feat: port CategoryTheory.Sites.Closed (#3356)

Co-authored-by: Moritz Firsching <firsching@google.com>
Co-authored-by: int-y1 <jason_yuen2007@hotmail.com>

Mathlib.lean

@@ -856,6 +856,7 @@ import Mathlib.CategoryTheory.Sigma.Basic
 import Mathlib.CategoryTheory.Simple
 import Mathlib.CategoryTheory.SingleObj
 import Mathlib.CategoryTheory.Sites.Adjunction
+import Mathlib.CategoryTheory.Sites.Closed
 import Mathlib.CategoryTheory.Sites.CoverLifting
 import Mathlib.CategoryTheory.Sites.CoverPreserving
 import Mathlib.CategoryTheory.Sites.Coverage

Mathlib/CategoryTheory/Sites/Closed.lean

@@ -0,0 +1,331 @@
+/-
+Copyright (c) 2020 Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta
+
+! This file was ported from Lean 3 source module category_theory.sites.closed
+! leanprover-community/mathlib commit 4cfc30e317caad46858393f1a7a33f609296cc30
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.CategoryTheory.Sites.SheafOfTypes
+import Mathlib.Order.Closure
+
+/-!
+# Closed sieves
+
+A natural closure operator on sieves is a closure operator on `Sieve X` for each `X` which commutes
+with pullback.
+We show that a Grothendieck topology `J` induces a natural closure operator, and define what the
+closed sieves are. The collection of `J`-closed sieves forms a presheaf which is a sheaf for `J`,
+and further this presheaf can be used to determine the Grothendieck topology from the sheaf
+predicate.
+Finally we show that a natural closure operator on sieves induces a Grothendieck topology, and hence
+that natural closure operators are in bijection with Grothendieck topologies.
+
+## Main definitions
+
+* `CategoryTheory.GrothendieckTopology.close`: Sends a sieve `S` on `X` to the set of arrows
+  which it covers. This has all the usual properties of a closure operator, as well as commuting
+  with pullback.
+* `CategoryTheory.GrothendieckTopology.closureOperator`: The bundled `ClosureOperator` given
+  by `CategoryTheory.GrothendieckTopology.close`.
+* `CategoryTheory.GrothendieckTopology.IsClosed`: A sieve `S` on `X` is closed for the topology `J`
+   if it contains every arrow it covers.
+* `CategoryTheory.Functor.closedSieves`: The presheaf sending `X` to the collection of `J`-closed
+  sieves on `X`. This is additionally shown to be a sheaf for `J`, and if this is a sheaf for a
+  different topology `J'`, then `J' ≤ J`.
+* `CategoryTheory.topologyOfClosureOperator`: A closure operator on the
+  set of sieves on every object which commutes with pullback additionally induces a Grothendieck
+  topology, giving a bijection with `CategoryTheory.GrothendieckTopology.closureOperator`.
+
+
+## Tags
+
+closed sieve, closure, Grothendieck topology
+
+## References
+
+* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]
+-/
+
+
+universe v u
+
+namespace CategoryTheory
+
+variable {C : Type u} [Category.{v} C]
+
+variable (J₁ J₂ : GrothendieckTopology C)
+
+namespace GrothendieckTopology
+
+/-- The `J`-closure of a sieve is the collection of arrows which it covers. -/
+@[simps]
+def close {X : C} (S : Sieve X) : Sieve X where
+  arrows _ f := J₁.Covers S f
+  downward_closed hS := J₁.arrow_stable _ _ hS
+#align category_theory.grothendieck_topology.close CategoryTheory.GrothendieckTopology.close
+
+/-- Any sieve is smaller than its closure. -/
+theorem le_close {X : C} (S : Sieve X) : S ≤ J₁.close S :=
+  fun _ _ hg => J₁.covering_of_eq_top (S.pullback_eq_top_of_mem hg)
+#align category_theory.grothendieck_topology.le_close CategoryTheory.GrothendieckTopology.le_close
+
+/-- A sieve is closed for the Grothendieck topology if it contains every arrow it covers.
+In the case of the usual topology on a topological space, this means that the open cover contains
+every open set which it covers.
+
+Note this has no relation to a closed subset of a topological space.
