibered_of_isIso
 
 theorem NatTrans.Equifibered.comp {F G H : J ⥤ C} {α : F ⟶ G} {β : G ⟶ H} (hα : α.Equifibered)
-    (hβ : β.Equifibered) : (α ≫ β).Equifibered := fun i j f => (hα f).pasteVert (hβ f)
+    (hβ : β.Equifibered) : (α ≫ β).Equifibered := fun i j f => (hα f).paste_vert (hβ f)
 #align category_theory.nat_trans.equifibered.comp CategoryTheory.NatTrans.Equifibered.comp
 
 /-- A (colimit) cocone over a diagram `F : J ⥤ C` is universal if it is stable under pullbacks. -/
@@ -217,14 +217,14 @@ theorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)
         dsimp
         simp
       haveI : is_iso (𝟙 X') := inferInstance
-      exact (is_pullback.of_vert_is_iso ⟨this⟩).pasteVert hl
+      exact (is_pullback.of_vert_is_iso ⟨this⟩).paste_vert hl
     · rw [← category.id_comp αY, ← iso.hom_inv_id_assoc e f]
       have : c'.inr ≫ e.hom = 𝟙 Y' ≫ (cofans X' Y').inr :=
         by
         dsimp
         simp
       haveI : is_iso (𝟙 Y') := inferInstance
-      exact (is_pullback.of_vert_is_iso ⟨this⟩).pasteVert hr
+      exact (is_pullback.of_vert_is_iso ⟨this⟩).paste_vert hr
   · rintro ⟨H₁, H₂⟩
     refine' ⟨is_colimit.of_iso_colimit _ <| (iso_binary_cofan_mk _).symm⟩
     let e₁ : X' ≅ _ := H₁.is_limit.cone_point_unique_up_to_iso (limits _ _)
@@ -270,7 +270,7 @@ instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inl : X ⟶ X ⨿ Y) :=
 instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inr : Y ⟶ X ⨿ Y) :=
   (FinitaryExtensive.mono_inr_of_isColimit (coprodIsCoprod X Y) : _)
 
-theorem BinaryCofan.isPullbackInitialToOfIsVanKampen [HasInitial C] {c : BinaryCofan X Y}
+theorem BinaryCofan.isPullback_initial_to_of_is_van_kampen [HasInitial C] {c : BinaryCofan X Y}
     (h : IsVanKampenColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
   by
   refine'
@@ -284,12 +284,13 @@ theorem BinaryCofan.isPullbackInitialToOfIsVanKampen [HasInitial C] {c : BinaryC
           (by
             dsimp
             infer_instance)).some
-#align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullbackInitialToOfIsVanKampen
+#align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_is_van_kampen
 
-theorem FinitaryExtensive.isPullbackInitialToBinaryCofan [FinitaryExtensive C] {c : BinaryCofan X Y}
-    (hc : IsColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
-  BinaryCofan.isPullbackInitialToOfIsVanKampen (FinitaryExtensive.van_kampen c hc)
-#align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullbackInitialToBinaryCofan
+theorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]
+    {c : BinaryCofan X Y} (hc : IsColimit c) :
+    IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
+  BinaryCofan.isPullback_initial_to_of_is_van_kampen (FinitaryExtensive.van_kampen c hc)
+#align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan
 
 theorem has_strict_initial_of_is_universal [HasInitial C]
     (H : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))) : HasStrictInitialObjects C :=
@@ -621,14 +622,14 @@ instance [HasPullbacks C] [FinitaryExtensive C] : FinitaryExtensive (D ⥤ C) :=
     is_van_kampen_colimit_of_evaluation _ c fun x =>
       finitary_extensive.van_kampen _ <| preserves_colimit.preserves hc⟩
 
-theorem finitaryExtensiveOfPreservesAndReflects (F : C ⥤ D) [FinitaryExtensive D]
+theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExtensive D]
     [HasFiniteCoproducts C] [PreservesLimitsOfShape WalkingCospan F]
     [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
     [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C :=
   ⟨fun X Y c hc => (FinitaryExtensive.van_kampen _ (isColimitOfPreserves F hc)).of_map F⟩
-#align category_theory.finitary_extensive_of_preserves_and_reflects CategoryTheory.finitaryExtensiveOfPreservesAndReflects
+#align category_theory.finitary_extensive_of_preserves_and_reflects CategoryTheory.finitaryExtensive_of_preserves_and_reflects
 
-theorem finitaryExtensiveOfPreservesAndReflectsIsomorphism (F : C ⥤ D) [FinitaryExtensive D]
+theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]
     [HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]
     [PreservesColimitsOfShape (Discrete WalkingPair) F] [ReflectsIsomorphisms F] :
     FinitaryExtensive C :=
@@ -638,7 +639,7 @@ theorem finitaryExtensiveOfPreservesAndReflectsIsomorphism (F : C ⥤ D) [Finita
   haveI : reflects_colimits_of_shape (discrete walking_pair) F :=
     reflects_colimits_of_shape_of_reflects_isomorphisms
   exact finitary_extensive_of_preserves_and_reflects F
-#align category_theory.finitary_extensive_of_preserves_and_reflects_isomorphism CategoryTheory.finitaryExtensiveOfPreservesAndReflectsIsomorphism
+#align category_theory.finitary_extensive_of_preserves_and_reflects_isomorphism CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism
 
 end Functor
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

This is a PR that's a temporary measure to improve performance of congr!/convert, and the implementation may change in a future PR with a new version of congr!.

Introduces two typeclasses that are meant to quickly evaluate in common cases of Subsingleton and IsEmpty. Makes congr! use these typeclasses rather than Subsingleton.

Local Subsingleton/IsEmpty instances are included as Fast instances. To get congr!/convert to reason about subsingleton types, you can add such instances to the local context. Or, you can apply Subsingleton.elim yourself.

Zulip discussion

@@ -764,6 +764,7 @@ theorem isPullback_initial_to_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {
     Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
   clear_value f
   subst this
+  have : ∀ i, Subsingleton (⊥_ C ⟶ (Discrete.functor f).obj i) := inferInstance
   convert isPullback_of_cofan_isVanKampen hc i.as j.as
   exact (if_neg (mt (Discrete.ext _ _) hi.symm)).symm
 

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

@@ -360,7 +360,7 @@ theorem IsUniversalColimit.map_reflective
         exact c.w _
   let cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' := by
     refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
-    exact (inv <| adj.counit.app c'.pt)
+    exact inv <| adj.counit.app c'.pt
     · rw [IsIso.inv_comp_eq, ← adj.counit_naturality_assoc f, ← cancel_mono (adj.counit.app <|
         Gl.obj c.pt), Category.assoc, Category.assoc, adj.left_triangle_components]
       erw [Category.comp_id]

These notions on functors are now Functor.Full, Functor.Faithful, Functor.EssSurj, Functor.IsEquivalence, Functor.ReflectsIsomorphisms. Deprecated aliases are introduced for the previous names.

@@ -221,7 +221,7 @@ theorem IsVanKampenColimit.of_mapCocone (G : C ⥤ D) {F : J ⥤ C} {c : Cocone
 #align category_theory.is_van_kampen_colimit.of_map CategoryTheory.IsVanKampenColimit.of_mapCocone
 
 theorem IsVanKampenColimit.mapCocone_iff (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
-    [IsEquivalence G] : IsVanKampenColimit (G.mapCocone c) ↔ IsVanKampenColimit c :=
+    [G.IsEquivalence] : IsVanKampenColimit (G.mapCocone c) ↔ IsVanKampenColimit c :=
   ⟨IsVanKampenColimit.of_mapCocone G, fun hc ↦ by
     let e : F ⋙ G ⋙ Functor.inv G ≅ F := NatIso.hcomp (Iso.refl F) G.asEquivalence.unitIso.symm
     apply IsVanKampenColimit.of_mapCocone G.inv
@@ -307,7 +307,7 @@ end Functor
 section reflective
 
 theorem IsUniversalColimit.map_reflective
-    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
+    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
     {F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
     (H : IsUniversalColimit c)
     [∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
@@ -410,7 +410,7 @@ theorem IsUniversalColimit.map_reflective
         pullback.lift_snd]
 
 theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
-    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
+    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
     {F : J ⥤ D} {c : Cocone (F ⋙ Gr)} (H : IsVanKampenColimit c)
     [∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
     [∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl]

These notions on functors are now Functor.Full, Functor.Faithful, Functor.EssSurj, Functor.IsEquivalence, Functor.ReflectsIsomorphisms. Deprecated aliases are introduced for the previous names.

@@ -416,7 +416,7 @@ section Functor
 
 theorem finitaryExtensive_of_reflective
     [HasFiniteCoproducts D] [HasPullbacksOfInclusions D] [FinitaryExtensive C]
-    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
+    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
     [∀ X Y (f : X ⟶ Gl.obj Y), HasPullback (Gr.map f) (adj.unit.app Y)]
     [∀ X Y (f : X ⟶ Gl.obj Y), PreservesLimit (cospan (Gr.map f) (adj.unit.app Y)) Gl]
     [PreservesPullbacksOfInclusions Gl] :
@@ -475,7 +475,7 @@ theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExten
 
 theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]
     [HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]
-    [PreservesColimitsOfShape (Discrete WalkingPair) F] [ReflectsIsomorphisms F] :
+    [PreservesColimitsOfShape (Discrete WalkingPair) F] [F.ReflectsIsomorphisms] :
     FinitaryExtensive C := by
   haveI : ReflectsLimitsOfShape WalkingCospan F := reflectsLimitsOfShapeOfReflectsIsomorphisms
   haveI : ReflectsColimitsOfShape (Discrete WalkingPair) F :=

@@ -340,7 +340,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
         change f x ≠ Sum.inl PUnit.unit ↔ f x ∈ Set.range Sum.inr
         trans f x = Sum.inr PUnit.unit
         · rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩) <;>
-            simp only [iff_self_iff, eq_self_iff_true, not_true, Ne.def, not_false_iff]
+            simp only [iff_self_iff, eq_self_iff_true, not_true, Ne, not_false_iff]
         · exact ⟨fun h => ⟨_, h.symm⟩,
             fun ⟨e, h⟩ => h.symm.trans (congr_arg Sum.inr <| Subsingleton.elim _ _)⟩
   · intro s

All these lemmas refer to the range of some function being open/range (i.e. isOpen or isClosed).

@@ -321,7 +321,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
         -- Porting note: this `(BinaryCofan.inl s).2` was unnecessary
         have := (BinaryCofan.inl s).2
         continuity
-      · convert f.2.1 _ openEmbedding_inl.open_range
+      · convert f.2.1 _ openEmbedding_inl.isOpen_range
         rename_i x
         exact ⟨fun h => ⟨_, h.symm⟩,
           fun ⟨e, h⟩ => h.symm.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)⟩
@@ -335,7 +335,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
         -- Porting note: this `(BinaryCofan.inr s).2` was unnecessary
         have := (BinaryCofan.inr s).2
         continuity
-      · convert f.2.1 _ openEmbedding_inr.open_range
+      · convert f.2.1 _ openEmbedding_inr.isOpen_range
         rename_i x
         change f x ≠ Sum.inl PUnit.unit ↔ f x ∈ Set.range Sum.inr
         trans f x = Sum.inr PUnit.unit

Includes some doc comments and real code: this is exhaustive, with two exceptions:

• some files are handled in #11409 instead
• I left FunProp/{ToStd,RefinedDiscTree}.lean, Tactic/NormNum and Tactic/Simps alone, as these seem likely enough to end up in std.

Follow-up to #11301, much shorter this time.

@@ -339,7 +339,7 @@ theorem IsUniversalColimit.map_reflective
   let c'' : Cocone (F' ⋙ Gr) := by
     refine
     { pt := pullback (Gr.map f) (adj.unit.app _)
-      ι := { app := λ j ↦ pullback.lift (Gr.map <| c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
+      ι := { app := fun j ↦ pullback.lift (Gr.map <| c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
              naturality := ?_ } }
     · rw [← Gr.map_comp, ← hc'']
       erw [← adj.unit_naturality]

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

@@ -227,7 +227,7 @@ theorem IsVanKampenColimit.mapCocone_iff (G : C ⥤ D) {F : J ⥤ C} {c : Cocone
     apply IsVanKampenColimit.of_mapCocone G.inv
     apply (IsVanKampenColimit.precompose_isIso_iff e.inv).mp
     refine hc.of_iso (Cocones.ext (G.asEquivalence.unitIso.app c.pt) ?_)
-    simp [Functor.asEquivalence]⟩
+    simp [e, Functor.asEquivalence]⟩
 
