[Merged by Bors] - feat: port/CategoryTheory.PEmpty

Mathlib.lean

@@ -259,6 +259,7 @@ import Mathlib.CategoryTheory.Monoidal.Category
 import Mathlib.CategoryTheory.NatIso
 import Mathlib.CategoryTheory.NatTrans
 import Mathlib.CategoryTheory.Opposites
+import Mathlib.CategoryTheory.PEmpty
 import Mathlib.CategoryTheory.PUnit
 import Mathlib.CategoryTheory.Pi.Basic
 import Mathlib.CategoryTheory.Products.Associator

Mathlib/CategoryTheory/PEmpty.lean

@@ -0,0 +1,86 @@
+/-
+Copyright (c) 2018 Scott Morrison. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Morrison, Bhavik Mehta
+
+! This file was ported from Lean 3 source module category_theory.pempty
+! leanprover-community/mathlib commit 2738d2ca56cbc63be80c3bd48e9ed90ad94e947d
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.CategoryTheory.DiscreteCategory
+
+/-!
+# The empty category
+
+Defines a category structure on `PEmpty`, and the unique functor `PEmpty ⥤ C` for any category `C`.
+-/
+
+-- Porting note: this file stressed Lean a good deal despite its length
+
+universe w v u
+-- morphism levels before object levels. See note [CategoryTheory universes].
+namespace CategoryTheory
+
+namespace Functor
+
+variable (C : Type u) [Category.{v} C]
+/- Porting note: `aesop_cat` could not close any of the goals `tidy` did. 
+Switched to more explict construction -/
+/-- Equivalence between two empty categories. -/
+def emptyEquivalence : Discrete.{w} PEmpty ≌ Discrete.{v} PEmpty where 
+  functor := 
+    { obj := PEmpty.elim ∘ Discrete.as
+      map := fun {X} _ _ => X.as.elim 
+      map_comp := fun {X} _ _ _ _ => X.as.elim }
+  inverse := 
+    { obj := PEmpty.elim ∘ Discrete.as
+      map := fun {X} _ _ => X.as.elim  
+      map_comp := fun {X} _ _ _ _ => X.as.elim }
+  unitIso := 
+    { hom := 
+        { app := fun X => X.as.elim 
+          naturality := fun {X} _ _ => X.as.elim }
+      inv := 
+        { app := fun X => X.as.elim 
+          naturality := fun {X} _ _ => X.as.elim } }
+  counitIso := 
+    { hom := 
+        { app := fun X => X.as.elim 
+          naturality := fun {X} _ _ => X.as.elim }
+      inv := 
+        { app := fun X => X.as.elim 
+          naturality := fun {X} _ _ => X.as.elim } }
+  functor_unitIso_comp := fun X => X.as.elim  
+#align category_theory.functor.empty_equivalence CategoryTheory.Functor.emptyEquivalence
+
+/-- The canonical functor out of the empty category. -/
+def empty : Discrete.{w} PEmpty ⥤ C :=
+  Discrete.functor PEmpty.elim
+#align category_theory.functor.empty CategoryTheory.Functor.empty
+
+variable {C}
+
+/-- Any two functors out of the empty category are isomorphic. -/
+def emptyExt (F G : Discrete.{w} PEmpty ⥤ C) : F ≅ G :=
+  Discrete.natIso fun x => x.as.elim
+#align category_theory.functor.empty_ext CategoryTheory.Functor.emptyExt
+
+/-- Any functor out of the empty category is isomorphic to the canonical functor from the empty
+category.
+-/
+def uniqueFromEmpty (F : Discrete.{w} PEmpty ⥤ C) : F ≅ empty C :=
+  emptyExt _ _
+#align category_theory.functor.unique_from_empty CategoryTheory.Functor.uniqueFromEmpty
+
+/-- Any two functors out of the empty category are *equal*. You probably want to use
+`emptyExt` instead of this.
+-/
+theorem empty_ext' (F G : Discrete.{w} PEmpty ⥤ C) : F = G :=
+  Functor.ext (fun x => x.as.elim) fun x _ _ => x.as.elim
+#align category_theory.functor.empty_ext' CategoryTheory.Functor.empty_ext'
+
+end Functor
+
+end CategoryTheory
+