[Merged by Bors] - feat(category_theory/sites): Sheaves of structures

src/category_theory/sites/sheaf.lean

@@ -0,0 +1,277 @@
+/-
+Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kevin Buzzard, Bhavik Mehta
+-/
+
+import category_theory.sites.sheaf_of_types
+import category_theory.limits.yoneda
+import category_theory.limits.preserves.shapes.equalizers
+import category_theory.limits.preserves.shapes.products
+import category_theory.concrete_category
+
+/-!
+# Sheaves taking values in a category
+
+If C is a category with a Grothendieck topology, we define the notion of a sheaf taking values in
+an arbitrary category `A`. We follow the definition in https://stacks.math.columbia.edu/tag/00VR,
+noting that the presheaf of sets "defined above" can be seen in the comments between tags 00VQ and
+00VR on the page https://stacks.math.columbia.edu/tag/00VL. The advantage of this definition is
+that we need no assumptions whatsoever on `A` other than the assumption that the morphisms in `C`
+and `A` live in the same universe.
+
+* An `A`-valued presheaf `P : Cᵒᵖ ⥤ A` is defined to be a sheaf (for the topology `J`) iff for
+  every `X : A`, the type-valued presheaves of sets given by sending `U : Cᵒᵖ` to `Hom_{A}(X, P U)`
+  are all sheaves of sets, see `category_theory.presheaf.is_sheaf`.
+* When `A = Type`, this recovers the basic definition of sheaves of sets, see
+  `category_theory.is_sheaf_iff_is_sheaf_of_type`.
+* An alternate definition when `C` is small, has pullbacks and `A` has products is given by an
+  equalizer condition `category_theory.presheaf.is_sheaf'`. This is equivalent to the earlier
+  definition, shown in `category_theory.presheaf.is_sheaf_iff_is_sheaf'`.
+* When `A = Type`, this is *definitionally* equal to the equalizer condition for presieves in
+  `category_theory.sites.sheaf_of_types`.
+* When `A` has limits and there is a functor `s : A ⥤ Type` which is faithful, reflects isomorphisms
+  and preserves limits, then `P : C^op ⥤ A` is a sheaf iff the underlying presheaf of types
+  `P ⋙ s : C^op ⥤ Type` is a sheaf (`category_theory.presheaf.is_sheaf_iff_is_sheaf_forget`).
+  Cf https://stacks.math.columbia.edu/tag/0073, which is a weaker version of this statement (it's
+  only over spaces, not sites) and https://stacks.math.columbia.edu/tag/00YR (a), which
+  additionally assumes filtered colimits.
+-/
+
+universes v v' u' u
+
+noncomputable theory
+
+namespace category_theory
+
+open opposite category_theory category limits sieve classical
+
+namespace presheaf
+
+variables {C : Type u} [category.{v} C]
+variables {A : Type u'} [category.{v} A]
+variables (J : grothendieck_topology C)
+
+-- We follow https://stacks.math.columbia.edu/tag/00VL definition 00VR
+
+/--
+A sheaf of A is a presheaf P : C^op => A such that for every X : A, the
+presheaf of types given by sending U : C to Hom_{A}(X, P U) is a sheaf of types.
+
+https://stacks.math.columbia.edu/tag/00VR
+-/
+def is_sheaf (P : Cᵒᵖ ⥤ A) : Prop :=
+∀ X : A, presieve.is_sheaf J (P ⋙ coyoneda.obj (op X))
+
+end presheaf
+
+variables {C : Type u} [category.{v} C]
+variables (J : grothendieck_topology C)
+variables (A : Type u') [category.{v} A]
+
+/-- The category of sheaves taking values in `A` on a grothendieck topology. -/
+@[derive category]
+def Sheaf : Type* :=
+{P : Cᵒᵖ ⥤ A // presheaf.is_sheaf J P}
+
+/-- The inclusion functor from sheaves to presheaves. -/
+@[simps {rhs_md := semireducible}, derive [full, faithful]]
+def Sheaf_to_presheaf : Sheaf J A ⥤ (Cᵒᵖ ⥤ A) :=
+full_subcategory_inclusion (presheaf.is_sheaf J)
+
+lemma is_sheaf_iff_is_sheaf_of_type (P : Cᵒᵖ ⥤ Type v) :
+  presheaf.is_sheaf J P ↔ presieve.is_sheaf J P :=
+begin
+  split,
+  { intros hP,
+    exact presieve.is_sheaf_iso J (coyoneda.iso_comp_punit _) (hP punit) },
+  { intros hP X Y S hS z hz,
+    refine ⟨λ x, (hP S hS).amalgamate (λ Z f hf, z f hf x) _, _, _⟩,
+    { intros Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ h,
+      exact congr_fun (hz g₁ g₂ hf₁ hf₂ h) x },
+    { intros Z f hf,
+      ext x,
+      apply presieve.is_sheaf_for.valid_glue },
+    { intros y hy,
+      ext x,
+      apply (hP S hS).is_separated_for.ext,
+      intros Y' f hf,
+      rw [presieve.is_sheaf_for.valid_glue _ _ _ hf, ← hy _ hf],
+      refl } }
+end
+
+/--
+The category of sheaves taking values in Type is the same as the category of set-valued sheaves.
+-/
+@[simps]
+def Sheaf_equiv_SheafOfTypes : Sheaf J (Type v) ≌ SheafOfTypes J :=
+{ functor :=
+  { obj := λ S, ⟨S.1, (is_sheaf_iff_is_sheaf_of_type _ _).1 S.2⟩,
+    map := λ S₁ S₂ f, f },
+  inverse :=
+  { obj := λ S, ⟨S.1, (is_sheaf_iff_is_sheaf_of_type _ _).2 S.2⟩,
+    map := λ S₁ S₂ f, f },
+  unit_iso := nat_iso.of_components (λ X, ⟨𝟙 _, 𝟙 _, by tidy, by tidy⟩) (by tidy),
+  counit_iso := nat_iso.of_components (λ X, ⟨𝟙 _, 𝟙 _, by tidy, by tidy⟩) (by tidy) }
+
+instance : inhabited (Sheaf (⊥ : grothendieck_topology C) (Type v)) :=
+⟨(Sheaf_equiv_SheafOfTypes _).inverse.obj (default _)⟩
+
+end category_theory
+
+namespace category_theory
+
+open opposite category_theory category limits sieve classical
+
+namespace presheaf
+
+-- Under here is the equalizer story, which is equivalent if A has products (and doesn't
+-- make sense otherwise). It's described in https://stacks.math.columbia.edu/tag/00VL,