 theorem IsUniversalColimit.whiskerEquivalence {K : Type*} [Category K] (e : J ≌ K)
     {F : K ⥤ C} {c : Cocone F} (hc : IsUniversalColimit c) :
@@ -331,7 +331,7 @@ theorem IsUniversalColimit.map_reflective
   have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j := by
     intro j
     rw [← cancel_mono (adj.counit.app <| F.obj j)]
-    dsimp
+    dsimp [α']
     simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
       Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
   have hc'' : ∀ j, α.app j ≫ Gl.map (c.ι.app j) = c'.ι.app j ≫ f := NatTrans.congr_app h
@@ -347,7 +347,7 @@ theorem IsUniversalColimit.map_reflective
       dsimp
       simp only [Category.assoc, Functor.map_comp, adj.right_triangle_components_assoc]
     · intros i j g
-      dsimp
+      dsimp [α']
       ext
       all_goals simp only [Category.comp_id, Category.id_comp, Category.assoc,
         ← Functor.map_comp, pullback.lift_fst, pullback.lift_snd, ← Functor.map_comp_assoc]
@@ -372,10 +372,10 @@ theorem IsUniversalColimit.map_reflective
           pullback.lift_fst, pullback.lift_snd, Category.assoc,
           Functor.mapCocone_ι_app, ← Gl.map_comp]
       · rw [IsIso.comp_inv_eq, adj.counit_naturality]
-        dsimp
+        dsimp [β]
         rw [Category.comp_id]
       · rw [Gl.map_comp, hα'', Category.assoc, hc'']
-        dsimp
+        dsimp [β]
         rw [Category.comp_id, Category.assoc]
   have : cf.hom ≫ (PreservesPullback.iso _ _ _).hom ≫ pullback.fst ≫ adj.counit.app _ = 𝟙 _ := by
     simp only [IsIso.inv_hom_id, Iso.inv_hom_id_assoc, Category.assoc, pullback.lift_fst_assoc]
@@ -393,7 +393,7 @@ theorem IsUniversalColimit.map_reflective
       Functor.map_comp, pullback.lift_snd]
   · intro j
     apply IsPullback.of_right _ _ (IsPullback.of_hasPullback _ _)
-    · dsimp
+    · dsimp [α']
       simp only [Category.comp_id, Category.id_comp, Category.assoc, Functor.map_comp,
         pullback.lift_fst]
       rw [← Category.comp_id (Gr.map f)]
@@ -426,7 +426,7 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
   have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j := by
     intro j
     rw [← cancel_mono (adj.counit.app <| F.obj j)]
-    dsimp
+    dsimp [α']
     simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
       Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
   let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
@@ -434,7 +434,7 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
   let hr := isColimitOfPreserves Gl (colimit.isColimit <| F' ⋙ Gr)
   have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map <| α'.app j) := by
     rw [hα'']
-    simp
+    simp [β]
   rw [this]
   have : f = (hl.coconePointUniqueUpToIso hr).hom ≫
     Gl.map (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) := by
@@ -449,7 +449,7 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
       apply hl.hom_ext
       intro j
       rw [hl.fac]
-      dsimp
+      dsimp [β]
       simp only [Category.comp_id, hα'', Category.assoc, Gl.map_comp]
       congr 1
       exact (NatTrans.congr_app h j).symm
@@ -550,26 +550,26 @@ theorem BinaryCofan.isVanKampen_mk {X Y : C} (c : BinaryCofan X Y)
   constructor
   · rintro ⟨h⟩
     let e := h.coconePointUniqueUpToIso (colimits _ _)
-    obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [hX]) (by simp [hY])
+    obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [e, hX]) (by simp [e, hY])
     constructor
     · rw [← Category.id_comp αX, ← Iso.hom_inv_id_assoc e f]
       haveI : IsIso (𝟙 X') := inferInstance
       have : c'.inl ≫ e.hom = 𝟙 X' ≫ (cofans X' Y').inl := by
-        dsimp
+        dsimp [e]
         simp
       exact (IsPullback.of_vert_isIso ⟨this⟩).paste_vert hl
     · rw [← Category.id_comp αY, ← Iso.hom_inv_id_assoc e f]
       haveI : IsIso (𝟙 Y') := inferInstance
       have : c'.inr ≫ e.hom = 𝟙 Y' ≫ (cofans X' Y').inr := by
-        dsimp
+        dsimp [e]
         simp
       exact (IsPullback.of_vert_isIso ⟨this⟩).paste_vert hr
   · rintro ⟨H₁, H₂⟩
     refine' ⟨IsColimit.ofIsoColimit _ <| (isoBinaryCofanMk _).symm⟩
     let e₁ : X' ≅ _ := H₁.isLimit.conePointUniqueUpToIso (limits _ _)
     let e₂ : Y' ≅ _ := H₂.isLimit.conePointUniqueUpToIso (limits _ _)
-    have he₁ : c'.inl = e₁.hom ≫ (cones f c.inl).fst := by simp
-    have he₂ : c'.inr = e₂.hom ≫ (cones f c.inr).fst := by simp
+    have he₁ : c'.inl = e₁.hom ≫ (cones f c.inl).fst := by simp [e₁]
+    have he₂ : c'.inr = e₂.hom ≫ (cones f c.inr).fst := by simp [e₂]
     rw [he₁, he₂]
     apply BinaryCofan.isColimitCompRightIso (BinaryCofan.mk _ _)
     apply BinaryCofan.isColimitCompLeftIso (BinaryCofan.mk _ _)
@@ -612,7 +612,7 @@ theorem isUniversalColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
     apply Functor.hext
     · exact fun i ↦ rfl
     · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
-      simp
+      simp [F']
   have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
     (Cofan.mk (pullback c₂.inr i) fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
     (Discrete.natTrans fun i ↦ α.app _) pullback.fst ?_ (NatTrans.equifibered_of_discrete _) ?_
@@ -687,7 +687,7 @@ theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
       apply Functor.hext
       · exact fun i ↦ rfl
       · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
-        simp
+        simp [F']
     clear_value F'
     subst this
     apply BinaryCofan.IsColimit.mk _ (fun {T} f₁ f₂ ↦ Hc.desc (Cofan.mk T (Fin.cases f₁
@@ -761,7 +761,7 @@ theorem isPullback_initial_to_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {
   classical
   let f : ι → C := F.obj ∘ Discrete.mk
   have : F = Discrete.functor f :=
-    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
   clear_value f
   subst this
   convert isPullback_of_cofan_isVanKampen hc i.as j.as
@@ -772,7 +772,7 @@ theorem mono_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {F : Discrete ι 
   classical
   let f : ι → C := F.obj ∘ Discrete.mk
   have : F = Discrete.functor f :=
-    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
   clear_value f
   subst this
   refine' PullbackCone.mono_of_isLimitMkIdId _ (IsPullback.isLimit _)

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

@@ -491,7 +491,7 @@ theorem FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin [FinitaryPreExtens
     {F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsUniversalColimit c := by
   let f : Fin n → C := F.obj ∘ Discrete.mk
   have : F = Discrete.functor f :=
-    Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+    Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
   clear_value f
   subst this
   induction' n with n IH
@@ -517,7 +517,7 @@ theorem FinitaryExtensive.isVanKampen_finiteCoproducts_Fin [FinitaryExtensive C]
     {F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsVanKampenColimit c := by
   let f : Fin n → C := F.obj ∘ Discrete.mk
   have : F = Discrete.functor f :=
-    Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+    Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
   clear_value f
   subst this
   induction' n with n IH
@@ -545,7 +545,7 @@ lemma FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C]
   classical
   let f : ι → C := F.obj ∘ Discrete.mk
   have : F = Discrete.functor f :=
-    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
   clear_value f
   subst this
   change Cofan f at c
@@ -565,7 +565,7 @@ lemma FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C]
   let e' : c.pt ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
     hc.coconePointUniqueUpToIso (getColimitCocone _).2 ≪≫ e
   have : coprod.inl ≫ e'.inv = c.ι.app ⟨i⟩ := by
-    simp only [Iso.trans_inv, coprod.desc_comp, colimit.ι_desc, BinaryCofan.mk_pt,
+    simp only [e', Iso.trans_inv, coprod.desc_comp, colimit.ι_desc, BinaryCofan.mk_pt,
       BinaryCofan.ι_app_left, BinaryCofan.mk_inl]
     exact colimit.comp_coconePointUniqueUpToIso_inv _ _
   clear_value e'

In this pull request, I have systematically eliminated the leading whitespace preceding the colon (:) within all unlabelled or unclassified porting notes. This adjustment facilitates a more efficient review process for the remaining notes by ensuring no entries are overlooked due to formatting inconsistencies.

@@ -288,7 +288,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) := by
 section TopCat
 
 /-- (Implementation) An auxiliary lemma for the proof that `TopCat` is finitary extensive. -/
--- Porting note : needs to add noncomputable modifier
+-- Porting note: needs to add noncomputable modifier
 noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
     (f : Z ⟶ TopCat.of (Sum PUnit.{u + 1} PUnit.{u + 1})) :
     IsColimit (BinaryCofan.mk
@@ -318,7 +318,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
             s.inl ⟨(x.1, PUnit.unit), x.2⟩
         · ext ⟨x, hx⟩
           exact dif_pos hx
-        -- Porting note : this `(BinaryCofan.inl s).2` was unnecessary
+        -- Porting note: this `(BinaryCofan.inl s).2` was unnecessary
         have := (BinaryCofan.inl s).2
         continuity
       · convert f.2.1 _ openEmbedding_inl.open_range
@@ -332,7 +332,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
             s.inr ⟨(x.1, PUnit.unit), (this _).resolve_left x.2⟩
         · ext ⟨x, hx⟩
           exact dif_neg hx
-        -- Porting note : this `(BinaryCofan.inr s).2` was unnecessary
+        -- Porting note: this `(BinaryCofan.inr s).2` was unnecessary
         have := (BinaryCofan.inr s).2
         continuity
       · convert f.2.1 _ openEmbedding_inr.open_range
@@ -348,7 +348,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
     change dite _ _ _ = _
     split_ifs with h
     · rfl
-    · cases (h hx) -- Porting note : in Lean3 it is `rfl`
+    · cases (h hx) -- Porting note: in Lean3 it is `rfl`
   · intro s
     ext ⟨⟨x, ⟨⟩⟩, hx⟩
     change dite _ _ _ = _

This was missing, but essentially what was proved in the analogous statement for Van Kampen colimits.

@@ -112,13 +112,18 @@ theorem IsVanKampenColimit.isUniversal {F : J ⥤ C} {c : Cocone F} (H : IsVanKa
   fun _ c' α f h hα => (H c' α f h hα).mpr
 #align category_theory.is_van_kampen_colimit.is_universal CategoryTheory.IsVanKampenColimit.isUniversal
 
-/-- A van Kampen colimit is a colimit. -/
-noncomputable def IsVanKampenColimit.isColimit {F : J ⥤ C} {c : Cocone F}
-    (h : IsVanKampenColimit c) : IsColimit c := by
-  refine' ((h c (𝟙 F) (𝟙 c.pt : _) (by rw [Functor.map_id, Category.comp_id, Category.id_comp])
-    (NatTrans.equifibered_of_isIso _)).mpr fun j => _).some
+/-- A universal colimit is a colimit. -/
+noncomputable def IsUniversalColimit.isColimit {F : J ⥤ C} {c : Cocone F}
+    (h : IsUniversalColimit c) : IsColimit c := by
+  refine ((h c (𝟙 F) (𝟙 c.pt : _) (by rw [Functor.map_id, Category.comp_id, Category.id_comp])
+    (NatTrans.equifibered_of_isIso _)) fun j => ?_).some
   haveI : IsIso (𝟙 c.pt) := inferInstance
   exact IsPullback.of_vert_isIso ⟨by erw [NatTrans.id_app, Category.comp_id, Category.id_comp]⟩
+
+/-- A van Kampen colimit is a colimit. -/
+noncomputable def IsVanKampenColimit.isColimit {F : J ⥤ C} {c : Cocone F}
+    (h : IsVanKampenColimit c) : IsColimit c :=
+  h.isUniversal.isColimit
 #align category_theory.is_van_kampen_colimit.is_colimit CategoryTheory.IsVanKampenColimit.isColimit
 
 theorem IsInitial.isVanKampenColimit [HasStrictInitialObjects C] {X : C} (h : IsInitial X) :

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -262,8 +262,9 @@ theorem IsVanKampenColimit.whiskerEquivalence {K : Type*} [Category K] (e : J 
       have : IsIso (𝟙 (Cocone.whisker e.functor c).pt) := inferInstance
       exact IsPullback.of_vert_isIso ⟨by simp⟩
     · intro H j
-      have : α.app j = F'.map (e.unit.app _) ≫ α.app _ ≫ F.map (e.counit.app (e.functor.obj j))
-      · simp [← Functor.map_comp]
+      have : α.app j
+          = F'.map (e.unit.app _) ≫ α.app _ ≫ F.map (e.counit.app (e.functor.obj j)) := by
+        simp [← Functor.map_comp]
       rw [← Category.id_comp f, this]
       refine IsPullback.paste_vert ?_ (H (e.functor.obj j))
       exact IsPullback.of_vert_isIso ⟨by simp⟩
@@ -315,23 +316,23 @@ theorem IsUniversalColimit.map_reflective
       (IsPullback.of_hasPullback _ _).isLimit⟩⟩
   let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
   have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
-  have hadj : ∀ X, Gl.map (adj.unit.app X) = inv (adj.counit.app _)
-  · intro X
+  have hadj : ∀ X, Gl.map (adj.unit.app X) = inv (adj.counit.app _) := by
+    intro X
     apply IsIso.eq_inv_of_inv_hom_id
     exact adj.left_triangle_components _
   haveI : ∀ X, IsIso (Gl.map (adj.unit.app X)) := by
     simp_rw [hadj]
     infer_instance
-  have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j
-  · intro j
+  have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j := by
+    intro j
     rw [← cancel_mono (adj.counit.app <| F.obj j)]
     dsimp
     simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
       Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
   have hc'' : ∀ j, α.app j ≫ Gl.map (c.ι.app j) = c'.ι.app j ≫ f := NatTrans.congr_app h
   let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
-  let c'' : Cocone (F' ⋙ Gr)
-  · refine
+  let c'' : Cocone (F' ⋙ Gr) := by
+    refine
     { pt := pullback (Gr.map f) (adj.unit.app _)
       ι := { app := λ j ↦ pullback.lift (Gr.map <| c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
              naturality := ?_ } }
@@ -352,8 +353,8 @@ theorem IsUniversalColimit.map_reflective
         rw [adj.counit_naturality, ← Category.assoc, Gr.map_comp_assoc]
         congr 1
         exact c.w _
-  let cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c''
-  · refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
+  let cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' := by
+    refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
     exact (inv <| adj.counit.app c'.pt)
     · rw [IsIso.inv_comp_eq, ← adj.counit_naturality_assoc f, ← cancel_mono (adj.counit.app <|
         Gl.obj c.pt), Category.assoc, Category.assoc, adj.left_triangle_components]
@@ -371,10 +372,10 @@ theorem IsUniversalColimit.map_reflective
       · rw [Gl.map_comp, hα'', Category.assoc, hc'']
         dsimp
         rw [Category.comp_id, Category.assoc]
-  have : cf.hom ≫ (PreservesPullback.iso _ _ _).hom ≫ pullback.fst ≫ adj.counit.app _ = 𝟙 _
-  · simp only [IsIso.inv_hom_id, Iso.inv_hom_id_assoc, Category.assoc, pullback.lift_fst_assoc]
-  have : IsIso cf
-  · apply @Cocones.cocone_iso_of_hom_iso (i := ?_)
+  have : cf.hom ≫ (PreservesPullback.iso _ _ _).hom ≫ pullback.fst ≫ adj.counit.app _ = 𝟙 _ := by
+    simp only [IsIso.inv_hom_id, Iso.inv_hom_id_assoc, Category.assoc, pullback.lift_fst_assoc]
+  have : IsIso cf := by
+    apply @Cocones.cocone_iso_of_hom_iso (i := ?_)
     rw [← IsIso.eq_comp_inv] at this
     rw [this]
     infer_instance
@@ -417,8 +418,8 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
   intro ⟨hc'⟩ j
   let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
   have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
-  have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j
-  · intro j
+  have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j := by
+    intro j
     rw [← cancel_mono (adj.counit.app <| F.obj j)]
     dsimp
     simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
@@ -426,13 +427,13 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
   let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
   let hl := (IsColimit.precomposeHomEquiv β c').symm hc'
   let hr := isColimitOfPreserves Gl (colimit.isColimit <| F' ⋙ Gr)
-  have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map <| α'.app j)
-  · rw [hα'']
+  have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map <| α'.app j) := by
+    rw [hα'']
     simp
   rw [this]
   have : f = (hl.coconePointUniqueUpToIso hr).hom ≫
-    Gl.map (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩)
-  · symm
+    Gl.map (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) := by
+    symm
     convert @IsColimit.coconePointUniqueUpToIso_hom_desc _ _ _ _ ((F' ⋙ Gr) ⋙ Gl)
       (Gl.mapCocone ⟨_, (whiskerRight α' Gr ≫ c.2 : _)⟩) _ _ hl hr using 2
     · apply hr.hom_ext
@@ -480,8 +481,8 @@ theorem hasStrictInitial_of_isUniversal [HasInitial C]
 
 theorem isVanKampenColimit_of_isEmpty [HasStrictInitialObjects C] [IsEmpty J] {F : J ⥤ C}
     (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by
-  have : IsInitial c.pt
-  · have := (IsColimit.precomposeInvEquiv (Functor.uniqueFromEmpty _) _).symm
+  have : IsInitial c.pt := by
+    have := (IsColimit.precomposeInvEquiv (Functor.uniqueFromEmpty _) _).symm
       (hc.whiskerEquivalence (equivalenceOfIsEmpty (Discrete PEmpty.{1}) J))
     exact IsColimit.ofIsoColimit this (Cocones.ext (Iso.refl c.pt) (fun {X} ↦ isEmptyElim X))
   replace this := IsInitial.isVanKampenColimit this
@@ -602,8 +603,8 @@ theorem isUniversalColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
     IsUniversalColimit (extendCofan c₁ c₂) := by
   intro F c α i e hα H
   let F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
-  have : F = Discrete.functor F'
-  · apply Functor.hext
+  have : F = Discrete.functor F' := by
+    apply Functor.hext
     · exact fun i ↦ rfl
     · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
       simp
@@ -677,8 +678,8 @@ theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
         Category.assoc, extendCofan_pt, Functor.const_obj_obj, NatTrans.comp_app, extendCofan_ι_app,
         Fin.cases_succ, Functor.const_map_app] using congr_app e ⟨j.succ⟩
   · let F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
-    have : F = Discrete.functor F'
-    · apply Functor.hext
+    have : F = Discrete.functor F' := by
+      apply Functor.hext
       · exact fun i ↦ rfl
       · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
         simp

