feat(topology/sheaves): The empty component of a sheaf is terminal (#10578)

Author: erdOne

src/category_theory/sites/sheaf.lean

@@ -168,6 +168,18 @@ def Sheaf_equiv_SheafOfTypes : Sheaf J (Type w) ≌ SheafOfTypes J :=
 instance : inhabited (Sheaf (⊥ : grothendieck_topology C) (Type w)) :=
 ⟨(Sheaf_equiv_SheafOfTypes _).inverse.obj (default _)⟩
 
+variables {J} {A}
+
+/-- If the empty sieve is a cover of `X`, then `F(X)` is terminal. -/
+def Sheaf.is_terminal_of_bot_cover (F : Sheaf J A) (X : C) (H : ⊥ ∈ J X) :
+  is_terminal (F.1.obj (op X)) :=
+begin
+  apply_with is_terminal.of_unique { instances := ff },
+  intro Y,
+  choose t h using F.2 Y _ H (by tidy) (by tidy),
+  exact ⟨⟨t⟩, λ a, h.2 a (by tidy)⟩
+end
+
 end category_theory
 
 namespace category_theory

src/topology/sheaves/sheaf_condition/sites.lean

@@ -514,6 +514,15 @@ variables {C : Type u} [category.{v} C] [limits.has_products C]
 variables {X : Top.{v}} {ι : Type*} {B : ι → opens X}
 variables (F : presheaf C X) (F' : sheaf C X) (h : opens.is_basis (set.range B))
 
+/-- The empty component of a sheaf is terminal -/
+def is_terminal_of_empty (F : sheaf C X) : limits.is_terminal (F.val.obj (op ∅)) :=
+((presheaf.Sheaf_spaces_to_sheaf_sites C X).obj F).is_terminal_of_bot_cover ∅ (by tidy)
+
+/-- A variant of `is_terminal_of_empty` that is easier to `apply`. -/
+def is_terminal_of_eq_empty (F : X.sheaf C) {U : opens X} (h : U = ∅) :
+  limits.is_terminal (F.val.obj (op U)) :=
+by convert F.is_terminal_of_empty
+
 /-- If a family `B` of open sets forms a basis of the topology on `X`, and if `F'`
     is a sheaf on `X`, then a homomorphism between a presheaf `F` on `X` and `F'`
     is equivalent to a homomorphism between their restrictions to the indexing type