+-- between 00VQ and 00VR.
+
+variables {C : Type v} [small_category C]
+variables {A : Type u} [category.{v} A]
+variables (J : grothendieck_topology C)
+variables {U : C} (R : presieve U)
+variables (P : Cᵒᵖ ⥤ A)
+
+section
+
+variables [has_products A]
+
+/--
+The middle object of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram
+of https://stacks.math.columbia.edu/tag/00VM.
+-/
+def first_obj : A :=
+∏ (λ (f : Σ V, {f : V ⟶ U // R f}), P.obj (op f.1))
+
+/--
+The left morphism of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram
+of https://stacks.math.columbia.edu/tag/00VM.
+-/
+def fork_map : P.obj (op U) ⟶ first_obj R P :=
+pi.lift (λ f, P.map f.2.1.op)
+
+variables [has_pullbacks C]
+
+/--
+The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag/00VM, which
+contains the data used to check a family of elements for a presieve is compatible.
+-/
+def second_obj : A :=
+∏ (λ (fg : (Σ V, {f : V ⟶ U // R f}) × (Σ W, {g : W ⟶ U // R g})),
+  P.obj (op (pullback fg.1.2.1 fg.2.2.1)))
+
+/-- The map `pr₀*` of https://stacks.math.columbia.edu/tag/00VM. -/
+def first_map : first_obj R P ⟶ second_obj R P :=
+pi.lift (λ fg, pi.π _ _ ≫ P.map pullback.fst.op)
+
+/-- The map `pr₁*` of https://stacks.math.columbia.edu/tag/00VM. -/
+def second_map : first_obj R P ⟶ second_obj R P :=
+pi.lift (λ fg, pi.π _ _ ≫ P.map pullback.snd.op)
+
+lemma w : fork_map R P ≫ first_map R P = fork_map R P ≫ second_map R P :=
+begin
+  apply limit.hom_ext,
+  rintro ⟨⟨Y, f, hf⟩, ⟨Z, g, hg⟩⟩,
+  simp only [first_map, second_map, fork_map, limit.lift_π, limit.lift_π_assoc, assoc,
+    fan.mk_π_app, subtype.coe_mk, subtype.val_eq_coe],
+  rw [← P.map_comp, ← op_comp, pullback.condition],
+  simp,
+end
+
+/--
+An alternative definition of the sheaf condition in terms of equalizers. This is shown to be
+equivalent in `category_theory.presheaf.is_sheaf_iff_is_sheaf'`.
+-/
+def is_sheaf' (P : Cᵒᵖ ⥤ A) : Prop := ∀ (U : C) (R : presieve U) (hR : generate R ∈ J U),
+nonempty (is_limit (fork.of_ι _ (w R P)))
+
+/-- (Implementation). An auxiliary lemma to convert between sheaf conditions. -/
+def is_sheaf_for_is_sheaf_for' (P : Cᵒᵖ ⥤ A) (s : A ⥤ Type v)
+  [Π J, preserves_limits_of_shape (discrete J) s] (U : C) (R : presieve U) :
+  is_limit (s.map_cone (fork.of_ι _ (w R P))) ≃
+    is_limit (fork.of_ι _ (equalizer.presieve.w (P ⋙ s) R)) :=
+begin
+  apply equiv.trans (is_limit_map_cone_fork_equiv _ _) _,
+  apply (is_limit.postcompose_hom_equiv _ _).symm.trans (is_limit.equiv_iso_limit _),
+  { apply nat_iso.of_components _ _,
+    { rintro (_ | _),
+      { apply preserves_product.iso s },
+      { apply preserves_product.iso s } },
+    { rintro _ _ (_ | _),
+      { ext : 1,
+        dsimp [equalizer.presieve.first_map, first_map],
+        simp only [limit.lift_π, map_lift_pi_comparison, assoc, fan.mk_π_app, functor.map_comp],
+        erw pi_comparison_comp_π_assoc },
+      { ext : 1,
+        dsimp [equalizer.presieve.second_map, second_map],
+        simp only [limit.lift_π, map_lift_pi_comparison, assoc, fan.mk_π_app, functor.map_comp],
+        erw pi_comparison_comp_π_assoc },
+      { dsimp,
+        simp } } },
+  { refine fork.ext (iso.refl _) _,
+    dsimp [equalizer.fork_map, fork_map],
+    simp }
+end
+
+/-- The equalizer definition of a sheaf given by `is_sheaf'` is equivalent to `is_sheaf`. -/
+theorem is_sheaf_iff_is_sheaf' :
+  is_sheaf J P ↔ is_sheaf' J P :=
+begin
+  split,
+  { intros h U R hR,
+    refine ⟨_⟩,
+    apply coyoneda_jointly_reflects_limits,
+    intro X,
+    have q : presieve.is_sheaf_for (P ⋙ coyoneda.obj X) _ := h X.unop _ hR,
+    rw ←presieve.is_sheaf_for_iff_generate at q,
+    rw equalizer.presieve.sheaf_condition at q,
+    replace q := classical.choice q,
+    apply (is_sheaf_for_is_sheaf_for' _ _ _ _).symm q },
+  { intros h U X S hS,
+    rw equalizer.presieve.sheaf_condition,
+    refine ⟨_⟩,
+    refine is_sheaf_for_is_sheaf_for' _ _ _ _ _,
+    apply is_limit_of_preserves,
+    apply classical.choice (h _ S _),
+    simpa }
+end
+
+end
+
+section concrete
+
+variables [has_pullbacks C]
+
+/--
+For a concrete category `(A, s)` where the forgetful functor `s : A ⥤ Type v` preserves limits and
+reflects isomorphisms, and `A` has limits, an `A`-valued presheaf `P : Cᵒᵖ ⥤ A` is a sheaf iff its
+underlying `Type`-valued presheaf `P ⋙ s : Cᵒᵖ ⥤ Type` is a sheaf.
+
+Note this lemma applies for "algebraic" categories, eg groups, abelian groups and rings, but not
+for the category of topological spaces, topological rings, etc since reflecting isomorphisms doesn't
+hold.
+-/
+lemma is_sheaf_iff_is_sheaf_forget (s : A ⥤ Type v)
+  [has_limits A] [preserves_limits s] [reflects_isomorphisms s] :
+  is_sheaf J P ↔ is_sheaf J (P ⋙ s) :=
+begin
+  rw [is_sheaf_iff_is_sheaf', is_sheaf_iff_is_sheaf'],
+  apply forall_congr (λ U, _),
+  apply ball_congr (λ R hR, _),
+  letI : reflects_limits s := reflects_limits_of_reflects_isomorphisms,
+  have : is_limit (s.map_cone (fork.of_ι _ (w R P))) ≃ is_limit (fork.of_ι _ (w R (P ⋙ s))) :=
+    is_sheaf_for_is_sheaf_for' P s U R,
+  rw ←equiv.nonempty_iff_nonempty this,
+  split,
+  { exact nonempty.map (λ t, is_limit_of_preserves s t) },
+  { exact nonempty.map (λ t, is_limit_of_reflects s t) }
+end
+
+end concrete
+
+end presheaf
+
+end category_theory