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -564,8 +564,8 @@ lemma FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C]
         exact dif_neg j.prop }
   let e' : c.pt ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
     hc.coconePointUniqueUpToIso (getColimitCocone _).2 ≪≫ e
-  have : coprod.inl ≫ e'.inv = c.ι.app ⟨i⟩
-  · simp only [Iso.trans_inv, coprod.desc_comp, colimit.ι_desc, BinaryCofan.mk_pt,
+  have : coprod.inl ≫ e'.inv = c.ι.app ⟨i⟩ := by
+    simp only [Iso.trans_inv, coprod.desc_comp, colimit.ι_desc, BinaryCofan.mk_pt,
       BinaryCofan.ι_app_left, BinaryCofan.mk_inl]
     exact colimit.comp_coconePointUniqueUpToIso_inv _ _
   clear_value e'
@@ -606,8 +606,8 @@ lemma FinitaryPreExtensive.sigma_desc_iso [FinitaryPreExtensive C] {α : Type} [
   suffices IsColimit (Cofan.mk _ ((fun _ ↦ pullback.fst) : (a : α) → pullback f (π a) ⟶ _)) by
     change IsIso (this.coconePointUniqueUpToIso (getColimitCocone _).2).inv
     infer_instance
-  let : IsColimit (Cofan.mk X π)
-  · refine @IsColimit.ofPointIso (t := Cofan.mk X π) (P := coproductIsCoproduct Z) ?_
+  let this : IsColimit (Cofan.mk X π) := by
+    refine @IsColimit.ofPointIso (t := Cofan.mk X π) (P := coproductIsCoproduct Z) ?_
     convert hπ
     simp [coproductIsCoproduct]
   refine (FinitaryPreExtensive.isUniversal_finiteCoproducts this

This PR contains various prerequisites in order to show that under suitable assumptions, a localized category of a category that has finite products also has finite products:

• the equivalence of categories (J → C) ≌ (Discrete J ⥤ C)
• more API for the existence of limits as a consequence of a right adjoint to the constant functor C ⥤ (J ⥤ C).
• the typeclass MorphismProperty.IsStableUnderFiniteProducts

@@ -318,7 +318,7 @@ theorem IsUniversalColimit.map_reflective
   have hadj : ∀ X, Gl.map (adj.unit.app X) = inv (adj.counit.app _)
   · intro X
     apply IsIso.eq_inv_of_inv_hom_id
-    exact adj.left_triangle_components
+    exact adj.left_triangle_components _
   haveI : ∀ X, IsIso (Gl.map (adj.unit.app X)) := by
     simp_rw [hadj]
     infer_instance

Now we use letters X and Y for topological spaces, not Greek letters.

I didn't replace all letters; some lemmas need a large number of different letters. Volunteers for the last instances welcome.

@@ -387,7 +387,7 @@ instance finitaryExtensive_TopCat : FinitaryExtensive TopCat.{u} := by
       refine' ⟨⟨l, _⟩, ContinuousMap.ext fun a => (hl a).symm, TopCat.isTerminalPUnit.hom_ext _ _,
         fun {l'} h₁ _ => ContinuousMap.ext fun x =>
           hl' x (l' x) (ConcreteCategory.congr_hom h₁ x).symm⟩
-      apply (embedding_inl (α := X') (β := Y')).toInducing.continuous_iff.mpr
+      apply (embedding_inl (X := X') (Y := Y')).toInducing.continuous_iff.mpr
       convert s.fst.2 using 1
       exact (funext hl).symm
     · refine' ⟨⟨hαY.symm⟩, ⟨PullbackCone.isLimitAux' _ _⟩⟩
@@ -404,7 +404,7 @@ instance finitaryExtensive_TopCat : FinitaryExtensive TopCat.{u} := by
       refine' ⟨⟨l, _⟩, ContinuousMap.ext fun a => (hl a).symm, TopCat.isTerminalPUnit.hom_ext _ _,
         fun {l'} h₁ _ =>
           ContinuousMap.ext fun x => hl' x (l' x) (ConcreteCategory.congr_hom h₁ x).symm⟩
-      apply (embedding_inr (α := X') (β := Y')).toInducing.continuous_iff.mpr
+      apply (embedding_inr (X := X') (Y := Y')).toInducing.continuous_iff.mpr
       convert s.fst.2 using 1
       exact (funext hl).symm
   · intro Z f

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
+import Mathlib.CategoryTheory.Adjunction.Limits
 import Mathlib.CategoryTheory.Limits.Shapes.CommSq
-import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
 import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
 import Mathlib.CategoryTheory.Limits.FunctorCategory
 import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

See Zulip thread for the discussion.

@@ -322,9 +322,9 @@ theorem IsUniversalColimit.map_reflective
   haveI : ∀ X, IsIso (Gl.map (adj.unit.app X)) := by
     simp_rw [hadj]
     infer_instance
-  have hα'' : ∀ j, Gl.map (Gr.map $ α'.app j) = adj.counit.app _ ≫ α.app j
+  have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j
   · intro j
-    rw [← cancel_mono (adj.counit.app $ F.obj j)]
+    rw [← cancel_mono (adj.counit.app <| F.obj j)]
     dsimp
     simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
       Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
@@ -333,7 +333,7 @@ theorem IsUniversalColimit.map_reflective
   let c'' : Cocone (F' ⋙ Gr)
   · refine
     { pt := pullback (Gr.map f) (adj.unit.app _)
-      ι := { app := λ j ↦ pullback.lift (Gr.map $ c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
+      ι := { app := λ j ↦ pullback.lift (Gr.map <| c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
              naturality := ?_ } }
     · rw [← Gr.map_comp, ← hc'']
       erw [← adj.unit_naturality]
@@ -354,8 +354,8 @@ theorem IsUniversalColimit.map_reflective
         exact c.w _
   let cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c''
   · refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
-    exact (inv $ adj.counit.app c'.pt)
-    · rw [IsIso.inv_comp_eq, ← adj.counit_naturality_assoc f, ← cancel_mono (adj.counit.app $
+    exact (inv <| adj.counit.app c'.pt)
+    · rw [IsIso.inv_comp_eq, ← adj.counit_naturality_assoc f, ← cancel_mono (adj.counit.app <|
         Gl.obj c.pt), Category.assoc, Category.assoc, adj.left_triangle_components]
       erw [Category.comp_id]
       rfl
@@ -379,7 +379,7 @@ theorem IsUniversalColimit.map_reflective
     rw [this]
     infer_instance
   have ⟨Hc''⟩ := H c'' (whiskerRight α' Gr) pullback.snd ?_ (hα'.whiskerRight Gr) ?_
-  · exact ⟨IsColimit.precomposeHomEquiv β c' $
+  · exact ⟨IsColimit.precomposeHomEquiv β c' <|
       (isColimitOfPreserves Gl Hc'').ofIsoColimit (asIso cf).symm⟩
   · ext j
     dsimp
@@ -417,16 +417,16 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
   intro ⟨hc'⟩ j
   let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
   have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
-  have hα'' : ∀ j, Gl.map (Gr.map $ α'.app j) = adj.counit.app _ ≫ α.app j
+  have hα'' : ∀ j, Gl.map (Gr.map <| α'.app j) = adj.counit.app _ ≫ α.app j
   · intro j
-    rw [← cancel_mono (adj.counit.app $ F.obj j)]
+    rw [← cancel_mono (adj.counit.app <| F.obj j)]
     dsimp
     simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
       Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
   let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
   let hl := (IsColimit.precomposeHomEquiv β c').symm hc'
-  let hr := isColimitOfPreserves Gl (colimit.isColimit $ F' ⋙ Gr)
-  have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map $ α'.app j)
+  let hr := isColimitOfPreserves Gl (colimit.isColimit <| F' ⋙ Gr)
+  have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map <| α'.app j)
   · rw [hα'']
     simp
   rw [this]
@@ -448,9 +448,9 @@ theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
       congr 1
       exact (NatTrans.congr_app h j).symm
   rw [this]
-  have := ((H (colimit.cocone $ F' ⋙ Gr) (whiskerRight α' Gr)
+  have := ((H (colimit.cocone <| F' ⋙ Gr) (whiskerRight α' Gr)
     (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) ?_ (hα'.whiskerRight Gr)).mp
-    ⟨(getColimitCocone $ F' ⋙ Gr).2⟩ j).map Gl
+    ⟨(getColimitCocone <| F' ⋙ Gr).2⟩ j).map Gl
   convert IsPullback.paste_vert _ this
   refine IsPullback.of_vert_isIso ⟨?_⟩
   rw [← IsIso.inv_comp_eq, ← Category.assoc, NatIso.inv_inv_app]
@@ -729,9 +729,9 @@ theorem isPullback_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {X : ι →
         else eqToHom (if_neg h) ≫ initial.to (X j))
       (Cofan.inj c i) (Cofan.inj c j) := by
   refine (hc (Cofan.mk (X i) (f := fun k ↦ if k = i then X i else ⊥_ C)
-    (fun k ↦ if h : k = i then (eqToHom $ if_pos h) else (eqToHom $ if_neg h) ≫ initial.to _))
-    (Discrete.natTrans (fun k ↦ if h : k.1 = i then (eqToHom $ (if_pos h).trans
-      (congr_arg X h.symm)) else (eqToHom $ if_neg h) ≫ initial.to _))
+    (fun k ↦ if h : k = i then (eqToHom <| if_pos h) else (eqToHom <| if_neg h) ≫ initial.to _))
+    (Discrete.natTrans (fun k ↦ if h : k.1 = i then (eqToHom <| (if_pos h).trans
+      (congr_arg X h.symm)) else (eqToHom <| if_neg h) ≫ initial.to _))
     (c.inj i) ?_ (NatTrans.equifibered_of_discrete _)).mp ⟨?_⟩ ⟨j⟩
   · ext ⟨k⟩
     simp only [Discrete.functor_obj, Functor.const_obj_obj, NatTrans.comp_app,

See Zulip thread for the discussion.

@@ -434,7 +434,7 @@ theorem finitaryExtensive_of_reflective
       · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp
     rw [this]
     rintro Z ⟨_|_⟩ f <;> dsimp <;> infer_instance
-  refine ((FinitaryExtensive.vanKampen _ (colimit.isColimit $ pair X Y ⋙ _)).map_reflective
+  refine ((FinitaryExtensive.vanKampen _ (colimit.isColimit <| pair X Y ⋙ _)).map_reflective
     adj).of_iso (IsColimit.uniqueUpToIso ?_ ?_)
   · exact isColimitOfPreserves Gl (colimit.isColimit _)
   · exact (IsColimit.precomposeHomEquiv _ _).symm hc

Subset of #9319

@@ -608,8 +608,8 @@ theorem isUniversalColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
     · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
       simp
   have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
-    (Cofan.mk (pullback c₂.inr i) <| fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
-    (Discrete.natTrans <| fun i ↦ α.app _) pullback.fst ?_ (NatTrans.equifibered_of_discrete _) ?_
+    (Cofan.mk (pullback c₂.inr i) fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
+    (Discrete.natTrans fun i ↦ α.app _) pullback.fst ?_ (NatTrans.equifibered_of_discrete _) ?_
   rotate_left
   · simpa only [Functor.const_obj_obj, pair_obj_right, Discrete.functor_obj, Category.assoc,
       extendCofan_pt, Functor.const_obj_obj, NatTrans.comp_app, extendCofan_ι_app,
@@ -662,8 +662,8 @@ theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
   refine ⟨?_, isUniversalColimit_extendCofan f t₁.isUniversal t₂.isUniversal c α i e hα⟩
   intro ⟨Hc⟩ ⟨j⟩
   have t₂' := (@t₂ (pair (F.obj ⟨0⟩) (∐ fun (j : Fin n) ↦ F.obj ⟨j.succ⟩))
-    (BinaryCofan.mk (P := c.pt) (c.ι.app _) (Sigma.desc <| fun b ↦ c.ι.app _))
-    (mapPair (α.app _) (Sigma.desc <| fun b ↦ α.app _ ≫ c₁.inj _)) i ?_
+    (BinaryCofan.mk (P := c.pt) (c.ι.app _) (Sigma.desc fun b ↦ c.ι.app _))
+    (mapPair (α.app _) (Sigma.desc fun b ↦ α.app _ ≫ c₁.inj _)) i ?_
     (mapPair_equifibered _)).mp ⟨?_⟩
   rotate_left
   · ext ⟨⟨⟩⟩
@@ -709,7 +709,7 @@ theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
   induction' j using Fin.inductionOn with j _
   · exact t₂' ⟨WalkingPair.left⟩
   · have t₁' := (@t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩)) (Cofan.mk _ _) (Discrete.natTrans
-      <| fun i ↦ α.app _) (Sigma.desc (fun j ↦ α.app _ ≫ c₁.inj _)) ?_
+      fun i ↦ α.app _) (Sigma.desc (fun j ↦ α.app _ ≫ c₁.inj _)) ?_
       (NatTrans.equifibered_of_discrete _)).mp ⟨coproductIsCoproduct _⟩ ⟨j⟩
     rotate_left
     · ext ⟨j⟩

Subset of #9319

@@ -553,7 +553,7 @@ lemma FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C]
   let e : ∐ f ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
   { hom := Sigma.desc (fun j ↦ if h : j = i then eqToHom (congr_arg f h) ≫ coprod.inl else
       Sigma.ι (fun j : ({i}ᶜ : Set ι) ↦ f j) ⟨j, h⟩ ≫ coprod.inr)
-    inv := coprod.desc (Sigma.ι f i) (Sigma.desc <| fun j ↦ Sigma.ι f j)
+    inv := coprod.desc (Sigma.ι f i) (Sigma.desc fun j ↦ Sigma.ι f j)
     hom_inv_id := by aesop_cat
     inv_hom_id := by
       ext j
@@ -612,7 +612,7 @@ lemma FinitaryPreExtensive.sigma_desc_iso [FinitaryPreExtensive C] {α : Type} [
     simp [coproductIsCoproduct]
   refine (FinitaryPreExtensive.isUniversal_finiteCoproducts this
     (Cofan.mk _ ((fun _ ↦ pullback.fst) : (a : α) → pullback f (π a) ⟶ _))
-    (Discrete.natTrans <| fun i ↦ pullback.snd) f ?_
+    (Discrete.natTrans fun i ↦ pullback.snd) f ?_
     (NatTrans.equifibered_of_discrete _) ?_).some
   · ext
     simp [pullback.condition]

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

@@ -3,13 +3,12 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
+import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 import Mathlib.CategoryTheory.Limits.Shapes.CommSq
+import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
 import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
-import Mathlib.CategoryTheory.Limits.Shapes.Types
-import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 import Mathlib.CategoryTheory.Limits.FunctorCategory
 import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
-import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 
 #align_import category_theory.extensive from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
 

Only Prop-values fields should be capitalized, not P-valued fields where P is Prop-valued.

