feat: port CategoryTheory.Limits.KanExtension (#2601)

Co-authored-by: Johan Commelin <johan@commelin.net>

Author: adamtopaz

Mathlib.lean

@@ -304,6 +304,7 @@ import Mathlib.CategoryTheory.Limits.FullSubcategory
 import Mathlib.CategoryTheory.Limits.FunctorCategory
 import Mathlib.CategoryTheory.Limits.HasLimits
 import Mathlib.CategoryTheory.Limits.IsLimit
+import Mathlib.CategoryTheory.Limits.KanExtension
 import Mathlib.CategoryTheory.Limits.Preserves.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Finite
 import Mathlib.CategoryTheory.Limits.Preserves.Limits

Mathlib/CategoryTheory/Limits/KanExtension.lean

@@ -0,0 +1,372 @@
+/-
+Copyright (c) 2021 Adam Topaz. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta, Adam Topaz
+
+! This file was ported from Lean 3 source module category_theory.limits.kan_extension
+! leanprover-community/mathlib commit c9c9fa15fec7ca18e9ec97306fb8764bfe988a7e
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.CategoryTheory.Limits.Shapes.Terminal
+import Mathlib.CategoryTheory.PUnit
+import Mathlib.CategoryTheory.StructuredArrow
+
+/-!
+
+# Kan extensions
+
+This file defines the right and left Kan extensions of a functor.
+They exist under the assumption that the target category has enough limits
+resp. colimits.
+
+The main definitions are `Ran ι` and `Lan ι`, where `ι : S ⥤ L` is a functor.
+Namely, `Ran ι` is the right Kan extension, while `Lan ι` is the left Kan extension,
+both as functors `(S ⥤ D) ⥤ (L ⥤ D)`.
+
+To access the right resp. left adjunction associated to these, use `Ran.adjunction`
+resp. `Lan.adjunction`.
+
+# Projects
+
+A lot of boilerplate could be generalized by defining and working with pseudofunctors.
+
+-/
+
+
+noncomputable section
+
+namespace CategoryTheory
+
+open Limits
+
+universe v v₁ v₂ v₃ u₁ u₂ u₃
+
+variable {S : Type u₁} {L : Type u₂} {D : Type u₃}
+
+variable [Category.{v₁} S] [Category.{v₂} L] [Category.{v₃} D]
+
+variable (ι : S ⥤ L)
+
+namespace Ran
+
+attribute [local simp] StructuredArrow.proj
+
+/-- The diagram indexed by `Ran.index ι x` used to define `Ran`. -/
+abbrev diagram (F : S ⥤ D) (x : L) : StructuredArrow x ι ⥤ D :=
+  StructuredArrow.proj x ι ⋙ F
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran.diagram CategoryTheory.Ran.diagram
+
+variable {ι}
+
+/-- A cone over `Ran.diagram ι F x` used to define `Ran`. -/
+@[simp]
+def cone {F : S ⥤ D} {G : L ⥤ D} (x : L) (f : ι ⋙ G ⟶ F) : Cone (diagram ι F x)
+    where
+  pt := G.obj x
+  π :=
+    { app := fun i => G.map i.hom ≫ f.app i.right
+      naturality := by
+        rintro ⟨⟨il⟩, ir, i⟩ ⟨⟨jl⟩, jr, j⟩ ⟨⟨⟨fl⟩⟩, fr, ff⟩
+        dsimp at *
+        dsimp at ff
+        simp only [Category.id_comp, Category.assoc] at *
+        rw [ff]
+        have := f.naturality
+        aesop_cat }
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran.cone CategoryTheory.Ran.cone
+
+variable (ι)
+
+/-- An auxiliary definition used to define `Ran`. -/
+@[simps]
+def loc (F : S ⥤ D) [h : ∀ x, HasLimit (diagram ι F x)] : L ⥤ D
+    where
+  obj x := limit (diagram ι F x)
+  map {X Y} f :=
+    haveI : HasLimit <| StructuredArrow.map f ⋙ diagram ι F X := h Y
+    limit.pre (diagram ι F X) (StructuredArrow.map f)
+  map_id := by
+    intro l
+    haveI : HasLimit (StructuredArrow.map (𝟙 _) ⋙ diagram ι F l) := h _
+    dsimp
+    ext j
+    simp only [Category.id_comp, limit.pre_π]
+    congr 1