+-/
+def IsClosed {X : C} (S : Sieve X) : Prop :=
+  ∀ ⦃Y : C⦄ (f : Y ⟶ X), J₁.Covers S f → S f
+#align category_theory.grothendieck_topology.is_closed CategoryTheory.GrothendieckTopology.IsClosed
+
+/-- If `S` is `J₁`-closed, then `S` covers exactly the arrows it contains. -/
+theorem covers_iff_mem_of_isClosed {X : C} {S : Sieve X} (h : J₁.IsClosed S) {Y : C} (f : Y ⟶ X) :
+    J₁.Covers S f ↔ S f :=
+  ⟨h _, J₁.arrow_max _ _⟩
+#align category_theory.grothendieck_topology.covers_iff_mem_of_closed CategoryTheory.GrothendieckTopology.covers_iff_mem_of_isClosed
+
+/-- Being `J`-closed is stable under pullback. -/
+theorem isClosed_pullback {X Y : C} (f : Y ⟶ X) (S : Sieve X) :
+    J₁.IsClosed S → J₁.IsClosed (S.pullback f) :=
+  fun hS Z g hg => hS (g ≫ f) (by rwa [J₁.covers_iff, Sieve.pullback_comp])
+#align category_theory.grothendieck_topology.is_closed_pullback CategoryTheory.GrothendieckTopology.isClosed_pullback
+
+/-- The closure of a sieve `S` is the largest closed sieve which contains `S` (justifying the name
+"closure").
+-/
+theorem le_close_of_isClosed {X : C} {S T : Sieve X} (h : S ≤ T) (hT : J₁.IsClosed T) :
+    J₁.close S ≤ T :=
+  fun _ f hf => hT _ (J₁.superset_covering (Sieve.pullback_monotone f h) hf)
+#align category_theory.grothendieck_topology.le_close_of_is_closed CategoryTheory.GrothendieckTopology.le_close_of_isClosed
+
+/-- The closure of a sieve is closed. -/
+theorem close_isClosed {X : C} (S : Sieve X) : J₁.IsClosed (J₁.close S) :=
+  fun _ g hg => J₁.arrow_trans g _ S hg fun _ hS => hS
+#align category_theory.grothendieck_topology.close_is_closed CategoryTheory.GrothendieckTopology.close_isClosed
+
+/-- The sieve `S` is closed iff its closure is equal to itself. -/
+theorem isClosed_iff_close_eq_self {X : C} (S : Sieve X) : J₁.IsClosed S ↔ J₁.close S = S := by
+  constructor
+  · intro h
+    apply le_antisymm
+    · intro Y f hf
+      rw [← J₁.covers_iff_mem_of_isClosed h]
+      apply hf
+    · apply J₁.le_close
+  · intro e
+    rw [← e]
+    apply J₁.close_isClosed
+#align category_theory.grothendieck_topology.is_closed_iff_close_eq_self CategoryTheory.GrothendieckTopology.isClosed_iff_close_eq_self
+
+theorem close_eq_self_of_isClosed {X : C} {S : Sieve X} (hS : J₁.IsClosed S) : J₁.close S = S :=
+  (J₁.isClosed_iff_close_eq_self S).1 hS
+#align category_theory.grothendieck_topology.close_eq_self_of_is_closed CategoryTheory.GrothendieckTopology.close_eq_self_of_isClosed
+
+/-- Closing under `J` is stable under pullback. -/
+theorem pullback_close {X Y : C} (f : Y ⟶ X) (S : Sieve X) :
+    J₁.close (S.pullback f) = (J₁.close S).pullback f := by
+  apply le_antisymm
+  · refine' J₁.le_close_of_isClosed (Sieve.pullback_monotone _ (J₁.le_close S)) _
+    apply J₁.isClosed_pullback _ _ (J₁.close_isClosed _)
+  · intro Z g hg
+    change _ ∈ J₁ _
+    rw [← Sieve.pullback_comp]
+    apply hg
+#align category_theory.grothendieck_topology.pullback_close CategoryTheory.GrothendieckTopology.pullback_close
+
+@[mono]
+theorem monotone_close {X : C} : Monotone (J₁.close : Sieve X → Sieve X) :=
+  fun _ S₂ h => J₁.le_close_of_isClosed (h.trans (J₁.le_close _)) (J₁.close_isClosed S₂)
+#align category_theory.grothendieck_topology.monotone_close CategoryTheory.GrothendieckTopology.monotone_close
+
+@[simp]
+theorem close_close {X : C} (S : Sieve X) : J₁.close (J₁.close S) = J₁.close S :=
+  le_antisymm (J₁.le_close_of_isClosed le_rfl (J₁.close_isClosed S))
+    (J₁.monotone_close (J₁.le_close _))
+#align category_theory.grothendieck_topology.close_close CategoryTheory.GrothendieckTopology.close_close
+
+/--
+The sieve `S` is in the topology iff its closure is the maximal sieve. This shows that the closure
+operator determines the topology.