Rather than fixing Nonempty := in constructors, I just deleted the line as the instance can almost always be found automatically.

@@ -80,24 +80,24 @@ attribute [instance] PreservesPullbacksOfInclusions.preservesPullbackInl
 and binary coproducts are universal. -/
 class FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
-  [HasPullbacksOfInclusions : HasPullbacksOfInclusions C]
+  [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]
   /-- In a finitary extensive category, all coproducts are van Kampen-/
   universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c
 
 attribute [instance] FinitaryPreExtensive.hasFiniteCoproducts
-attribute [instance] FinitaryPreExtensive.HasPullbacksOfInclusions
+attribute [instance] FinitaryPreExtensive.hasPullbacksOfInclusions
 
 /-- A category is (finitary) extensive if it has finite coproducts,
 and binary coproducts are van Kampen. -/
 class FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
-  [HasPullbacksOfInclusions : HasPullbacksOfInclusions C]
+  [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]
   /-- In a finitary extensive category, all coproducts are van Kampen-/
   van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c
 #align category_theory.finitary_extensive CategoryTheory.FinitaryExtensive
 
 attribute [instance] FinitaryExtensive.hasFiniteCoproducts
-attribute [instance] FinitaryExtensive.HasPullbacksOfInclusions
+attribute [instance] FinitaryExtensive.hasPullbacksOfInclusions
 
 theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
     (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by

I've also got a change to make this required, but I'd like to land this first.

@@ -310,7 +310,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
       (fun h => s.inl <| eX.symm ⟨x, h⟩) fun h => s.inr <| eY.symm ⟨x, (this x).resolve_left h⟩, _⟩
     rw [continuous_iff_continuousAt]
     intro x
-    by_cases f x = Sum.inl PUnit.unit
+    by_cases h : f x = Sum.inl PUnit.unit
     · revert h x
       apply (IsOpen.continuousOn_iff _).mp
       · rw [continuousOn_iff_continuous_restrict]

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

@@ -85,7 +85,7 @@ theorem NatTrans.equifibered_of_discrete {ι : Type*} {F G : Discrete ι ⥤ C}
     (α : F ⟶ G) : NatTrans.Equifibered α := by
   rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
   simp only [Discrete.functor_map_id]
-  refine IsPullback.of_horiz_isIso ⟨by rw [Category.id_comp, Category.comp_id]⟩
+  exact IsPullback.of_horiz_isIso ⟨by rw [Category.id_comp, Category.comp_id]⟩
 
 end NatTrans
 

@@ -77,7 +77,7 @@ class PreservesPullbacksOfInclusions {C : Type*} [Category C] {D : Type*} [Categ
 attribute [instance] PreservesPullbacksOfInclusions.preservesPullbackInl
 
 /-- A category is (finitary) pre-extensive if it has finite coproducts,
-and binary coproducts are van Kampen. -/
+and binary coproducts are universal. -/
 class FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
   [HasPullbacksOfInclusions : HasPullbacksOfInclusions C]

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -64,7 +64,8 @@ theorem NatTrans.Equifibered.comp {F G H : J ⥤ C} {α : F ⟶ G} {β : G ⟶ H
 #align category_theory.nat_trans.equifibered.comp CategoryTheory.NatTrans.Equifibered.comp
 
 theorem NatTrans.Equifibered.whiskerRight {F G : J ⥤ C} {α : F ⟶ G} (hα : Equifibered α)
-    (H : C ⥤ D) [PreservesLimitsOfShape WalkingCospan H] : Equifibered (whiskerRight α H) :=
+    (H : C ⥤ D) [∀ (i j : J) (f : j ⟶ i), PreservesLimit (cospan (α.app i) (G.map f)) H] :
+    Equifibered (whiskerRight α H) :=
   fun _ _ f => (hα f).map H
 #align category_theory.nat_trans.equifibered.whisker_right CategoryTheory.NatTrans.Equifibered.whiskerRight
 
@@ -192,9 +193,20 @@ theorem IsVanKampenColimit.precompose_isIso_iff {F G : J ⥤ C} (α : F ⟶ G) [
     (Cocones.ext (Iso.refl _) (by simp)),
     IsVanKampenColimit.precompose_isIso α⟩
 
+theorem IsUniversalColimit.of_mapCocone (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
+    [PreservesLimitsOfShape WalkingCospan G] [ReflectsColimitsOfShape J G]
+    (hc : IsUniversalColimit (G.mapCocone c)) : IsUniversalColimit c :=
+  fun F' c' α f h hα H ↦
+    ⟨ReflectsColimit.reflects (hc (G.mapCocone c') (whiskerRight α G) (G.map f)
+    (by ext j; simpa using G.congr_map (NatTrans.congr_app h j))
+    (hα.whiskerRight G) (fun j ↦ (H j).map G)).some⟩
+
 theorem IsVanKampenColimit.of_mapCocone (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
-    [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
-    [PreservesColimitsOfShape J G] [ReflectsColimitsOfShape J G]
+    [∀ (i j : J) (X : C) (f : X ⟶ F.obj j) (g : i ⟶ j), PreservesLimit (cospan f (F.map g)) G]
+    [∀ (i : J) (X : C) (f : X ⟶ c.pt), PreservesLimit (cospan f (c.ι.app i)) G]
+    [ReflectsLimitsOfShape WalkingCospan G]
+    [PreservesColimitsOfShape J G]
+    [ReflectsColimitsOfShape J G]
     (H : IsVanKampenColimit (G.mapCocone c)) : IsVanKampenColimit c := by
   intro F' c' α f h hα
   refine' (Iff.trans _ (H (G.mapCocone c') (whiskerRight α G) (G.map f)
@@ -290,14 +302,18 @@ end Functor
 section reflective
 
 theorem IsUniversalColimit.map_reflective
-    [HasPullbacks C] [HasPullbacks D]
     {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
-    [PreservesLimitsOfShape WalkingCospan Gl] {F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
-    (H : IsUniversalColimit c) :
+    {F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
+    (H : IsUniversalColimit c)
+    [∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
+    [∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl] :
     IsUniversalColimit (Gl.mapCocone c) := by
   have := adj.rightAdjointPreservesLimits
   have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits
   intros F' c' α f h hα hc'
+  have : HasPullback (Gl.map (Gr.map f)) (Gl.map (adj.unit.app c.pt)) :=
+    ⟨⟨_, isLimitPullbackConeMapOfIsLimit _ pullback.condition
+      (IsPullback.of_hasPullback _ _).isLimit⟩⟩
   let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
   have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
   have hadj : ∀ X, Gl.map (adj.unit.app X) = inv (adj.counit.app _)
@@ -389,10 +405,11 @@ theorem IsUniversalColimit.map_reflective
         pullback.lift_snd]
 
 theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
-    [HasPullbacks C] [HasPullbacks D]
     {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
-    [PreservesLimitsOfShape WalkingCospan Gl]
-    {F : J ⥤ D} {c : Cocone (F ⋙ Gr)} (H : IsVanKampenColimit c) :
+    {F : J ⥤ D} {c : Cocone (F ⋙ Gr)} (H : IsVanKampenColimit c)
+    [∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
+    [∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl]
+    [∀ X i (f : X ⟶ c.pt), PreservesLimit (cospan f (c.ι.app i)) Gl] :
     IsVanKampenColimit (Gl.mapCocone c) := by
   have := adj.rightAdjointPreservesLimits
   have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -61,26 +61,43 @@ section Extensive
 
 variable {X Y : C}
 
+/-- A category has pullback of inclusions if it has all pullbacks along coproduct injections. -/
+class HasPullbacksOfInclusions (C : Type u) [Category.{v} C] [HasBinaryCoproducts C] : Prop where
+  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
+
+attribute [instance] HasPullbacksOfInclusions.hasPullbackInl
+
+/--
+A functor preserves pullback of inclusions if it preserves all pullbacks along coproduct injections.
+-/
+class PreservesPullbacksOfInclusions {C : Type*} [Category C] {D : Type*} [Category D]
+    (F : C ⥤ D) [HasBinaryCoproducts C] where
+  [preservesPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), PreservesLimit (cospan coprod.inl f) F]
+
+attribute [instance] PreservesPullbacksOfInclusions.preservesPullbackInl
+
 /-- A category is (finitary) pre-extensive if it has finite coproducts,
 and binary coproducts are van Kampen. -/
 class FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
-  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
+  [HasPullbacksOfInclusions : HasPullbacksOfInclusions C]
   /-- In a finitary extensive category, all coproducts are van Kampen-/
   universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c
 
-attribute [instance] FinitaryPreExtensive.hasFiniteCoproducts FinitaryPreExtensive.hasPullbackInl
+attribute [instance] FinitaryPreExtensive.hasFiniteCoproducts
+attribute [instance] FinitaryPreExtensive.HasPullbacksOfInclusions
 
 /-- A category is (finitary) extensive if it has finite coproducts,
 and binary coproducts are van Kampen. -/
 class FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
-  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
+  [HasPullbacksOfInclusions : HasPullbacksOfInclusions C]
   /-- In a finitary extensive category, all coproducts are van Kampen-/
   van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c
 #align category_theory.finitary_extensive CategoryTheory.FinitaryExtensive
 
-attribute [instance] FinitaryExtensive.hasFiniteCoproducts FinitaryExtensive.hasPullbackInl
+attribute [instance] FinitaryExtensive.hasFiniteCoproducts
+attribute [instance] FinitaryExtensive.HasPullbacksOfInclusions
 
 theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
     (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by
@@ -95,11 +112,14 @@ theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingP
   exact FinitaryExtensive.van_kampen' c hc
 #align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.vanKampen
 
-namespace FinitaryPreExtensive
+namespace HasPullbacksOfInclusions
+
+instance (priority := 100) [HasBinaryCoproducts C] [HasPullbacks C] :
+    HasPullbacksOfInclusions C := ⟨⟩
 
-variable [FinitaryPreExtensive C] {X Y Z : C} (f : Z ⟶ X ⨿ Y)
+variable [HasBinaryCoproducts C] [HasPullbacksOfInclusions C] {X Y Z : C} (f : Z ⟶ X ⨿ Y)
 
-instance hasPullbackInl' :
+instance preservesPullbackInl' :
     HasPullback f coprod.inl :=
   hasPullback_symmetry _ _
 
@@ -116,7 +136,35 @@ instance hasPullbackInr :
     HasPullback coprod.inr f :=
   hasPullback_symmetry _ _
 
-end FinitaryPreExtensive
+end HasPullbacksOfInclusions
+
+namespace PreservesPullbacksOfInclusions
+
+variable {D : Type*} [Category D] [HasBinaryCoproducts C] (F : C ⥤ D)
+
+noncomputable
+instance (priority := 100) [PreservesLimitsOfShape WalkingCospan F] :
+    PreservesPullbacksOfInclusions F := ⟨⟩
+
+variable [PreservesPullbacksOfInclusions F] {X Y Z : C} (f : Z ⟶ X ⨿ Y)
+
+noncomputable
+instance preservesPullbackInl' :
+    PreservesLimit (cospan f coprod.inl) F :=
+  preservesPullbackSymmetry _ _ _
+
+noncomputable
+instance preservesPullbackInr' :
+    PreservesLimit (cospan f coprod.inr) F := by
+  apply preservesLimitOfIsoDiagram (K₁ := cospan (f ≫ (coprod.braiding X Y).hom) coprod.inl)
+  apply cospanExt (Iso.refl _) (Iso.refl _) (coprod.braiding X Y).symm <;> simp
+
+noncomputable
+instance preservesPullbackInr :
+    PreservesLimit (cospan coprod.inr f) F :=
+  preservesPullbackSymmetry _ _ _
+
+end PreservesPullbacksOfInclusions
 
 instance (priority := 100) FinitaryExtensive.toFinitaryPreExtensive [FinitaryExtensive C] :
     FinitaryPreExtensive C :=
@@ -153,7 +201,7 @@ instance (priority := 100) hasStrictInitialObjects_of_finitaryPreExtensive
 #align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryPreExtensive
 
 theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]
-    [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
+    [HasPullbacksOfInclusions C]
     (T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :
     FinitaryExtensive C ↔ IsVanKampenColimit c₀ := by
   refine' ⟨fun H => H.van_kampen' c₀ hc₀, fun H => _⟩
@@ -366,16 +414,26 @@ end TopCat
 
 section Functor
 
-theorem finitaryExtensive_of_reflective [HasFiniteCoproducts D] [HasPullbacks D]
-    [FinitaryExtensive C] [HasPullbacks C]
+theorem finitaryExtensive_of_reflective
+    [HasFiniteCoproducts D] [HasPullbacksOfInclusions D] [FinitaryExtensive C]
     {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
-    [PreservesLimitsOfShape WalkingCospan Gl] :
+    [∀ X Y (f : X ⟶ Gl.obj Y), HasPullback (Gr.map f) (adj.unit.app Y)]
+    [∀ X Y (f : X ⟶ Gl.obj Y), PreservesLimit (cospan (Gr.map f) (adj.unit.app Y)) Gl]
+    [PreservesPullbacksOfInclusions Gl] :
     FinitaryExtensive D := by
   have : PreservesColimitsOfSize Gl := adj.leftAdjointPreservesColimits
   constructor
   intros X Y c hc
   apply (IsVanKampenColimit.precompose_isIso_iff
     (isoWhiskerLeft _ (asIso adj.counit) ≪≫ Functor.rightUnitor _).hom).mp
+  have : ∀ (Z : C) (i : Discrete WalkingPair) (f : Z ⟶ (colimit.cocone (pair X Y ⋙ Gr)).pt),
+        PreservesLimit (cospan f ((colimit.cocone (pair X Y ⋙ Gr)).ι.app i)) Gl := by
+    have : pair X Y ⋙ Gr = pair (Gr.obj X) (Gr.obj Y) := by
+      apply Functor.hext
+      · rintro ⟨⟨⟩⟩ <;> rfl
+      · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp
+    rw [this]
+    rintro Z ⟨_|_⟩ f <;> dsimp <;> infer_instance
   refine ((FinitaryExtensive.vanKampen _ (colimit.isColimit $ pair X Y ⋙ _)).map_reflective
     adj).of_iso (IsColimit.uniqueUpToIso ?_ ?_)
   · exact isColimitOfPreserves Gl (colimit.isColimit _)
@@ -387,12 +445,32 @@ instance finitaryExtensive_functor [HasPullbacks C] [FinitaryExtensive C] :
   ⟨fun c hc => isVanKampenColimit_of_evaluation _ c fun _ =>
     FinitaryExtensive.vanKampen _ <| PreservesColimit.preserves hc⟩
 
