Merge remote-tracking branch 'origin/master' into coq-8.17

Construction/Comma/Adjunction.v

@@ -56,7 +56,7 @@ Context {G : C ⟶ D}.
 Definition πD := comma_proj1.
 Definition πC := comma_proj2.
 
-#[local] Notation "⟨πD,πC⟩" := comma_proj (at level 100).
+#[local] Notation "⟨πD,πC⟩" := comma_proj (at level 0).
 
 Record lawvere_equiv := {
   lawvere_iso : F ↓ Id[C] ≅[Cat] Id[D] ↓ G;

Functor/Applicative.v

@@ -36,4 +36,4 @@ Arguments Applicative {_ _} F.
 Arguments pure {C _ F _ _ _}.
 
 Notation "pure[ F ]" := (@pure _ _ F _ _ _)
-  (at level 9, format "pure[ F ]") : morphism_scope.
+  (at level 0, format "pure[ F ]") : morphism_scope.

Functor/Diagonal.v

@@ -26,12 +26,12 @@ Program Instance Diagonal_Product `(C : Category) : C ⟶ C ∏ C.
 
 End Diagonal_Product.
 
-Notation "Δ[ J ]( c )" := (Diagonal J c) (at level 90, format "Δ[ J ]( c )") : functor_scope.
+Notation "Δ[ J ]( c )" := (Diagonal J c) (at level 0, format "Δ[ J ]( c )") : functor_scope.
 
 Require Import Category.Instance.One.
 
-Notation "Δ( c )" := (Diagonal _ c) (at level 90, format "Δ( c )") : functor_scope.
-Notation "=( c )" := (Diagonal 1 c) (at level 90, format "=( c )") : functor_scope.
+Notation "Δ( c )" := (Diagonal _ c) (at level 0, format "Δ( c )") : functor_scope.
+Notation "=( c )" := (Diagonal 1 c) (at level 0, format "=( c )") : functor_scope.
 
 Definition Δ {C J : Category} := @Diagonal C J.
 

Functor/Product/Internal.v

@@ -25,4 +25,4 @@ Next Obligation.
 Qed.
 
 Notation "×( C )" := (@InternalProductFunctor C _)
-  (at level 90, format "×( C )") : functor_scope.
+  (at level 0, format "×( C )") : functor_scope.

Functor/Structure/Monoidal/Pure.v

@@ -53,4 +53,4 @@ End Pure.
 Arguments pure {C _ F _ _ A}.
 
 Notation "pure[ F ]" := (@pure _ _ F _ _ _)
-  (at level 9, format "pure[ F ]") : morphism_scope.
+  (at level 0, format "pure[ F ]") : morphism_scope.

Instance/Cones/Limit.v

@@ -11,7 +11,7 @@ Program Definition Limit_Cones `(F : J ⟶ C) `{T : @Terminal (Cones F)} :
   Limit F := {|
   limit_cone := @terminal_obj _ T;
   ump_limits := fun N =>
-    {| unique_obj := `1 @one _ T N
-     ; unique_property := `2 @one _ T N
+    {| unique_obj := `1 (@one _ T N)
+     ; unique_property := `2 (@one _ T N)
      ; uniqueness       := fun v H => @one_unique _ T N (@one _ T N) (v; H) |}
 |}.

Instance/Coq/Applicative.v

@@ -35,11 +35,11 @@ Arguments ap   {F _ _ _} _ x.
 
 Coercion Applicative_Functor `{@Applicative F} : Coq ⟶ Coq := F.
 
-Notation "pure[ F ]" := (@pure F _ _) (at level 19, F at next level).
-Notation "pure[ F G ]" := (@pure (fun x => F (G x)) _ _) (at level 9).
+Notation "pure[ F ]" := (@pure F _ _) (at level 0, F at next level).
+Notation "pure[ F G ]" := (@pure (fun x => F (G x)) _ _) (at level 0).
 
-Notation "ap[ F ]" := (@ap F _ _ _) (at level 9).
-Notation "ap[ F G ]" := (@ap (fun x => F (G x)) _ _ _) (at level 9).
+Notation "ap[ F ]" := (@ap F _ _ _) (at level 0).
+Notation "ap[ F G ]" := (@ap (fun x => F (G x)) _ _ _) (at level 0).
 