src/category_theory/sites/sheaf_of_types.lean

@@ -895,7 +895,7 @@ def SheafOfTypes (J : grothendieck_topology C) : Type (max u (v+1)) :=
 {P : Cᵒᵖ ⥤ Type v // presieve.is_sheaf J P}
 
 /-- The inclusion functor from sheaves to presheaves. -/
-@[simps, derive [full, faithful]]
+@[simps {rhs_md := semireducible}, derive [full, faithful]]
 def SheafOfTypes_to_presheaf : SheafOfTypes J ⥤ (Cᵒᵖ ⥤ Type v) :=
 full_subcategory_inclusion (presieve.is_sheaf J)
 

src/category_theory/sites/sieves.lean

@@ -225,6 +225,9 @@ def gi_generate : galois_insertion (generate : presieve X → sieve X) arrows :=
 lemma le_generate (R : presieve X) : R ≤ generate R :=
 gi_generate.gc.le_u_l R
 
+@[simp] lemma generate_sieve (S : sieve X) : generate S = S :=
+gi_generate.l_u_eq S
+
 /-- If the identity arrow is in a sieve, the sieve is maximal. -/
 lemma id_mem_iff_eq_top : S (𝟙 X) ↔ S = ⊤ :=
 ⟨λ h, top_unique $ λ Y f _, by simpa using downward_closed _ h f,