+-/
+theorem close_eq_top_iff_mem {X : C} (S : Sieve X) : J₁.close S = ⊤ ↔ S ∈ J₁ X := by
+  constructor
+  · intro h
+    apply J₁.transitive (J₁.top_mem X)
+    intro Y f hf
+    change J₁.close S f
+    rwa [h]
+  · intro hS
+    rw [eq_top_iff]
+    intro Y f _
+    apply J₁.pullback_stable _ hS
+#align category_theory.grothendieck_topology.close_eq_top_iff_mem CategoryTheory.GrothendieckTopology.close_eq_top_iff_mem
+
+/-- A Grothendieck topology induces a natural family of closure operators on sieves. -/
+@[simps!]
+def closureOperator (X : C) : ClosureOperator (Sieve X) :=
+  ClosureOperator.mk' J₁.close
+    (fun _ S₂ h => J₁.le_close_of_isClosed (h.trans (J₁.le_close _)) (J₁.close_isClosed S₂))
+    J₁.le_close fun S => J₁.le_close_of_isClosed le_rfl (J₁.close_isClosed S)
+#align category_theory.grothendieck_topology.closure_operator CategoryTheory.GrothendieckTopology.closureOperator
+
+@[simp]
+theorem closed_iff_closed {X : C} (S : Sieve X) :
+    S ∈ (J₁.closureOperator X).closed ↔ J₁.IsClosed S :=
+  (J₁.isClosed_iff_close_eq_self S).symm
+#align category_theory.grothendieck_topology.closed_iff_closed CategoryTheory.GrothendieckTopology.closed_iff_closed
+
+end GrothendieckTopology
+
+/--
+The presheaf sending each object to the set of `J`-closed sieves on it. This presheaf is a `J`-sheaf
+(and will turn out to be a subobject classifier for the category of `J`-sheaves).
+-/
+@[simps]
+def Functor.closedSieves : Cᵒᵖ ⥤ Type max v u where
+  obj X := { S : Sieve X.unop // J₁.IsClosed S }
+  map f S := ⟨S.1.pullback f.unop, J₁.isClosed_pullback f.unop _ S.2⟩
+#align category_theory.functor.closed_sieves CategoryTheory.Functor.closedSieves
+
+/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
+The proof of this is adapted from [MM92], Chatper III, Section 7, Lemma 1.
+-/
+theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) := by
+  intro X S hS
+  rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
+  refine' ⟨_, _⟩
+  · rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂
+    simp only [Functor.closedSieves_obj]
+    ext Y
+    intro f
+    dsimp only [Subtype.coe_mk]
+    rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]
+    have q : ∀ ⦃Z : C⦄ (g : Z ⟶ X) (_ : S g), M.pullback g = N.pullback g :=
+      fun Z g hg => congr_arg Subtype.val ((hM₂ g hg).trans (hN₂ g hg).symm)
+    have MSNS : M ⊓ S = N ⊓ S := by
+      ext Z
+      intro g
+      rw [Sieve.inter_apply, Sieve.inter_apply]
+      simp only [and_comm]
+      apply and_congr_right
+      intro hg
+      rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]
+    constructor
+    · intro hf
+      rw [J₁.covers_iff]
+      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
+      rw [← MSNS]
+      apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
+    · intro hf
+      rw [J₁.covers_iff]
+      apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
+      rw [MSNS]
+      apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)
+  · intro x hx
+    rw [Presieve.compatible_iff_sieveCompatible] at hx
+    let M := Sieve.bind S fun Y f hf => (x f hf).1
+    have : ∀ ⦃Y⦄ (f : Y ⟶ X) (hf : S f), M.pullback f = (x f hf).1 := by
+      intro Y f hf
+      apply le_antisymm
+      · rintro Z u ⟨W, g, f', hf', hg : (x f' hf').1 _, c⟩
+        rw [Sieve.pullback_eq_top_iff_mem,
+          ← show (x (u ≫ f) _).1 = (x f hf).1.pullback u from congr_arg Subtype.val (hx f u hf)]
+        conv_lhs => congr; congr; rw [← c] -- Porting note: Originally `simp_rw [← c]`
+        rw [show (x (g ≫ f') _).1 = _ from congr_arg Subtype.val (hx f' g hf')]
+        apply Sieve.pullback_eq_top_of_mem _ hg
+      · apply Sieve.le_pullback_bind S fun Y f hf => (x f hf).1
+    refine' ⟨⟨_, J₁.close_isClosed M⟩, _⟩
+    · intro Y f hf
+      simp only [Functor.closedSieves_obj]
+      ext1
+      dsimp
+      rw [← J₁.pullback_close, this _ hf]
+      apply le_antisymm (J₁.le_close_of_isClosed le_rfl (x f hf).2) (J₁.le_close _)
+#align category_theory.classifier_is_sheaf CategoryTheory.classifier_isSheaf
+
+/-- If presheaf of `J₁`-closed sieves is a `J₂`-sheaf then `J₁ ≤ J₂`. Note the converse is true by
+`classifier_isSheaf` and `isSheaf_of_le`.
