feat(CategoryTheory/Sites): add the Grothendieck topology generated by a precoverage

Mathlib.lean

@@ -3033,6 +3033,7 @@ public import Mathlib.CategoryTheory.Sites.Over
 public import Mathlib.CategoryTheory.Sites.Plus
 public import Mathlib.CategoryTheory.Sites.Point.Basic
 public import Mathlib.CategoryTheory.Sites.Precoverage
+public import Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
 public import Mathlib.CategoryTheory.Sites.Preserves
 public import Mathlib.CategoryTheory.Sites.PreservesLocallyBijective
 public import Mathlib.CategoryTheory.Sites.PreservesSheafification

Mathlib/CategoryTheory/Sites/Coverage.lean

@@ -5,8 +5,7 @@ Authors: Adam Topaz
 -/
 module
 
-public import Mathlib.CategoryTheory.Sites.Precoverage
-public import Mathlib.CategoryTheory.Sites.Sheaf
+public import Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
 
 /-!
 
@@ -227,6 +226,36 @@ lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T
   · apply Saturate.top
   · assumption
 
+lemma saturate_iff_saturate_toPrecoverage (K : Coverage C) {X : C} {S : Sieve X} :
+    K.Saturate X S ↔ K.toPrecoverage.Saturate X S := by
+  constructor <;> intro hS
+  · induction hS with
+    | of _ _ hS => exact .of _ _ hS
+    | top => exact .top _
+    | transitive _ _ _ _ _ ih1 ih2 => exact .transitive _ _ _ ih1 ih2
+  · induction hS with
+    | of _ _ hS => exact .of _ _ hS
+    | top => exact .top _
+    | pullback _ R hR Y f ih =>
+      clear hR
+      induction ih generalizing Y with
+      | of X S hS =>
+        obtain ⟨R,hR1,hR2⟩ := K.pullback f S hS
+        suffices Sieve.generate R ≤ (Sieve.generate S).pullback f from
+          saturate_of_superset _ this (Saturate.of _ _ hR1)
+        rintro Z g ⟨W, i, e, h1, h2⟩
+        obtain ⟨WW, ii, ee, hh1, hh2⟩ := hR2 h1
+        refine ⟨WW, i ≫ ii, ee, hh1, ?_⟩
+        simp only [hh2, reassoc_of% h2, Category.assoc]
+      | top X => apply Saturate.top
+      | transitive X R S _ hS H1 _ =>
+        apply Saturate.transitive
+        · apply H1 _ f
+        intro Z g hg
+        rw [← Sieve.pullback_comp]
+        exact hS hg
+    | transitive _ _ _ _ _ ih1 ih2 => exact .transitive _ _ _ ih1 ih2
+
 /--
 The Grothendieck topology associated to a coverage `K`.
 It is defined *inductively* as follows:
@@ -239,28 +268,10 @@ The pullback compatibility condition for a coverage ensures that the
 associated Grothendieck topology is pullback stable, and so an additional constructor
 in the inductive construction is not needed.
 -/
-def toGrothendieck (K : Coverage C) : GrothendieckTopology C where
-  sieves := Saturate K
-  top_mem' := .top
-  pullback_stable' := by
-    intro X Y S f hS
-    induction hS generalizing Y with
-    | of X S hS =>
-      obtain ⟨R,hR1,hR2⟩ := K.pullback f S hS
-      suffices Sieve.generate R ≤ (Sieve.generate S).pullback f from
-        saturate_of_superset _ this (Saturate.of _ _ hR1)
-      rintro Z g ⟨W, i, e, h1, h2⟩
-      obtain ⟨WW, ii, ee, hh1, hh2⟩ := hR2 h1
-      refine ⟨WW, i ≫ ii, ee, hh1, ?_⟩
-      simp only [hh2, reassoc_of% h2, Category.assoc]
-    | top X => apply Saturate.top
-    | transitive X R S _ hS H1 _ =>
-      apply Saturate.transitive
-      · apply H1 f
-      intro Z g hg
-      rw [← Sieve.pullback_comp]
-      exact hS hg
-  transitive' _ _ hS _ hR := .transitive _ _ _ hS hR
+def toGrothendieck (K : Coverage C) : GrothendieckTopology C := by
+  apply K.toPrecoverage.toGrothendieck.copy K.Saturate
+  ext
+  exact K.saturate_iff_saturate_toPrecoverage.symm
 
 lemma mem_toGrothendieck {K : Coverage C} {X : C} {S : Sieve X} :
     S ∈ K.toGrothendieck X ↔ Saturate K X S := .rfl
@@ -368,29 +379,31 @@ lemma Pretopology.toGrothendieck_toCoverage [HasPullbacks C] (J : Pretopology C)
   · refine Coverage.saturate_of_superset _ ?_ (.of _ _ hR)
     rwa [Sieve.generate_le_iff]
 