+noncomputable
+instance {C} [Category C] {D} [Category D] (F : C ⥤ D)
+    {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [IsIso f] : PreservesLimit (cospan f g) F :=
+  have := hasPullback_of_left_iso f g
+  preservesLimitOfPreservesLimitCone (IsPullback.of_hasPullback f g).isLimit
+    ((isLimitMapConePullbackConeEquiv _ pullback.condition).symm
+      (IsPullback.of_vert_isIso ⟨by simp only [← F.map_comp, pullback.condition]⟩).isLimit)
+
+noncomputable
+instance {C} [Category C] {D} [Category D] (F : C ⥤ D)
+    {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [IsIso g] : PreservesLimit (cospan f g) F :=
+  preservesPullbackSymmetry _ _ _
+
 theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExtensive D]
-    [HasFiniteCoproducts C] [∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
-    [PreservesLimitsOfShape WalkingCospan F]
+    [HasFiniteCoproducts C] [HasPullbacksOfInclusions C]
+    [PreservesPullbacksOfInclusions F]
     [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
-    [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C :=
-  ⟨fun _ hc => (FinitaryExtensive.vanKampen _ (isColimitOfPreserves F hc)).of_mapCocone F⟩
+    [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C := by
+  constructor
+  intros X Y c hc
+  refine IsVanKampenColimit.of_iso ?_ (hc.uniqueUpToIso (coprodIsCoprod X Y)).symm
+  have (i : Discrete WalkingPair) (Z : C) (f : Z ⟶ X ⨿ Y) :
+    PreservesLimit (cospan f ((BinaryCofan.mk coprod.inl coprod.inr).ι.app i)) F := by
+    rcases i with ⟨_|_⟩ <;> dsimp <;> infer_instance
+  refine (FinitaryExtensive.vanKampen _
+    (isColimitOfPreserves F (coprodIsCoprod X Y))).of_mapCocone F
 #align category_theory.finitary_extensive_of_preserves_and_reflects CategoryTheory.finitaryExtensive_of_preserves_and_reflects
 
 theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -133,6 +133,18 @@ theorem IsInitial.isVanKampenColimit [HasStrictInitialObjects C] {X : C} (h : Is
 
 section Functor
 
+theorem IsUniversalColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (hc : IsUniversalColimit c)
+    (e : c ≅ c') : IsUniversalColimit c' := by
+  intro F' c'' α f h hα H
+  have : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι := by
+    ext j
+    exact e.inv.2 j
+  apply hc c'' α (f ≫ e.inv.1) (by rw [Functor.map_comp, ← reassoc_of% h, this]) hα
+  intro j
+  rw [← Category.comp_id (α.app j)]
+  have : IsIso e.inv.hom := Functor.map_isIso (Cocones.forget _) e.inv
+  exact (H j).paste_vert (IsPullback.of_vert_isIso ⟨by simp⟩)
+
 theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKampenColimit c)
     (e : c ≅ c') : IsVanKampenColimit c' := by
   intro F' c'' α f h hα
@@ -162,6 +174,18 @@ theorem IsVanKampenColimit.precompose_isIso {F G : J ⥤ C} (α : F ⟶ G) [IsIs
   rw [← IsPullback.paste_vert_iff this _, Category.comp_id]
   exact (congr_app e j).symm
 
+theorem IsUniversalColimit.precompose_isIso {F G : J ⥤ C} (α : F ⟶ G) [IsIso α]
+    {c : Cocone G} (hc : IsUniversalColimit c) :
+    IsUniversalColimit ((Cocones.precompose α).obj c) := by
+  intros F' c' α' f e hα H
+  apply (hc c' (α' ≫ α) f ((Category.assoc _ _ _).trans e)
+    (hα.comp (NatTrans.equifibered_of_isIso _)))
+  intro j
+  simp only [Functor.const_obj_obj, NatTrans.comp_app,
+    Cocones.precompose_obj_pt, Cocones.precompose_obj_ι]
+  rw [← Category.comp_id f]
+  exact (H j).paste_vert (IsPullback.of_vert_isIso ⟨Category.comp_id _⟩)
+
 theorem IsVanKampenColimit.precompose_isIso_iff {F G : J ⥤ C} (α : F ⟶ G) [IsIso α]
     {c : Cocone G} : IsVanKampenColimit ((Cocones.precompose α).obj c) ↔ IsVanKampenColimit c :=
   ⟨fun hc ↦ IsVanKampenColimit.of_iso (IsVanKampenColimit.precompose_isIso (inv α) hc)
@@ -189,6 +213,28 @@ theorem IsVanKampenColimit.mapCocone_iff (G : C ⥤ D) {F : J ⥤ C} {c : Cocone
     refine hc.of_iso (Cocones.ext (G.asEquivalence.unitIso.app c.pt) ?_)
     simp [Functor.asEquivalence]⟩
 
+theorem IsUniversalColimit.whiskerEquivalence {K : Type*} [Category K] (e : J ≌ K)
+    {F : K ⥤ C} {c : Cocone F} (hc : IsUniversalColimit c) :
+    IsUniversalColimit (c.whisker e.functor) := by
+  intro F' c' α f e' hα H
+  convert hc (c'.whisker e.inverse) (whiskerLeft e.inverse α ≫ (e.invFunIdAssoc F).hom) f ?_
+    ((hα.whiskerLeft _).comp (NatTrans.equifibered_of_isIso _)) ?_ using 1
+  · exact (IsColimit.whiskerEquivalenceEquiv e.symm).nonempty_congr
+  · convert congr_arg (whiskerLeft e.inverse) e'
+    ext
+    simp
+  · intro k
+    rw [← Category.comp_id f]
+    refine (H (e.inverse.obj k)).paste_vert ?_
+    have : IsIso (𝟙 (Cocone.whisker e.functor c).pt) := inferInstance
+    exact IsPullback.of_vert_isIso ⟨by simp⟩
+
+theorem IsUniversalColimit.whiskerEquivalence_iff {K : Type*} [Category K] (e : J ≌ K)
+    {F : K ⥤ C} {c : Cocone F} :
+    IsUniversalColimit (c.whisker e.functor) ↔ IsUniversalColimit c :=
+  ⟨fun hc ↦ ((hc.whiskerEquivalence e.symm).precompose_isIso (e.invFunIdAssoc F).inv).of_iso
+      (Cocones.ext (Iso.refl _) (by simp)), IsUniversalColimit.whiskerEquivalence e⟩
+
 theorem IsVanKampenColimit.whiskerEquivalence {K : Type*} [Category K] (e : J ≌ K)
     {F : K ⥤ C} {c : Cocone F} (hc : IsVanKampenColimit c) :
     IsVanKampenColimit (c.whisker e.functor) := by
@@ -659,6 +705,62 @@ theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
       BinaryCofan.ι_app_right, BinaryCofan.mk_inr, colimit.ι_desc,
       Discrete.natTrans_app] using t₁'.paste_horiz (t₂' ⟨WalkingPair.right⟩)
 
+theorem isPullback_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {X : ι → C}
+    {c : Cofan X} (hc : IsVanKampenColimit c) (i j : ι) [DecidableEq ι] :
+    IsPullback (P := (if j = i then X i else ⊥_ C))
+      (if h : j = i then eqToHom (if_pos h) else eqToHom (if_neg h) ≫ initial.to (X i))
+      (if h : j = i then eqToHom ((if_pos h).trans (congr_arg X h.symm))
+        else eqToHom (if_neg h) ≫ initial.to (X j))
+      (Cofan.inj c i) (Cofan.inj c j) := by
+  refine (hc (Cofan.mk (X i) (f := fun k ↦ if k = i then X i else ⊥_ C)
+    (fun k ↦ if h : k = i then (eqToHom $ if_pos h) else (eqToHom $ if_neg h) ≫ initial.to _))
+    (Discrete.natTrans (fun k ↦ if h : k.1 = i then (eqToHom $ (if_pos h).trans
+      (congr_arg X h.symm)) else (eqToHom $ if_neg h) ≫ initial.to _))
+    (c.inj i) ?_ (NatTrans.equifibered_of_discrete _)).mp ⟨?_⟩ ⟨j⟩
+  · ext ⟨k⟩
+    simp only [Discrete.functor_obj, Functor.const_obj_obj, NatTrans.comp_app,
+      Discrete.natTrans_app, Cofan.mk_pt, Cofan.mk_ι_app, Functor.const_map_app]
+    split
+    · subst ‹k = i›; rfl
+    · simp
+  · refine mkCofanColimit _ (fun t ↦ (eqToHom (if_pos rfl).symm) ≫ t.inj i) ?_ ?_
+    · intro t j
+      simp only [Cofan.mk_pt, cofan_mk_inj]
+      split
+      · subst ‹j = i›; simp
+      · rw [Category.assoc, ← IsIso.eq_inv_comp]
+        exact initialIsInitial.hom_ext _ _
+    · intro t m hm
+      simp [← hm i]
+
+theorem isPullback_initial_to_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {F : Discrete ι ⥤ C}
+    {c : Cocone F} (hc : IsVanKampenColimit c) (i j : Discrete ι) (hi : i ≠ j) :
+    IsPullback (initial.to _) (initial.to _) (c.ι.app i) (c.ι.app j) := by
+  classical
+  let f : ι → C := F.obj ∘ Discrete.mk
+  have : F = Discrete.functor f :=
+    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+  clear_value f
+  subst this
+  convert isPullback_of_cofan_isVanKampen hc i.as j.as
+  exact (if_neg (mt (Discrete.ext _ _) hi.symm)).symm
+
+theorem mono_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {F : Discrete ι ⥤ C}
+    {c : Cocone F} (hc : IsVanKampenColimit c) (i : Discrete ι) : Mono (c.ι.app i) := by
+  classical
+  let f : ι → C := F.obj ∘ Discrete.mk
+  have : F = Discrete.functor f :=
+    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+  clear_value f
+  subst this
+  refine' PullbackCone.mono_of_isLimitMkIdId _ (IsPullback.isLimit _)
+  nth_rw 1 [← Category.id_comp (c.ι.app i)]
+  convert IsPullback.paste_vert _ (isPullback_of_cofan_isVanKampen hc i.as i.as)
+  swap
+  · exact (eqToHom (if_pos rfl).symm)
+  · simp
+  · exact IsPullback.of_vert_isIso ⟨by simp⟩
+
 end FiniteCoproducts
 
 end CategoryTheory

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -8,6 +8,7 @@ import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
 import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 import Mathlib.CategoryTheory.Limits.FunctorCategory
+import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
 import Mathlib.CategoryTheory.Limits.VanKampen
 
 #align_import category_theory.extensive from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
@@ -31,7 +32,9 @@ import Mathlib.CategoryTheory.Limits.VanKampen
 - `CategoryTheory.FinitaryExtensive_TopCat`:
   The category `Top` is extensive.
 - `CategoryTheory.FinitaryExtensive_functor`: The category `C ⥤ D` is extensive if `D`
-  has all pullbacks and is extensive
+  has all pullbacks and is extensive.
+- `CategoryTheory.FinitaryExtensive.isVanKampen_finiteCoproducts`: Finite coproducts in a
+  finitary extensive category are van Kampen.
 ## TODO
 
 Show that the following are finitary extensive:
@@ -58,17 +61,26 @@ section Extensive
 
 variable {X Y : C}
 
-/-- A category is (finitary) extensive if it has finite coproducts,
-and binary coproducts are van Kampen.
+/-- A category is (finitary) pre-extensive if it has finite coproducts,
+and binary coproducts are van Kampen. -/
+class FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where
+  [hasFiniteCoproducts : HasFiniteCoproducts C]
+  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
+  /-- In a finitary extensive category, all coproducts are van Kampen-/
+  universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c
 
-TODO: Show that this is iff all finite coproducts are van Kampen. -/
+attribute [instance] FinitaryPreExtensive.hasFiniteCoproducts FinitaryPreExtensive.hasPullbackInl
+
+/-- A category is (finitary) extensive if it has finite coproducts,
+and binary coproducts are van Kampen. -/
 class FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
+  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
   /-- In a finitary extensive category, all coproducts are van Kampen-/
   van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c
 #align category_theory.finitary_extensive CategoryTheory.FinitaryExtensive
 
-attribute [instance] FinitaryExtensive.hasFiniteCoproducts
+attribute [instance] FinitaryExtensive.hasFiniteCoproducts FinitaryExtensive.hasPullbackInl
 
 theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
     (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by
@@ -83,6 +95,33 @@ theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingP
   exact FinitaryExtensive.van_kampen' c hc
 #align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.vanKampen
 
+namespace FinitaryPreExtensive
+
+variable [FinitaryPreExtensive C] {X Y Z : C} (f : Z ⟶ X ⨿ Y)
+
+instance hasPullbackInl' :
+    HasPullback f coprod.inl :=
+  hasPullback_symmetry _ _
+
+instance hasPullbackInr' :
+    HasPullback f coprod.inr := by
+  have : IsPullback (𝟙 _) (f ≫ (coprod.braiding X Y).hom) f (coprod.braiding Y X).hom :=
+    IsPullback.of_horiz_isIso ⟨by simp⟩
+  have := (IsPullback.of_hasPullback (f ≫ (coprod.braiding X Y).hom) coprod.inl).paste_horiz this
+  simp only [coprod.braiding_hom, Category.comp_id, colimit.ι_desc, BinaryCofan.mk_pt,
+    BinaryCofan.ι_app_left, BinaryCofan.mk_inl] at this
+  exact ⟨⟨⟨_, this.isLimit⟩⟩⟩
+
+instance hasPullbackInr :
+    HasPullback coprod.inr f :=
+  hasPullback_symmetry _ _
+
+end FinitaryPreExtensive
+
+instance (priority := 100) FinitaryExtensive.toFinitaryPreExtensive [FinitaryExtensive C] :
+    FinitaryPreExtensive C :=
+  ⟨fun c hc ↦ (FinitaryExtensive.van_kampen' c hc).isUniversal⟩
+
 theorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
     (hc : IsColimit c) : Mono c.inr :=
   BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
@@ -105,18 +144,19 @@ theorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C
   BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
 #align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan
 