+-/
+theorem le_topology_of_closedSieves_isSheaf {J₁ J₂ : GrothendieckTopology C}
+    (h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)) : J₁ ≤ J₂ := by
+  intro X S hS
+  rw [← J₂.close_eq_top_iff_mem]
+  have : J₂.IsClosed (⊤ : Sieve X) := by
+    intro Y f _
+    trivial
+  suffices (⟨J₂.close S, J₂.close_isClosed S⟩ : Subtype _) = ⟨⊤, this⟩ by
+    rw [Subtype.ext_iff] at this
+    exact this
+  apply (h S hS).isSeparatedFor.ext
+  · intro Y f hf
+    simp only [Functor.closedSieves_obj]
+    ext1
+    dsimp
+    rw [Sieve.pullback_top, ← J₂.pullback_close, S.pullback_eq_top_of_mem hf,
+      J₂.close_eq_top_iff_mem]
+    apply J₂.top_mem
+#align category_theory.le_topology_of_closed_sieves_is_sheaf CategoryTheory.le_topology_of_closedSieves_isSheaf
+
+/-- If being a sheaf for `J₁` is equivalent to being a sheaf for `J₂`, then `J₁ = J₂`. -/
+theorem topology_eq_iff_same_sheaves {J₁ J₂ : GrothendieckTopology C} :
+    J₁ = J₂ ↔ ∀ P : Cᵒᵖ ⥤ Type max v u, Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P := by
+  constructor
+  · rintro rfl
+    intro P
+    rfl
+  · intro h
+    apply le_antisymm
+    · apply le_topology_of_closedSieves_isSheaf
+      rw [h]
+      apply classifier_isSheaf
+    · apply le_topology_of_closedSieves_isSheaf
+      rw [← h]
+      apply classifier_isSheaf
+#align category_theory.topology_eq_iff_same_sheaves CategoryTheory.topology_eq_iff_same_sheaves
+
+/--
+A closure (increasing, inflationary and idempotent) operation on sieves that commutes with pullback
+induces a Grothendieck topology.
+In fact, such operations are in bijection with Grothendieck topologies.
+-/
+@[simps]
+def topologyOfClosureOperator (c : ∀ X : C, ClosureOperator (Sieve X))
+    (hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), c _ (S.pullback f) = (c _ S).pullback f) :
+    GrothendieckTopology C where
+  sieves X := { S | c X S = ⊤ }
+  top_mem' X := top_unique ((c X).le_closure _)
+  pullback_stable' X Y S f hS := by
+    rw [Set.mem_setOf_eq] at hS
+    rw [Set.mem_setOf_eq, hc, hS, Sieve.pullback_top]
+  transitive' X S hS R hR := by
+    rw [Set.mem_setOf_eq] at hS
+    rw [Set.mem_setOf_eq, ← (c X).idempotent, eq_top_iff, ← hS]
+    apply (c X).monotone fun Y f hf => _
+    intros Y f hf
+    rw [Sieve.pullback_eq_top_iff_mem, ← hc]
+    apply hR hf
+#align category_theory.topology_of_closure_operator CategoryTheory.topologyOfClosureOperator
+
+/--
+The topology given by the closure operator `J.close` on a Grothendieck topology is the same as `J`.
+-/
+theorem topologyOfClosureOperator_self :
+    (topologyOfClosureOperator J₁.closureOperator fun X Y => J₁.pullback_close) = J₁ := by
+  ext (X S)
+  apply GrothendieckTopology.close_eq_top_iff_mem
+#align category_theory.topology_of_closure_operator_self CategoryTheory.topologyOfClosureOperator_self
+
+theorem topologyOfClosureOperator_close (c : ∀ X : C, ClosureOperator (Sieve X))
+    (pb : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), c Y (S.pullback f) = (c X S).pullback f) (X : C)
+    (S : Sieve X) : (topologyOfClosureOperator c pb).close S = c X S := by
+  ext Y
+  intro f
+  change c _ (Sieve.pullback f S) = ⊤ ↔ c _ S f
+  rw [pb, Sieve.pullback_eq_top_iff_mem]
+#align category_theory.topology_of_closure_operator_close CategoryTheory.topologyOfClosureOperator_close
+
+end CategoryTheory