-namespace Precoverage
-
 /-- A precoverage with pullbacks defines a coverage. -/
 @[simps toPrecoverage]
-def toCoverage (J : Precoverage C) [J.HasPullbacks] [J.IsStableUnderBaseChange] :
+def Precoverage.toCoverage (J : Precoverage C) [J.HasPullbacks] [J.IsStableUnderBaseChange] :
     Coverage C where
   __ := J
   pullback X Y f S hS := by
     have : S.HasPullbacks f := J.hasPullbacks_of_mem _ hS
     exact ⟨S.pullbackArrows f, J.pullbackArrows_mem _ hS,
       Presieve.FactorsThruAlong.pullbackArrows f S⟩
 
-/-- A precoverage with pullbacks defines a Grothendieck topology. -/
-def toGrothendieck (J : Precoverage C) [J.HasPullbacks] [J.IsStableUnderBaseChange] :
-    GrothendieckTopology C :=
-  J.toCoverage.toGrothendieck
-
-lemma mem_toGrothendieck_iff {J : Precoverage C} [J.HasPullbacks] [J.IsStableUnderBaseChange]
-    {X : C} {S : Sieve X} :
-    S ∈ toGrothendieck J X ↔ J.toCoverage.Saturate X S :=
-  .rfl
-
-end Precoverage
+lemma Precoverage.toGrothendieck_toCoverage {J : Precoverage C} [J.HasPullbacks]
+    [J.IsStableUnderBaseChange] :
+    J.toCoverage.toGrothendieck = J.toGrothendieck := by
+  refine le_antisymm ?_ ?_
+  · intro _ _ hS
+    induction hS with
+    | of _ _ hS => exact generate_mem_toGrothendieck hS
+    | top => exact J.toGrothendieck.top_mem _
+    | transitive _ _ _ _ _ ih1 ih2 => exact J.toGrothendieck.transitive ih1 _ ih2
+  · intro _ _ hS
+    induction hS with
+    | of _ _ hS => exact .of _ _ hS
+    | top => exact J.toCoverage.toGrothendieck.top_mem _
+    | pullback _ _ _ _ _ ih => exact J.toCoverage.toGrothendieck.pullback_stable _ ih
+    | transitive _ _ _ _ _ ih1 ih2 => exact J.toCoverage.toGrothendieck.transitive ih1 _ ih2
 
 namespace GrothendieckTopology
 
@@ -420,8 +433,13 @@ end GrothendieckTopology
 defined. -/
 lemma Precoverage.toGrothendieck_le_iff_le_toPrecoverage {K : Precoverage C}
     {J : GrothendieckTopology C} [K.HasPullbacks] [K.IsStableUnderBaseChange] :
-    K.toGrothendieck ≤ J ↔ K ≤ J.toPrecoverage :=
-  (Coverage.gi C).gc _ _
+    K.toGrothendieck ≤ J ↔ K ≤ J.toPrecoverage := by
+  rw [← toGrothendieck_toCoverage]
+  exact (Coverage.gi C).gc _ _
+
+lemma Coverage.toGrothendieck_toPrecoverage (J : Coverage C) :
+    J.toPrecoverage.toGrothendieck = J.toGrothendieck := by
+  rw [Coverage.toGrothendieck, GrothendieckTopology.copy_eq]
 