-instance (priority := 100) hasStrictInitialObjects_of_finitaryExtensive [FinitaryExtensive C] :
-    HasStrictInitialObjects C :=
-  hasStrictInitial_of_isUniversal (FinitaryExtensive.vanKampen _
+instance (priority := 100) hasStrictInitialObjects_of_finitaryPreExtensive
+    [FinitaryPreExtensive C] : HasStrictInitialObjects C :=
+  hasStrictInitial_of_isUniversal (FinitaryPreExtensive.universal' _
     ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr (by
       dsimp
-      infer_instance)).some).isUniversal
-#align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive
+      infer_instance)).some)
+#align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryPreExtensive
 
 theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]
+    [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
     (T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :
     FinitaryExtensive C ↔ IsVanKampenColimit c₀ := by
-  refine' ⟨fun H => H.2 c₀ hc₀, fun H => _⟩
+  refine' ⟨fun H => H.van_kampen' c₀ hc₀, fun H => _⟩
   constructor
   simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢
   intro X Y c hc X' Y' c' αX αY f hX hY
@@ -348,7 +388,8 @@ instance finitaryExtensive_functor [HasPullbacks C] [FinitaryExtensive C] :
     FinitaryExtensive.vanKampen _ <| PreservesColimit.preserves hc⟩
 
 theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExtensive D]
-    [HasFiniteCoproducts C] [PreservesLimitsOfShape WalkingCospan F]
+    [HasFiniteCoproducts C] [∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
+    [PreservesLimitsOfShape WalkingCospan F]
     [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
     [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C :=
   ⟨fun _ hc => (FinitaryExtensive.vanKampen _ (isColimitOfPreserves F hc)).of_mapCocone F⟩
@@ -366,6 +407,141 @@ theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [F
 
 end Functor
 
+section FiniteCoproducts
+
+theorem FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin [FinitaryPreExtensive C] {n : ℕ}
+    {F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsUniversalColimit c := by
+  let f : Fin n → C := F.obj ∘ Discrete.mk
+  have : F = Discrete.functor f :=
+    Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+  clear_value f
+  subst this
+  induction' n with n IH
+  · exact (isVanKampenColimit_of_isEmpty _ hc).isUniversal
+  · apply IsUniversalColimit.of_iso _
+      ((extendCofanIsColimit f (coproductIsCoproduct _) (coprodIsCoprod _ _)).uniqueUpToIso hc)
+    apply @isUniversalColimit_extendCofan _ _ _ _ _ _ _ _ ?_
+    · apply IH
+      exact coproductIsCoproduct _
+    · apply FinitaryPreExtensive.universal'
+      exact coprodIsCoprod _ _
+    · dsimp
+      infer_instance
+
+theorem FinitaryPreExtensive.isUniversal_finiteCoproducts [FinitaryPreExtensive C] {ι : Type*}
+    [Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsUniversalColimit c := by
+  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin ι
+  apply (IsUniversalColimit.whiskerEquivalence_iff (Discrete.equivalence e).symm).mp
+  apply FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin
+  exact (IsColimit.whiskerEquivalenceEquiv (Discrete.equivalence e).symm) hc
+
+theorem FinitaryExtensive.isVanKampen_finiteCoproducts_Fin [FinitaryExtensive C] {n : ℕ}
+    {F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsVanKampenColimit c := by
+  let f : Fin n → C := F.obj ∘ Discrete.mk
+  have : F = Discrete.functor f :=
+    Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+  clear_value f
+  subst this
+  induction' n with n IH
+  · exact isVanKampenColimit_of_isEmpty _ hc
+  · apply IsVanKampenColimit.of_iso _
+      ((extendCofanIsColimit f (coproductIsCoproduct _) (coprodIsCoprod _ _)).uniqueUpToIso hc)
+    apply @isVanKampenColimit_extendCofan _ _ _ _ _ _ _ _ ?_
+    · apply IH
+      exact coproductIsCoproduct _
+    · apply FinitaryExtensive.van_kampen'
+      exact coprodIsCoprod _ _
+    · dsimp
+      infer_instance
+
+theorem FinitaryExtensive.isVanKampen_finiteCoproducts [FinitaryExtensive C] {ι : Type*}
+    [Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsVanKampenColimit c := by
+  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin ι
+  apply (IsVanKampenColimit.whiskerEquivalence_iff (Discrete.equivalence e).symm).mp
+  apply FinitaryExtensive.isVanKampen_finiteCoproducts_Fin
+  exact (IsColimit.whiskerEquivalenceEquiv (Discrete.equivalence e).symm) hc
+
+lemma FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C] {ι : Type*}
+    [Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) (i : Discrete ι) {X : C}
+    (g : X ⟶ _) : HasPullback g (c.ι.app i) := by
+  classical
+  let f : ι → C := F.obj ∘ Discrete.mk
+  have : F = Discrete.functor f :=
+    Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp)
+  clear_value f
+  subst this
+  change Cofan f at c
+  obtain ⟨i⟩ := i
+  let e : ∐ f ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
+  { hom := Sigma.desc (fun j ↦ if h : j = i then eqToHom (congr_arg f h) ≫ coprod.inl else
+      Sigma.ι (fun j : ({i}ᶜ : Set ι) ↦ f j) ⟨j, h⟩ ≫ coprod.inr)
+    inv := coprod.desc (Sigma.ι f i) (Sigma.desc <| fun j ↦ Sigma.ι f j)
+    hom_inv_id := by aesop_cat
+    inv_hom_id := by
+      ext j
+      · simp
+      · simp only [coprod.desc_comp, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
+          eqToHom_refl, Category.id_comp, dite_true, BinaryCofan.mk_pt, BinaryCofan.ι_app_right,
+          BinaryCofan.mk_inr, colimit.ι_desc_assoc, Discrete.functor_obj, Category.comp_id]
+        exact dif_neg j.prop }
+  let e' : c.pt ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
+    hc.coconePointUniqueUpToIso (getColimitCocone _).2 ≪≫ e
+  have : coprod.inl ≫ e'.inv = c.ι.app ⟨i⟩
+  · simp only [Iso.trans_inv, coprod.desc_comp, colimit.ι_desc, BinaryCofan.mk_pt,
+      BinaryCofan.ι_app_left, BinaryCofan.mk_inl]
+    exact colimit.comp_coconePointUniqueUpToIso_inv _ _
+  clear_value e'
+  rw [← this]
+  have : IsPullback (𝟙 _) (g ≫ e'.hom) g e'.inv := IsPullback.of_horiz_isIso ⟨by simp⟩
+  exact ⟨⟨⟨_, ((IsPullback.of_hasPullback (g ≫ e'.hom) coprod.inl).paste_horiz this).isLimit⟩⟩⟩
+
+lemma FinitaryExtensive.mono_ι [FinitaryExtensive C] {ι : Type*} [Finite ι] {F : Discrete ι ⥤ C}
+    {c : Cocone F} (hc : IsColimit c) (i : Discrete ι) :
+    Mono (c.ι.app i) :=
+  mono_of_cofan_isVanKampen (isVanKampen_finiteCoproducts hc) _
+
+instance [FinitaryExtensive C] {ι : Type*} [Finite ι] (X : ι → C) (i : ι) :
+    Mono (Sigma.ι X i) :=
+  FinitaryExtensive.mono_ι (coproductIsCoproduct _) ⟨i⟩
+
+lemma FinitaryExtensive.isPullback_initial_to [FinitaryExtensive C]
+    {ι : Type*} [Finite ι] {F : Discrete ι ⥤ C}
+    {c : Cocone F} (hc : IsColimit c) (i j : Discrete ι) (e : i ≠ j) :
+    IsPullback (initial.to _) (initial.to _) (c.ι.app i) (c.ι.app j) :=
+  isPullback_initial_to_of_cofan_isVanKampen (isVanKampen_finiteCoproducts hc) i j e
+
+lemma FinitaryExtensive.isPullback_initial_to_sigma_ι [FinitaryExtensive C] {ι : Type*} [Finite ι]
+    (X : ι → C) (i j : ι) (e : i ≠ j) :
+    IsPullback (initial.to _) (initial.to _) (Sigma.ι X i) (Sigma.ι X j) :=
+  FinitaryExtensive.isPullback_initial_to (coproductIsCoproduct _) ⟨i⟩ ⟨j⟩
+    (ne_of_apply_ne Discrete.as e)
+
+instance FinitaryPreExtensive.hasPullbacks_of_inclusions [FinitaryPreExtensive C] {X Z : C}
+    {α : Type*} (f : X ⟶ Z) {Y : (a : α) → C} (i : (a : α) → Y a ⟶ Z) [Finite α]
+    [hi : IsIso (Sigma.desc i)] (a : α) : HasPullback f (i a) := by
+  apply FinitaryPreExtensive.hasPullbacks_of_is_coproduct (c := Cofan.mk Z i)
+  exact @IsColimit.ofPointIso (t := Cofan.mk Z i) (P := _) hi
+
+lemma FinitaryPreExtensive.sigma_desc_iso [FinitaryPreExtensive C] {α : Type} [Finite α] {X : C}
+    {Z : α → C} (π : (a : α) → Z a ⟶ X) {Y : C} (f : Y ⟶ X) (hπ : IsIso (Sigma.desc π)) :
+    IsIso (Sigma.desc ((fun _ ↦ pullback.fst) : (a : α) → pullback f (π a) ⟶ _)) := by
+  suffices IsColimit (Cofan.mk _ ((fun _ ↦ pullback.fst) : (a : α) → pullback f (π a) ⟶ _)) by
+    change IsIso (this.coconePointUniqueUpToIso (getColimitCocone _).2).inv
+    infer_instance
+  let : IsColimit (Cofan.mk X π)
+  · refine @IsColimit.ofPointIso (t := Cofan.mk X π) (P := coproductIsCoproduct Z) ?_
+    convert hπ
+    simp [coproductIsCoproduct]
+  refine (FinitaryPreExtensive.isUniversal_finiteCoproducts this
+    (Cofan.mk _ ((fun _ ↦ pullback.fst) : (a : α) → pullback f (π a) ⟶ _))
+    (Discrete.natTrans <| fun i ↦ pullback.snd) f ?_
+    (NatTrans.equifibered_of_discrete _) ?_).some
+  · ext
+    simp [pullback.condition]
+  · exact fun j ↦ IsPullback.of_hasPullback f (π j.as)
+
+end FiniteCoproducts
+
 end Extensive
 
 end CategoryTheory

Port of https://github.com/leanprover-community/mathlib/pull/17637

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -9,6 +9,7 @@ import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 import Mathlib.CategoryTheory.Limits.FunctorCategory
 import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
+import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 
 #align_import category_theory.extensive from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
 
@@ -25,7 +26,7 @@ import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProduct
 
 ## References
 - https://ncatlab.org/nlab/show/van+Kampen+colimit
-- *Adhesive categories*, Stephen Lack and Paweł Sobociński
+- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]
 
 -/
 
@@ -240,6 +241,164 @@ theorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J
 
 end Functor
 
+section reflective
+
+theorem IsUniversalColimit.map_reflective
+    [HasPullbacks C] [HasPullbacks D]
+    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
+    [PreservesLimitsOfShape WalkingCospan Gl] {F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
+    (H : IsUniversalColimit c) :
+    IsUniversalColimit (Gl.mapCocone c) := by
+  have := adj.rightAdjointPreservesLimits
+  have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits
+  intros F' c' α f h hα hc'
+  let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
+  have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
+  have hadj : ∀ X, Gl.map (adj.unit.app X) = inv (adj.counit.app _)
+  · intro X
+    apply IsIso.eq_inv_of_inv_hom_id
+    exact adj.left_triangle_components
+  haveI : ∀ X, IsIso (Gl.map (adj.unit.app X)) := by
+    simp_rw [hadj]
+    infer_instance
+  have hα'' : ∀ j, Gl.map (Gr.map $ α'.app j) = adj.counit.app _ ≫ α.app j
+  · intro j
+    rw [← cancel_mono (adj.counit.app $ F.obj j)]
+    dsimp
+    simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
+      Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
+  have hc'' : ∀ j, α.app j ≫ Gl.map (c.ι.app j) = c'.ι.app j ≫ f := NatTrans.congr_app h
+  let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
+  let c'' : Cocone (F' ⋙ Gr)
+  · refine
+    { pt := pullback (Gr.map f) (adj.unit.app _)
+      ι := { app := λ j ↦ pullback.lift (Gr.map $ c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
+             naturality := ?_ } }
+    · rw [← Gr.map_comp, ← hc'']
+      erw [← adj.unit_naturality]
+      rw [Gl.map_comp, hα'']
+      dsimp
+      simp only [Category.assoc, Functor.map_comp, adj.right_triangle_components_assoc]
+    · intros i j g
+      dsimp
+      ext
+      all_goals simp only [Category.comp_id, Category.id_comp, Category.assoc,
+        ← Functor.map_comp, pullback.lift_fst, pullback.lift_snd, ← Functor.map_comp_assoc]
+      · congr 1
+        exact c'.w _
+      · rw [α.naturality_assoc]
+        dsimp
+        rw [adj.counit_naturality, ← Category.assoc, Gr.map_comp_assoc]
+        congr 1
+        exact c.w _
+  let cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c''
+  · refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
+    exact (inv $ adj.counit.app c'.pt)
+    · rw [IsIso.inv_comp_eq, ← adj.counit_naturality_assoc f, ← cancel_mono (adj.counit.app $
+        Gl.obj c.pt), Category.assoc, Category.assoc, adj.left_triangle_components]
+      erw [Category.comp_id]
+      rfl
+    · intro j
+      rw [← Category.assoc, Iso.comp_inv_eq]
+      ext
+      all_goals simp only [PreservesPullback.iso_hom_fst, PreservesPullback.iso_hom_snd,
+          pullback.lift_fst, pullback.lift_snd, Category.assoc,
+          Functor.mapCocone_ι_app, ← Gl.map_comp]
+      · rw [IsIso.comp_inv_eq, adj.counit_naturality]
+        dsimp
+        rw [Category.comp_id]
+      · rw [Gl.map_comp, hα'', Category.assoc, hc'']
+        dsimp
+        rw [Category.comp_id, Category.assoc]
+  have : cf.hom ≫ (PreservesPullback.iso _ _ _).hom ≫ pullback.fst ≫ adj.counit.app _ = 𝟙 _
+  · simp only [IsIso.inv_hom_id, Iso.inv_hom_id_assoc, Category.assoc, pullback.lift_fst_assoc]
+  have : IsIso cf
+  · apply @Cocones.cocone_iso_of_hom_iso (i := ?_)
+    rw [← IsIso.eq_comp_inv] at this
+    rw [this]
+    infer_instance
+  have ⟨Hc''⟩ := H c'' (whiskerRight α' Gr) pullback.snd ?_ (hα'.whiskerRight Gr) ?_
+  · exact ⟨IsColimit.precomposeHomEquiv β c' $
+      (isColimitOfPreserves Gl Hc'').ofIsoColimit (asIso cf).symm⟩
+  · ext j
+    dsimp
+    simp only [Category.comp_id, Category.id_comp, Category.assoc,
+      Functor.map_comp, pullback.lift_snd]
+  · intro j
+    apply IsPullback.of_right _ _ (IsPullback.of_hasPullback _ _)
+    · dsimp
+      simp only [Category.comp_id, Category.id_comp, Category.assoc, Functor.map_comp,
+        pullback.lift_fst]
+      rw [← Category.comp_id (Gr.map f)]
+      refine ((hc' j).map Gr).paste_vert (IsPullback.of_vert_isIso ⟨?_⟩)
+      rw [← adj.unit_naturality, Category.comp_id, ← Category.assoc,
+        ← Category.id_comp (Gr.map ((Gl.mapCocone c).ι.app j))]
+      congr 1
+      rw [← cancel_mono (Gr.map (adj.counit.app (F.obj j)))]
+      dsimp
+      simp only [Category.comp_id, Adjunction.right_triangle_components, Category.id_comp,
+        Category.assoc]
+    · dsimp
+      simp only [Category.comp_id, Category.id_comp, Category.assoc, Functor.map_comp,
+        pullback.lift_snd]
+
+theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
+    [HasPullbacks C] [HasPullbacks D]
+    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
+    [PreservesLimitsOfShape WalkingCospan Gl]
+    {F : J ⥤ D} {c : Cocone (F ⋙ Gr)} (H : IsVanKampenColimit c) :
+    IsVanKampenColimit (Gl.mapCocone c) := by
+  have := adj.rightAdjointPreservesLimits
+  have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits
+  intro F' c' α f h hα
+  refine ⟨?_, H.isUniversal.map_reflective adj c' α f h hα⟩
+  intro ⟨hc'⟩ j
+  let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
+  have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
+  have hα'' : ∀ j, Gl.map (Gr.map $ α'.app j) = adj.counit.app _ ≫ α.app j
+  · intro j
+    rw [← cancel_mono (adj.counit.app $ F.obj j)]
+    dsimp
+    simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
+      Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
+  let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
+  let hl := (IsColimit.precomposeHomEquiv β c').symm hc'
+  let hr := isColimitOfPreserves Gl (colimit.isColimit $ F' ⋙ Gr)
+  have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map $ α'.app j)
+  · rw [hα'']
+    simp
+  rw [this]
+  have : f = (hl.coconePointUniqueUpToIso hr).hom ≫
+    Gl.map (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩)
+  · symm
+    convert @IsColimit.coconePointUniqueUpToIso_hom_desc _ _ _ _ ((F' ⋙ Gr) ⋙ Gl)
+      (Gl.mapCocone ⟨_, (whiskerRight α' Gr ≫ c.2 : _)⟩) _ _ hl hr using 2
+    · apply hr.hom_ext
+      intro j
+      rw [hr.fac, Functor.mapCocone_ι_app, ← Gl.map_comp, colimit.cocone_ι, colimit.ι_desc]
+      rfl
+    · clear_value α'
+      apply hl.hom_ext
+      intro j
+      rw [hl.fac]
+      dsimp
+      simp only [Category.comp_id, hα'', Category.assoc, Gl.map_comp]
+      congr 1
+      exact (NatTrans.congr_app h j).symm
+  rw [this]
+  have := ((H (colimit.cocone $ F' ⋙ Gr) (whiskerRight α' Gr)
+    (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) ?_ (hα'.whiskerRight Gr)).mp
+    ⟨(getColimitCocone $ F' ⋙ Gr).2⟩ j).map Gl
+  convert IsPullback.paste_vert _ this
+  refine IsPullback.of_vert_isIso ⟨?_⟩
+  rw [← IsIso.inv_comp_eq, ← Category.assoc, NatIso.inv_inv_app]
+  exact IsColimit.comp_coconePointUniqueUpToIso_hom hl hr _
+  · clear_value α'
+    ext j
+    simp
+
+end reflective
+
 section Initial
 
 theorem hasStrictInitial_of_isUniversal [HasInitial C]