 Infix "<*>" := ap (at level 29, left associativity).
 Notation "x <**> F" := (ap F x) (at level 29, left associativity).

Theory/Coq/Applicative.v

@@ -28,16 +28,16 @@ Module ApplicativeNotations.
 Export FunctorNotations.
 
 Notation "pure[ M ]" := (@pure M _ _)
-  (at level 9, format "pure[ M ]").
+  (at level 0, format "pure[ M ]").
 Notation "pure[ M N ]" := (@pure (fun X => M (N X)) _ _)
-  (at level 9, format "pure[ M  N ]").
+  (at level 0, format "pure[ M  N ]").
 
 Notation "ap[ M ]" := (@ap M _ _ _)
-  (at level 9, format "ap[ M ]").
+  (at level 0, format "ap[ M ]").
 Notation "ap[ M N ]" := (@ap (fun X => M (N X)) _ _ _)
-  (at level 9, format "ap[ M  N ]").
+  (at level 0, format "ap[ M  N ]").
 Notation "ap[ M N O ]" := (@ap (fun X => M (N (O X))) _ _ _)
-  (at level 9, format "ap[ M  N  O ]").
+  (at level 0, format "ap[ M  N  O ]").
 
 Infix "<*>" := ap (at level 29, left associativity).
 Infix "<*[ M ]>" := (@ap M _ _ _)

Theory/Coq/Functor.v

@@ -32,22 +32,22 @@ Notation "x <&> f" := (fmap f x)
   (at level 29, left associativity, only parsing).
 
 Notation "fmap[ M ]" := (@fmap M _ _ _)
-  (at level 9, format "fmap[ M ]").
+  (at level 0, format "fmap[ M ]").
 Notation "fmap[ M N ]" := (@fmap (λ X, M (N X)) _ _ _)
-  (at level 9, format "fmap[ M  N ]").
+  (at level 0, format "fmap[ M  N ]").
 Notation "fmap[ M N O ]" := (@fmap (λ X, M (N (O X))) _ _ _)
-  (at level 9, format "fmap[ M  N  O ]").
+  (at level 0, format "fmap[ M  N  O ]").
 
 Infix ">$<" := contramap (at level 29, left associativity, only parsing).
 Notation "x >&< f" :=
   (contramap f x) (at level 29, left associativity, only parsing).
 
 Notation "contramap[ M ]" := (@contramap M _ _ _)
-  (at level 9, format "contramap[ M ]").
+  (at level 0, format "contramap[ M ]").
 Notation "contramap[ M N ]" := (@contramap (λ X, M (N X)) _ _ _)
-  (at level 9, format "contramap[ M  N ]").
+  (at level 0, format "contramap[ M  N ]").
 Notation "contramap[ M N O ]" := (@contramap (λ X, M (N (O X))) _ _ _)
-  (at level 9, format "contramap[ M  N  O ]").
+  (at level 0, format "contramap[ M  N  O ]").
 
 End FunctorNotations.
 

Theory/Functor.v

@@ -57,9 +57,9 @@ Notation "x <&> f" := (fmap f x)
   (at level 29, left associativity, only parsing) : morphism_scope.
 
 Notation "fobj[ F ]" := (@fobj _ _ F%functor)
-  (at level 9, format "fobj[ F ]") : object_scope.
+  (at level 0, format "fobj[ F ]") : object_scope.
 Notation "fmap[ F ]" := (@fmap _ _ F%functor _ _)
-  (at level 9, format "fmap[ F ]") : morphism_scope.
+  (at level 0, format "fmap[ F ]") : morphism_scope.
 
 #[export] Hint Rewrite @fmap_id : categories.
 
@@ -168,7 +168,7 @@ Program Definition Id {C : Category} : C ⟶ C := {|
 
 Arguments Id {C} /.
 
-Notation "Id[ C ]" := (@Id C) (at level 9, format "Id[ C ]") : functor_scope.
+Notation "Id[ C ]" := (@Id C) (at level 0, format "Id[ C ]") : functor_scope.
 