 open Coverage
 
@@ -435,72 +453,20 @@ the associated Grothendieck topology.
 theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type*) :
     Presieve.IsSheaf K.toGrothendieck P ↔
     (∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) := by
+  rw [← toGrothendieck_toPrecoverage, Precoverage.isSheaf_toGrothendieck_iff]
   constructor
-  · intro H X R hR
-    rw [Presieve.isSheafFor_iff_generate]
-    apply H _ <| Saturate.of _ _ hR
   · intro H X S hS
-    -- This is the key point of the proof:
-    -- We must generalize the induction in the correct way.
-    suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
-      simpa using this (f := 𝟙 _)
-    induction hS with
-    | of X S hS =>
-      intro Y f
-      obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
-      apply Presieve.isSheafFor_of_factorsThru (S := T)
-      · intro Z g hg
-        obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
-        exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
-      · apply H; assumption
-      · intro Z g _
-        obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
-        exact ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
-    | top => intros; simpa using Presieve.isSheafFor_top_sieve _
-    | transitive X R S _ _ H1 H2 =>
-      intro Y f
-      simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
-      choose H1 H1' using H1
-      choose H2 H2' using H2
-      refine ⟨?_, fun x hx => ?_⟩
-      · intro x t₁ t₂ h₁ h₂
-        refine (H1 f).ext (fun Z g hg => ?_)
-        refine (H2 hg (𝟙 _)).ext (fun ZZ gg hgg => ?_)
-        simp only [Sieve.pullback_id, Sieve.pullback_apply] at hgg
-        simp only [← types_comp_apply]
-        rw [← P.map_comp, ← op_comp, h₁, h₂]
-        simpa only [Sieve.pullback_apply, Category.assoc] using hgg
-      let y : ∀ ⦃Z : C⦄ (g : Z ⟶ Y),
-        ((S.pullback (g ≫ f)).pullback (𝟙 _)).arrows.FamilyOfElements P :=
-        fun Z g ZZ gg hgg => x (gg ≫ g) (by simpa using hgg)
-      have hy : ∀ ⦃Z : C⦄ (g : Z ⟶ Y), (y g).Compatible := by
-        intro Z g Y₁ Y₂ ZZ g₁ g₂ f₁ f₂ h₁ h₂ h
-        rw [hx]
-        rw [reassoc_of% h]
-      choose z hz using fun ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (hg : R.pullback f g) =>
-        H2' hg (𝟙 _) (y g) (hy g)
-      let q : (R.pullback f).arrows.FamilyOfElements P := fun Z g hg => z hg
-      have hq : q.Compatible := by
-        intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
-        apply (H2 h₁ g₁).ext
-        intro ZZ gg hgg
-        simp only [← types_comp_apply]
-        rw [← P.map_comp, ← P.map_comp, ← op_comp, ← op_comp, hz, hz]
-        · dsimp [y]; congr 1; simp only [Category.assoc, h]
-        · simpa [reassoc_of% h] using hgg
-        · simpa using hgg
-      obtain ⟨t, ht⟩ := H1' f q hq
-      refine ⟨t, fun Z g hg => ?_⟩
-      refine (H1 (g ≫ f)).ext (fun ZZ gg hgg => ?_)
-      rw [← types_comp_apply _ (P.map gg.op), ← P.map_comp, ← op_comp, ht]
-      on_goal 2 => simpa using hgg
-      refine (H2 hgg (𝟙 _)).ext (fun ZZZ ggg hggg => ?_)
-      rw [← types_comp_apply _ (P.map ggg.op), ← P.map_comp, ← op_comp, hz]
-      on_goal 2 => simpa using hggg
-      refine (H2 hgg ggg).ext (fun ZZZZ gggg _ => ?_)
-      rw [← types_comp_apply _ (P.map gggg.op), ← P.map_comp, ← op_comp]
-      apply hx
-      simp
+    simpa [← Presieve.isSheafFor_iff_generate] using H (f := 𝟙 X) S hS
+  · intro H X Y f S hS
+    obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
+    apply Presieve.isSheafFor_of_factorsThru (S := T)
+    · intro Z g hg
+      obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
+      exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
+    · apply H; assumption
+    · intro Z g _
+      obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1
+      exact ⟨R, (H _ hR1).isSeparatedFor, hR2⟩
 
 /--
 A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a

Mathlib/CategoryTheory/Sites/Over.lean

@@ -332,6 +332,8 @@ abbrev Sheaf.over {A : Type u'} [Category.{v'} A] (F : Sheaf J A) (X : C) :
 
 section
 
+-- TODO: Generalize this section to arbitrary precoverages.
+
 variable (K : Precoverage C) [K.HasPullbacks] [K.IsStableUnderBaseChange]
 
 /-- The Grothendieck topology on `Over X`, obtained from localizing the topology generated
@@ -341,7 +343,8 @@ lemma over_toGrothendieck_eq_toGrothendieck_comap_forget (X : C) :
   refine le_antisymm ?_ ?_
   · intro ⟨Y, right, (s : Y ⟶ X)⟩ R hR
     obtain ⟨(R : Sieve Y), rfl⟩ := (Sieve.overEquiv _).symm.surjective R
-    simp only [GrothendieckTopology.mem_over_iff, Equiv.apply_symm_apply] at hR
+    simp only [GrothendieckTopology.mem_over_iff, Equiv.apply_symm_apply,
+      ← Precoverage.toGrothendieck_toCoverage] at hR
     induction hR with
     | of Z S hS =>
       rw [Sieve.overEquiv_symm_generate]
@@ -359,7 +362,7 @@ lemma over_toGrothendieck_eq_toGrothendieck_comap_forget (X : C) :
     rw [Precoverage.mem_comap_iff] at hR
     rw [GrothendieckTopology.mem_toPrecoverage_iff, GrothendieckTopology.mem_over_iff,
       Sieve.overEquiv, Equiv.coe_fn_mk, ← Sieve.generate_map_eq_functorPushforward]
-    exact Coverage.Saturate.of _ _ hR
+    exact Precoverage.Saturate.of _ _ hR
 
 end