Port of https://github.com/leanprover-community/mathlib/pull/17637

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -28,11 +28,13 @@ import Mathlib.CategoryTheory.Limits.VanKampen
 - `CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen`: In extensive categories,
   sums are disjoint, i.e. the pullback of `X ⟶ X ⨿ Y` and `Y ⟶ X ⨿ Y` is the initial object.
 - `CategoryTheory.types.finitaryExtensive`: The category of types is extensive.
-
+- `CategoryTheory.FinitaryExtensive_TopCat`:
+  The category `Top` is extensive.
+- `CategoryTheory.FinitaryExtensive_functor`: The category `C ⥤ D` is extensive if `D`
+  has all pullbacks and is extensive
 ## TODO
 
 Show that the following are finitary extensive:
-- the categories of sheaves over a site
 - `Scheme`
 - `AffineScheme` (`CommRingᵒᵖ`)
 
@@ -47,9 +49,10 @@ open CategoryTheory.Limits
 
 namespace CategoryTheory
 
-universe v' u' v u
+universe v' u' v u v'' u''
 
 variable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]
+variable {D : Type u''} [Category.{v''} D]
 
 section Extensive
 
@@ -275,7 +278,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
 set_option linter.uppercaseLean3 false in
 #align category_theory.finitary_extensive_Top_aux CategoryTheory.finitaryExtensiveTopCatAux
 
-instance : FinitaryExtensive TopCat.{u} := by
+instance finitaryExtensive_TopCat : FinitaryExtensive TopCat.{u} := by
   rw [finitaryExtensive_iff_of_isTerminal TopCat.{u} _ TopCat.isTerminalPUnit _
       (TopCat.binaryCofanIsColimit _ _)]
   apply BinaryCofan.isVanKampen_mk _ _ (fun X Y => TopCat.binaryCofanIsColimit X Y) _
@@ -323,11 +326,23 @@ end TopCat
 
 section Functor
 
-universe v'' u''
-
-variable {D : Type u''} [Category.{v''} D]
-
-instance [HasPullbacks C] [FinitaryExtensive C] : FinitaryExtensive (D ⥤ C) :=
+theorem finitaryExtensive_of_reflective [HasFiniteCoproducts D] [HasPullbacks D]
+    [FinitaryExtensive C] [HasPullbacks C]
+    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
+    [PreservesLimitsOfShape WalkingCospan Gl] :
+    FinitaryExtensive D := by
+  have : PreservesColimitsOfSize Gl := adj.leftAdjointPreservesColimits
+  constructor
+  intros X Y c hc
+  apply (IsVanKampenColimit.precompose_isIso_iff
+    (isoWhiskerLeft _ (asIso adj.counit) ≪≫ Functor.rightUnitor _).hom).mp
+  refine ((FinitaryExtensive.vanKampen _ (colimit.isColimit $ pair X Y ⋙ _)).map_reflective
+    adj).of_iso (IsColimit.uniqueUpToIso ?_ ?_)
+  · exact isColimitOfPreserves Gl (colimit.isColimit _)
+  · exact (IsColimit.precomposeHomEquiv _ _).symm hc
+
+instance finitaryExtensive_functor [HasPullbacks C] [FinitaryExtensive C] :
+    FinitaryExtensive (D ⥤ C) :=
   haveI : HasFiniteCoproducts (D ⥤ C) := ⟨fun _ => Limits.functorCategoryHasColimitsOfShape⟩
   ⟨fun c hc => isVanKampenColimit_of_evaluation _ c fun _ =>
     FinitaryExtensive.vanKampen _ <| PreservesColimit.preserves hc⟩

Also moves the definition of van Kampen colimits from CategoryTheory/Extensive.lean into a new file.

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

Also moves the definition of van Kampen colimits from CategoryTheory/Extensive.lean into a new file.

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -8,6 +8,7 @@ import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
 import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 import Mathlib.CategoryTheory.Limits.FunctorCategory
+import Mathlib.CategoryTheory.Limits.VanKampen
 
 #align_import category_theory.extensive from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
 
@@ -16,9 +17,6 @@ import Mathlib.CategoryTheory.Limits.FunctorCategory
 # Extensive categories
 
 ## Main definitions
-- `CategoryTheory.IsVanKampenColimit`: A (colimit) cocone over a diagram `F : J ⥤ C` is van
-  Kampen if for every cocone `c'` over the pullback of the diagram `F' : J ⥤ C'`,
-  `c'` is colimiting iff `c'` is the pullback of `c`.
 - `CategoryTheory.FinitaryExtensive`: A category is (finitary) extensive if it has finite
   coproducts, and binary coproducts are van Kampen.
 
@@ -53,70 +51,6 @@ universe v' u' v u
 
 variable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]
 
-/-- A natural transformation is equifibered if every commutative square of the following form is
-a pullback.
-```
-F(X) → F(Y)
- ↓      ↓
-G(X) → G(Y)
-```
--/
-def NatTrans.Equifibered {F G : J ⥤ C} (α : F ⟶ G) : Prop :=
-  ∀ ⦃i j : J⦄ (f : i ⟶ j), IsPullback (F.map f) (α.app i) (α.app j) (G.map f)
-#align category_theory.nat_trans.equifibered CategoryTheory.NatTrans.Equifibered
-
-theorem NatTrans.equifibered_of_isIso {F G : J ⥤ C} (α : F ⟶ G) [IsIso α] : Equifibered α :=
-  fun _ _ f => IsPullback.of_vert_isIso ⟨NatTrans.naturality _ f⟩
-#align category_theory.nat_trans.equifibered_of_is_iso CategoryTheory.NatTrans.equifibered_of_isIso
-
-theorem NatTrans.Equifibered.comp {F G H : J ⥤ C} {α : F ⟶ G} {β : G ⟶ H} (hα : Equifibered α)
-    (hβ : Equifibered β) : Equifibered (α ≫ β) :=
-  fun _ _ f => (hα f).paste_vert (hβ f)
-#align category_theory.nat_trans.equifibered.comp CategoryTheory.NatTrans.Equifibered.comp
-
-/-- A (colimit) cocone over a diagram `F : J ⥤ C` is universal if it is stable under pullbacks. -/
-def IsUniversalColimit {F : J ⥤ C} (c : Cocone F) : Prop :=
-  ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.pt ⟶ c.pt)
-    (_ : α ≫ c.ι = c'.ι ≫ (Functor.const J).map f) (_ : NatTrans.Equifibered α),
-    (∀ j : J, IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)) → Nonempty (IsColimit c')
-#align category_theory.is_universal_colimit CategoryTheory.IsUniversalColimit
-
-/-- A (colimit) cocone over a diagram `F : J ⥤ C` is van Kampen if for every cocone `c'` over the
-pullback of the diagram `F' : J ⥤ C'`, `c'` is colimiting iff `c'` is the pullback of `c`.
-
-TODO: Show that this is iff the functor `C ⥤ Catᵒᵖ` sending `x` to `C/x` preserves it.
-TODO: Show that this is iff the inclusion functor `C ⥤ Span(C)` preserves it.
--/
-def IsVanKampenColimit {F : J ⥤ C} (c : Cocone F) : Prop :=
-  ∀ ⦃F' : J ⥤ C⦄ (c' : Cocone F') (α : F' ⟶ F) (f : c'.pt ⟶ c.pt)
-    (_ : α ≫ c.ι = c'.ι ≫ (Functor.const J).map f) (_ : NatTrans.Equifibered α),
-    Nonempty (IsColimit c') ↔ ∀ j : J, IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)
-#align category_theory.is_van_kampen_colimit CategoryTheory.IsVanKampenColimit
-
-theorem IsVanKampenColimit.isUniversal {F : J ⥤ C} {c : Cocone F} (H : IsVanKampenColimit c) :
-    IsUniversalColimit c :=
-  fun _ c' α f h hα => (H c' α f h hα).mpr
-#align category_theory.is_van_kampen_colimit.is_universal CategoryTheory.IsVanKampenColimit.isUniversal
-
-/-- A van Kampen colimit is a colimit. -/
-noncomputable def IsVanKampenColimit.isColimit {F : J ⥤ C} {c : Cocone F}
-    (h : IsVanKampenColimit c) : IsColimit c := by
-  refine' ((h c (𝟙 F) (𝟙 c.pt : _) (by rw [Functor.map_id, Category.comp_id, Category.id_comp])
-    (NatTrans.equifibered_of_isIso _)).mpr fun j => _).some
-  haveI : IsIso (𝟙 c.pt) := inferInstance
-  exact IsPullback.of_vert_isIso ⟨by erw [NatTrans.id_app, Category.comp_id, Category.id_comp]⟩
-#align category_theory.is_van_kampen_colimit.is_colimit CategoryTheory.IsVanKampenColimit.isColimit
-
-theorem IsInitial.isVanKampenColimit [HasStrictInitialObjects C] {X : C} (h : IsInitial X) :
-    IsVanKampenColimit (asEmptyCocone X) := by
-  intro F' c' α f hf hα
-  have : F' = Functor.empty C := by apply Functor.hext <;> rintro ⟨⟨⟩⟩
-  subst this
-  haveI := h.isIso_to f
-  refine' ⟨by rintro _ ⟨⟨⟩⟩,
-    fun _ => ⟨IsColimit.ofIsoColimit h (Cocones.ext (asIso f).symm <| by rintro ⟨⟨⟩⟩)⟩⟩
-#align category_theory.is_initial.is_van_kampen_colimit CategoryTheory.IsInitial.isVanKampenColimit
-
 section Extensive
 
 variable {X Y : C}
@@ -146,98 +80,6 @@ theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingP
   exact FinitaryExtensive.van_kampen' c hc
 #align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.vanKampen
 
-theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') :
-    NatTrans.Equifibered α := by
-  rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩
-  all_goals
-    dsimp; simp only [Discrete.functor_map_id]
-    exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩
-#align category_theory.map_pair_equifibered CategoryTheory.mapPair_equifibered
-
-theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
-    IsVanKampenColimit c ↔
-      ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt)
-        (_ : αX ≫ c.inl = c'.inl ≫ f) (_ : αY ≫ c.inr = c'.inr ≫ f),
-        Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr := by
-  constructor
-  · introv H hαX hαY
-    rw [H c' (mapPair αX αY) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> assumption) (mapPair_equifibered _)]
-    constructor
-    · intro H
-      exact ⟨H _, H _⟩
-    · rintro H ⟨⟨⟩⟩
-      exacts [H.1, H.2]
-  · introv H F' hα h
-    let X' := F'.obj ⟨WalkingPair.left⟩
-    let Y' := F'.obj ⟨WalkingPair.right⟩
-    have : F' = pair X' Y' := by
-      apply Functor.hext
-      · rintro ⟨⟨⟩⟩ <;> rfl
-      · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp
-    clear_value X' Y'
-    subst this
-    change BinaryCofan X' Y' at c'
-    rw [H c' _ _ _ (NatTrans.congr_app hα ⟨WalkingPair.left⟩)
-        (NatTrans.congr_app hα ⟨WalkingPair.right⟩)]
-    constructor
-    · rintro H ⟨⟨⟩⟩
-      exacts [H.1, H.2]
-    · intro H
-      exact ⟨H _, H _⟩
-#align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.isVanKampen_iff
-
-theorem BinaryCofan.isVanKampen_mk {X Y : C} (c : BinaryCofan X Y)
-    (cofans : ∀ X Y : C, BinaryCofan X Y) (colimits : ∀ X Y, IsColimit (cofans X Y))
-    (cones : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), PullbackCone f g)
-    (limits : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), IsLimit (cones f g))
-    (h₁ : ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt)
-      (_ : αX ≫ c.inl = (cofans X' Y').inl ≫ f) (_ : αY ≫ c.inr = (cofans X' Y').inr ≫ f),
-      IsPullback (cofans X' Y').inl αX f c.inl ∧ IsPullback (cofans X' Y').inr αY f c.inr)
-    (h₂ : ∀ {Z : C} (f : Z ⟶ c.pt),
-      IsColimit (BinaryCofan.mk (cones f c.inl).fst (cones f c.inr).fst)) :
-    IsVanKampenColimit c := by
-  rw [BinaryCofan.isVanKampen_iff]
-  introv hX hY
-  constructor
-  · rintro ⟨h⟩
-    let e := h.coconePointUniqueUpToIso (colimits _ _)
-    obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [hX]) (by simp [hY])
-    constructor
-    · rw [← Category.id_comp αX, ← Iso.hom_inv_id_assoc e f]
-      haveI : IsIso (𝟙 X') := inferInstance
-      have : c'.inl ≫ e.hom = 𝟙 X' ≫ (cofans X' Y').inl := by
-        dsimp
-        simp
-      exact (IsPullback.of_vert_isIso ⟨this⟩).paste_vert hl
-    · rw [← Category.id_comp αY, ← Iso.hom_inv_id_assoc e f]
-      haveI : IsIso (𝟙 Y') := inferInstance
-      have : c'.inr ≫ e.hom = 𝟙 Y' ≫ (cofans X' Y').inr := by
-        dsimp
-        simp
-      exact (IsPullback.of_vert_isIso ⟨this⟩).paste_vert hr
-  · rintro ⟨H₁, H₂⟩
-    refine' ⟨IsColimit.ofIsoColimit _ <| (isoBinaryCofanMk _).symm⟩
-    let e₁ : X' ≅ _ := H₁.isLimit.conePointUniqueUpToIso (limits _ _)
-    let e₂ : Y' ≅ _ := H₂.isLimit.conePointUniqueUpToIso (limits _ _)
-    have he₁ : c'.inl = e₁.hom ≫ (cones f c.inl).fst := by simp
-    have he₂ : c'.inr = e₂.hom ≫ (cones f c.inr).fst := by simp
-    rw [he₁, he₂]
-    apply BinaryCofan.isColimitCompRightIso (BinaryCofan.mk _ _)
-    apply BinaryCofan.isColimitCompLeftIso (BinaryCofan.mk _ _)
-    exact h₂ f
-#align category_theory.binary_cofan.is_van_kampen_mk CategoryTheory.BinaryCofan.isVanKampen_mk
-
-theorem BinaryCofan.mono_inr_of_isVanKampen [HasInitial C] {X Y : C} {c : BinaryCofan X Y}
-    (h : IsVanKampenColimit c) : Mono c.inr := by
-  refine' PullbackCone.mono_of_isLimitMkIdId _ (IsPullback.isLimit _)
-  refine' (h (BinaryCofan.mk (initial.to Y) (𝟙 Y)) (mapPair (initial.to X) (𝟙 Y)) c.inr _
-      (mapPair_equifibered _)).mp ⟨_⟩ ⟨WalkingPair.right⟩
-  · ext ⟨⟨⟩⟩ <;> dsimp; simp
-  · exact ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr (by
-      dsimp
-      infer_instance)).some
-#align category_theory.binary_cofan.mono_inr_of_is_van_kampen CategoryTheory.BinaryCofan.mono_inr_of_isVanKampen
-
 theorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
     (hc : IsColimit c) : Mono c.inr :=
   BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
@@ -254,37 +96,12 @@ instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inl : X ⟶ X ⨿ Y) :=
 instance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inr : Y ⟶ X ⨿ Y) :=
   (FinitaryExtensive.mono_inr_of_isColimit (coprodIsCoprod X Y) : _)
 
