feat(CategoryTheory): the bicategory of profunctors

Mathlib.lean

@@ -3243,6 +3243,7 @@ public import Mathlib.CategoryTheory.Products.Basic
 public import Mathlib.CategoryTheory.Products.Bifunctor
 public import Mathlib.CategoryTheory.Products.Unitor
 public import Mathlib.CategoryTheory.Profunctor.Basic
+public import Mathlib.CategoryTheory.Profunctor.Bicategory
 public import Mathlib.CategoryTheory.Profunctor.Comp
 public import Mathlib.CategoryTheory.Quotient
 public import Mathlib.CategoryTheory.Quotient.Linear

Mathlib/CategoryTheory/Profunctor/Bicategory.lean

@@ -0,0 +1,83 @@
+/-
+Copyright (c) 2026 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+module
+
+public import Mathlib.CategoryTheory.Profunctor.Comp
+
+/-!
+# The Profunctor Bicategory
+
+This file defines the bicategory `ProfCat` whose objects are categories and whose 1-morphisms are
+profunctors.
+-/
+
+@[expose] public section
+
+universe w v u
+
+namespace CategoryTheory
+
+/-- The bicategory of categories where the 1-morphisms are profunctors. -/
+@[nolint checkUnivs]
+structure ProfCat where
+  of ::
+  /-- The objects of the bicategory are types... -/
+  obj : Type u
+  /-- ... bundled with a category instance. -/
+  [str : Category.{v} obj]
+
+initialize_simps_projections ProfCat (-str)
+
+instance : CoeSort ProfCat (Type u) :=
+  ⟨ProfCat.obj⟩
+
+attribute [instance] ProfCat.str
+
+open Limits Types Profunctor
+
+namespace Profunctor
+
+section
+
+@[reassoc (attr := simp)]
+lemma pentagon {C D E F G : Type u} [Category* C] [Category* D] [Category* E]
+  [Category* F] [Category* G] (P : Profunctor.{max u w} C D) (Q : Profunctor.{max u w} D E)
+    (R : Profunctor.{max u w} E F) (S : Profunctor.{max u w} F G) :
+    whiskerRight S (P.associator Q R).hom ≫
+        (P.associator (Q.comp R) S).hom ≫ P.whiskerLeft (Q.associator R S).hom =
+      ((P.comp Q).associator R S).hom ≫ (P.associator Q (R.comp S)).hom := by
+  ext _ _ ⟨_, ⟨_, ⟨_, _, _⟩, _⟩, _⟩
+  rfl
+
+set_option backward.isDefEq.respectTransparency false in
+attribute [local simp] Types.chosenCoend_def in
+@[reassoc (attr := simp)]
+lemma triangle {C D E : Type u} [Category* C] [Category.{u} D] [Category* E]
+    (P : Profunctor.{u} C D) (Q : Profunctor.{u} D E) :
+    (P.associator (Profunctor.id (C := D)) Q).hom ≫ P.whiskerLeft (Q.leftUnitor.hom) =
+      whiskerRight Q (P.rightUnitor.hom) := by
+  ext _ _ ⟨_, ⟨_, _, g⟩, _⟩
+  dsimp [chosenCoend.map_apply, Quot.map]
+  symm
+  apply Quot.sound
+  rw [coendRel_iff]
+  exact ⟨g, by simp⟩
+
+end
+
+end Profunctor
+
+instance : Bicategory ProfCat.{u, u} where
+  Hom X Y := Profunctor.{u} X Y
+  id X := .id
+  comp P Q := P.comp Q
+  whiskerLeft P _ _ f := P.whiskerLeft f
+  whiskerRight f R := whiskerRight R f
+  associator P Q R := P.associator Q R
+  leftUnitor P := P.leftUnitor
+  rightUnitor P := P.rightUnitor
+
+end CategoryTheory