 Program Definition Compose {C D E : Category}
         (F : D ⟶ E) (G : C ⟶ D) : C ⟶ E := {|

Theory/Sheaf.v

@@ -3,6 +3,8 @@ Require Import Category.Theory.Category.
 Require Import Category.Theory.Functor.
 Require Import Category.Construction.Opposite.
 Require Import Category.Instance.Fun.
+Require Import Category.Instance.Sets.
+Require Import Coq.Vectors.Vector.
 
 Generalizable All Variables.
 
@@ -16,3 +18,74 @@ Definition Presheaves {U C : Category} := [U^op, C].
 Definition Copresheaf (U C : Category) := U ⟶ C.
 
 Definition Copresheaves {U C : Category} := [U, C].
+
+(* Custom data type to express Forall propositions on vectors over Type. *)
+Inductive ForallT {A : Type} (P : A → Type) :
+  ∀ {n : nat}, t A n → Type :=
+  | ForallT_nil : ForallT (nil A)
+  | ForallT_cons (n : nat) (x : A) (v : t A n) :
+    P x → ForallT v → ForallT (cons A x n v).
+
+Inductive ExistsT {A : Type} (P : A → Type) :
+  ∀ {n : nat}, t A n → Type :=
+  | ExistsT_cons_hd (m : nat) (x : A) (v : t A m) :
+    P x → ExistsT (cons A x m v)
+  | ExistsT_cons_tl (m : nat) (x : A) (v : t A m) :
+    ExistsT v → ExistsT (cons A x m v).
+
+(* A coverage on a category C consists of a function assigning to each object
+   U ∈ C a collection of families of morphisms { fᵢ : Uᵢ → U } (i∈I), called
+   covering families, such that
+
+   if { fᵢ : Uᵢ → U } (i∈I) is a covering family
+   and g : V → U is a morphism,
+   then there exists a covering family { hⱼ : Vⱼ → V }
+   such that each composite g ∘ hⱼ factors through some fᵢ. *)
+
+Class Site (C : Category) := {
+  covering_family (u : C) := sigT (Vector.t (∃ uᵢ, uᵢ ~> u));
+
+  coverage (u : C) : covering_family u;
+
+  coverage_condition
+    (u  : C) (fs : covering_family u)
+    (v  : C) (g  : v ~> u) :
+    ∃ hs : covering_family v,
+      ForallT (A := ∃ vⱼ, vⱼ ~> v) (λ '(vⱼ; hⱼ),
+          ExistsT (A := ∃ uᵢ, uᵢ ~> u) (λ '(uᵢ; fᵢ),
+              ∃ k : vⱼ ~> uᵢ, fᵢ ∘ k ≈ g ∘ hⱼ)
+            (`2 fs))
+        (`2 hs)
+}.
+
+(* If { fᵢ : Uᵢ → U } (i∈I) is a family of morphisms with codomain U,
+   a presheaf X : Cᵒᵖ → Set is called a sheaf for this family if:
+
+   for any collection of elements xᵢ ∈ X(Uᵢ)
+   such that,
+     whenever g : V → Uᵢ and h : V → Uⱼ
+     are such that fᵢ ∘ g = fⱼ ∘ h,
+     we have X(g)(xᵢ) = X(h)(xⱼ),
+   then there exists a unique x ∈ X(U)
+   such that X(fᵢ)(x) = xᵢ for all i . *)
+
+Class Sheaf `{@Site C} (X : Presheaf C Sets) := {
+  restriction :
+    ∀ u : C,
+      let '(i; f) := coverage u in
+      ForallT
+        (A := ∃ uᵢ, uᵢ ~> u)
+        (λ '(uᵢ; fᵢ),
+            ∀ xᵢ : X uᵢ,
+              (∀ (v : C) (g : v ~> uᵢ),
+                  ForallT
+                    (A := ∃ uⱼ, uⱼ ~> u)
+                    (λ '(uⱼ; fⱼ),
+                      ∀ h : v ~> uⱼ,
+                        fᵢ ∘ g ≈ fⱼ ∘ h →
+                        ∀ xⱼ : X uⱼ,
+                          fmap[X] g xᵢ = fmap[X] h xⱼ)
+                    f) →
+              ∃! x : X u, fmap[X] fᵢ x = xᵢ)
+        f
+}.