-theorem BinaryCofan.isPullback_initial_to_of_isVanKampen [HasInitial C] {c : BinaryCofan X Y}
-    (h : IsVanKampenColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr := by
-  refine' ((h (BinaryCofan.mk (initial.to Y) (𝟙 Y)) (mapPair (initial.to X) (𝟙 Y)) c.inr _
-      (mapPair_equifibered _)).mp ⟨_⟩ ⟨WalkingPair.left⟩).flip
-  · ext ⟨⟨⟩⟩ <;> dsimp; simp
-  · exact ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr (by
-      dsimp
-      infer_instance)).some
-#align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen
-
 theorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]
     {c : BinaryCofan X Y} (hc : IsColimit c) :
     IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
   BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc)
 #align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan
 
-theorem hasStrictInitial_of_isUniversal [HasInitial C]
-    (H : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))) : HasStrictInitialObjects C :=
-  hasStrictInitialObjects_of_initial_is_strict
-    (by
-      intro A f
-      suffices IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A)) by
-        obtain ⟨l, h₁, h₂⟩ := Limits.BinaryCofan.IsColimit.desc' this (f ≫ initial.to A) (𝟙 A)
-        rcases (Category.id_comp _).symm.trans h₂ with rfl
-        exact ⟨⟨_, ((Category.id_comp _).symm.trans h₁).symm, initialIsInitial.hom_ext _ _⟩⟩
-      refine' (H (BinaryCofan.mk (𝟙 _) (𝟙 _)) (mapPair f f) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> simp)
-        (mapPair_equifibered _) _).some
-      rintro ⟨⟨⟩⟩ <;> dsimp <;>
-        exact IsPullback.of_horiz_isIso ⟨(Category.id_comp _).trans (Category.comp_id _).symm⟩)
-#align category_theory.has_strict_initial_of_is_universal CategoryTheory.hasStrictInitial_of_isUniversal
-
 instance (priority := 100) hasStrictInitialObjects_of_finitaryExtensive [FinitaryExtensive C] :
     HasStrictInitialObjects C :=
   hasStrictInitial_of_isUniversal (FinitaryExtensive.vanKampen _
@@ -510,57 +327,6 @@ universe v'' u''
 
 variable {D : Type u''} [Category.{v''} D]
 
-theorem NatTrans.Equifibered.whiskerRight {F G : J ⥤ C} {α : F ⟶ G} (hα : Equifibered α)
-    (H : C ⥤ D) [PreservesLimitsOfShape WalkingCospan H] : Equifibered (whiskerRight α H) :=
-  fun _ _ f => (hα f).map H
-#align category_theory.nat_trans.equifibered.whisker_right CategoryTheory.NatTrans.Equifibered.whiskerRight
-
-theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKampenColimit c)
-    (e : c ≅ c') : IsVanKampenColimit c' := by
-  intro F' c'' α f h hα
-  have : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι := by
-    ext j
-    exact e.inv.2 j
-  rw [H c'' α (f ≫ e.inv.1) (by rw [Functor.map_comp, ← reassoc_of% h, this]) hα]
-  apply forall_congr'
-  intro j
-  conv_lhs => rw [← Category.comp_id (α.app j)]
-  haveI : IsIso e.inv.hom := Functor.map_isIso (Cocones.forget _) e.inv
-  exact (IsPullback.of_vert_isIso ⟨by simp⟩).paste_vert_iff (NatTrans.congr_app h j).symm
-#align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso
-
-theorem IsVanKampenColimit.of_map {D : Type*} [Category D] (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
-    [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
-    [PreservesColimitsOfShape J G] [ReflectsColimitsOfShape J G]
-    (H : IsVanKampenColimit (G.mapCocone c)) : IsVanKampenColimit c := by
-  intro F' c' α f h hα
-  refine' (Iff.trans _ (H (G.mapCocone c') (whiskerRight α G) (G.map f)
-      (by ext j; simpa using G.congr_map (NatTrans.congr_app h j))
-      (hα.whiskerRight G))).trans (forall_congr' fun j => _)
-  · exact ⟨fun h => ⟨isColimitOfPreserves G h.some⟩, fun h => ⟨isColimitOfReflects G h.some⟩⟩
-  · exact IsPullback.map_iff G (NatTrans.congr_app h.symm j)
-#align category_theory.is_van_kampen_colimit.of_map CategoryTheory.IsVanKampenColimit.of_map
-
-theorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J D] (F : J ⥤ C ⥤ D)
-    (c : Cocone F) (hc : ∀ x : C, IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)) :
-    IsVanKampenColimit c := by
-  intro F' c' α f e hα
-  have := fun x => hc x (((evaluation C D).obj x).mapCocone c') (whiskerRight α _)
-      (((evaluation C D).obj x).map f)
-      (by
-        ext y
-        dsimp
-        exact NatTrans.congr_app (NatTrans.congr_app e y) x)
-      (hα.whiskerRight _)
-  constructor
-  · rintro ⟨hc'⟩ j
-    refine' ⟨⟨(NatTrans.congr_app e j).symm⟩, ⟨evaluationJointlyReflectsLimits _ _⟩⟩
-    refine' fun x => (isLimitMapConePullbackConeEquiv _ _).symm _
-    exact ((this x).mp ⟨PreservesColimit.preserves hc'⟩ _).isLimit
-  · exact fun H => ⟨evaluationJointlyReflectsColimits _ fun x =>
-      ((this x).mpr fun j => (H j).map ((evaluation C D).obj x)).some⟩
-#align category_theory.is_van_kampen_colimit_of_evaluation CategoryTheory.isVanKampenColimit_of_evaluation
-
 instance [HasPullbacks C] [FinitaryExtensive C] : FinitaryExtensive (D ⥤ C) :=
   haveI : HasFiniteCoproducts (D ⥤ C) := ⟨fun _ => Limits.functorCategoryHasColimitsOfShape⟩
   ⟨fun c hc => isVanKampenColimit_of_evaluation _ c fun _ =>
@@ -570,7 +336,7 @@ theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExten
     [HasFiniteCoproducts C] [PreservesLimitsOfShape WalkingCospan F]
     [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
     [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C :=
-  ⟨fun _ hc => (FinitaryExtensive.vanKampen _ (isColimitOfPreserves F hc)).of_map F⟩
+  ⟨fun _ hc => (FinitaryExtensive.vanKampen _ (isColimitOfPreserves F hc)).of_mapCocone F⟩
 #align category_theory.finitary_extensive_of_preserves_and_reflects CategoryTheory.finitaryExtensive_of_preserves_and_reflects
 
 theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -277,7 +277,7 @@ theorem hasStrictInitial_of_isUniversal [HasInitial C]
       intro A f
       suffices IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A)) by
         obtain ⟨l, h₁, h₂⟩ := Limits.BinaryCofan.IsColimit.desc' this (f ≫ initial.to A) (𝟙 A)
-        rcases(Category.id_comp _).symm.trans h₂ with rfl
+        rcases (Category.id_comp _).symm.trans h₂ with rfl
         exact ⟨⟨_, ((Category.id_comp _).symm.trans h₁).symm, initialIsInitial.hom_ext _ _⟩⟩
       refine' (H (BinaryCofan.mk (𝟙 _) (𝟙 _)) (mapPair f f) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> simp)
         (mapPair_equifibered _) _).some

@@ -518,14 +518,14 @@ theorem NatTrans.Equifibered.whiskerRight {F G : J ⥤ C} {α : F ⟶ G} (hα :
 theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKampenColimit c)
     (e : c ≅ c') : IsVanKampenColimit c' := by
   intro F' c'' α f h hα
-  have : c'.ι ≫ (Functor.const J).map e.inv.Hom = c.ι := by
+  have : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι := by
     ext j
     exact e.inv.2 j
   rw [H c'' α (f ≫ e.inv.1) (by rw [Functor.map_comp, ← reassoc_of% h, this]) hα]
   apply forall_congr'
   intro j
   conv_lhs => rw [← Category.comp_id (α.app j)]
-  haveI : IsIso e.inv.Hom := Functor.map_isIso (Cocones.forget _) e.inv
+  haveI : IsIso e.inv.hom := Functor.map_isIso (Cocones.forget _) e.inv
   exact (IsPullback.of_vert_isIso ⟨by simp⟩).paste_vert_iff (NatTrans.congr_app h j).symm
 #align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso
 

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

This has nice performance benefits.

@@ -529,7 +529,7 @@ theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKamp
   exact (IsPullback.of_vert_isIso ⟨by simp⟩).paste_vert_iff (NatTrans.congr_app h j).symm
 #align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso
 
-theorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
+theorem IsVanKampenColimit.of_map {D : Type*} [Category D] (G : C ⥤ D) {F : J ⥤ C} {c : Cocone F}
     [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]
     [PreservesColimitsOfShape J G] [ReflectsColimitsOfShape J G]
     (H : IsVanKampenColimit (G.mapCocone c)) : IsVanKampenColimit c := by

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.extensive
-! leanprover-community/mathlib commit 178a32653e369dce2da68dc6b2694e385d484ef1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.CommSq
 import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
@@ -14,6 +9,8 @@ import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 import Mathlib.CategoryTheory.Limits.FunctorCategory
 
+#align_import category_theory.extensive from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
+
 /-!
 
 # Extensive categories

This PR is the result of running

find . -type f -name "*.lean" -exec sed -i -E 's/^( +)\. /\1· /' {} \;
find . -type f -name "*.lean" -exec sed -i -E 'N;s/^( +·)\n +(.*)$/\1 \2/;P;D' {} \;

which firstly replaces . focusing dots with · and secondly removes isolated instances of such dots, unifying them with the following line. A new rule is placed in the style linter to verify this.

@@ -442,8 +442,8 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
     ext ⟨⟨x, ⟨⟩⟩, (hx : f x = Sum.inl PUnit.unit)⟩
     change dite _ _ _ = _
     split_ifs with h
-    . rfl
-    . cases (h hx) -- Porting note : in Lean3 it is `rfl`
+    · rfl
+    · cases (h hx) -- Porting note : in Lean3 it is `rfl`
   · intro s
     ext ⟨⟨x, ⟨⟩⟩, hx⟩
     change dite _ _ _ = _

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -301,7 +301,7 @@ theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFi
     FinitaryExtensive C ↔ IsVanKampenColimit c₀ := by
   refine' ⟨fun H => H.2 c₀ hc₀, fun H => _⟩
   constructor
-  simp_rw [BinaryCofan.isVanKampen_iff] at H⊢
+  simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢
   intro X Y c hc X' Y' c' αX αY f hX hY
   obtain ⟨d, hd, hd'⟩ :=
     Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)
@@ -326,7 +326,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) := by
         cases' h : s.fst x with val val
         · simp only [Types.binaryCoproductCocone_pt, Functor.const_obj_obj, Sum.inl.injEq,
             exists_unique_eq']
-        · apply_fun f  at h
+        · apply_fun f at h
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val : _).symm
       delta ExistsUnique at this
       choose l hl hl' using this
@@ -337,7 +337,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) := by
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases' h : s.fst x with val val
-        · apply_fun f  at h
+        · apply_fun f at h
           cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val : _).symm
         · simp only [Types.binaryCoproductCocone_pt, Functor.const_obj_obj, Sum.inr.injEq,
             exists_unique_eq']
@@ -474,7 +474,7 @@ instance : FinitaryExtensive TopCat.{u} := by
         intro x
         cases' h : s.fst x with val val
         · exact ⟨val, rfl, fun y h => Sum.inl_injective h.symm⟩
-        · apply_fun f  at h
+        · apply_fun f at h
           cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
             (ConcreteCategory.congr_hom hαY val : _).symm
       delta ExistsUnique at this
@@ -490,7 +490,7 @@ instance : FinitaryExtensive TopCat.{u} := by
       have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
         intro x
         cases' h : s.fst x with val val
-        · apply_fun f  at h
+        · apply_fun f at h
           cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
             (ConcreteCategory.congr_hom hαX val : _).symm
         · exact ⟨val, rfl, fun y h => Sum.inr_injective h.symm⟩

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -169,7 +169,7 @@ theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
     · intro H
       exact ⟨H _, H _⟩
     · rintro H ⟨⟨⟩⟩
-      exacts[H.1, H.2]
+      exacts [H.1, H.2]
   · introv H F' hα h
     let X' := F'.obj ⟨WalkingPair.left⟩
     let Y' := F'.obj ⟨WalkingPair.right⟩
@@ -184,7 +184,7 @@ theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
         (NatTrans.congr_app hα ⟨WalkingPair.right⟩)]
     constructor
     · rintro H ⟨⟨⟩⟩
-      exacts[H.1, H.2]
+      exacts [H.1, H.2]
     · intro H
       exact ⟨H _, H _⟩
 #align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.isVanKampen_iff
@@ -351,7 +351,7 @@ instance types.finitaryExtensive : FinitaryExtensive (Type u) := by
     have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by
       intro x
       rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
-      exacts[Or.inl rfl, Or.inr rfl]
+      exacts [Or.inl rfl, Or.inr rfl]
     let eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
       ⟨fun p => ⟨p.1.1, by convert p.2⟩, fun x => ⟨⟨_, _⟩, x.2⟩, fun _ => by ext; rfl,
         fun _ => by ext; rfl⟩
@@ -392,7 +392,7 @@ noncomputable def finitaryExtensiveTopCatAux (Z : TopCat.{u})
   have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by
     intro x
     rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
-    exacts[Or.inl rfl, Or.inr rfl]
+    exacts [Or.inl rfl, Or.inr rfl]
   letI eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
     ⟨fun p => ⟨p.1.1, p.2.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)⟩,
       fun x => ⟨⟨_, PUnit.unit⟩, x.2⟩, fun _ => by ext; rfl, fun _ => by ext; rfl⟩

Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com>