+    simp
+  map_comp := by
+    intro x y z f g
+    apply limit.hom_ext
+    intro j
+    -- Porting note: The fact that we need to add these instances all over the place
+    -- is certainly not ideal.
+    haveI : HasLimit (StructuredArrow.map f ⋙ diagram ι F _) := h _
+    haveI : HasLimit (StructuredArrow.map g ⋙ diagram ι F _) := h _
+    haveI : HasLimit (StructuredArrow.map (f ≫ g) ⋙ diagram ι F _) := h _
+    haveI : HasLimit (StructuredArrow.map g ⋙ StructuredArrow.map f ⋙ diagram ι F _) := h _
+    haveI : HasLimit ((StructuredArrow.map g ⋙ StructuredArrow.map f) ⋙ diagram ι F _) := h _
+    erw [limit.pre_pre, limit.pre_π, limit.pre_π]
+    congr 1
+    aesop_cat
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran.loc CategoryTheory.Ran.loc
+
+/-- An auxiliary definition used to define `Ran` and `Ran.adjunction`. -/
+@[simps]
+def equiv (F : S ⥤ D) [h : ∀ x, HasLimit (diagram ι F x)] (G : L ⥤ D) :
+    (G ⟶ loc ι F) ≃ (((whiskeringLeft _ _ _).obj ι).obj G ⟶ F)
+    where
+  toFun f :=
+    { app := fun x => f.app _ ≫ limit.π (diagram ι F (ι.obj x)) (StructuredArrow.mk (𝟙 _))
+      naturality := by
+        intro x y ff
+        dsimp only [whiskeringLeft]
+        simp only [Functor.comp_map, NatTrans.naturality_assoc, loc_map, Category.assoc]
+        congr 1
+        haveI : HasLimit (StructuredArrow.map (ι.map ff) ⋙ diagram ι F (ι.obj x)) := h _
+        erw [limit.pre_π]
+        let t : StructuredArrow.mk (𝟙 (ι.obj x)) ⟶
+          (StructuredArrow.map (ι.map ff)).obj (StructuredArrow.mk (𝟙 (ι.obj y))) :=
+          StructuredArrow.homMk ff ?_
+        convert (limit.w (diagram ι F (ι.obj x)) t).symm using 1
+        simp }
+  invFun f :=
+    { app := fun x => limit.lift (diagram ι F x) (cone _ f)
+      naturality := by
+        intro x y ff
+        apply limit.hom_ext
+        intros j
+        haveI : HasLimit (StructuredArrow.map ff ⋙ diagram ι F x) := h _
+        erw [limit.lift_pre, limit.lift_π, Category.assoc, limit.lift_π (cone _ f) j]
+        simp }
+  left_inv := by
+    intro x
+    ext k
+    apply limit.hom_ext
+    intros j
+    dsimp only [cone]
+    rw [limit.lift_π]
+    simp only [NatTrans.naturality_assoc, loc_map]
+    haveI : HasLimit (StructuredArrow.map j.hom ⋙ diagram ι F k) := h _
+    erw [limit.pre_π]
+    congr
+    rcases j with ⟨⟨⟩, _, _⟩
+    aesop_cat
+  right_inv := by aesop_cat
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran.equiv CategoryTheory.Ran.equiv
+
+end Ran
+
+/-- The right Kan extension of a functor. -/
+@[simps!]
+def ran [∀ X, HasLimitsOfShape (StructuredArrow X ι) D] : (S ⥤ D) ⥤ L ⥤ D :=
+  Adjunction.rightAdjointOfEquiv (fun F G => (Ran.equiv ι G F).symm) (by {
+    -- Porting note: was `tidy`
+    intros X' X Y f g
+    ext t
+    apply limit.hom_ext
+    intros j
+    dsimp [Ran.equiv]
+    simp })
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran CategoryTheory.ran
+
+namespace Ran
+
+variable (D)
+
+/-- The adjunction associated to `Ran`. -/
+def adjunction [∀ X, HasLimitsOfShape (StructuredArrow X ι) D] :
+    (whiskeringLeft _ _ D).obj ι ⊣ ran ι :=
+  Adjunction.adjunctionOfEquivRight _ _
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran.adjunction CategoryTheory.Ran.adjunction
+
+theorem reflective [Full ι] [Faithful ι] [∀ X, HasLimitsOfShape (StructuredArrow X ι) D] :
+    IsIso (adjunction D ι).counit := by
+  suffices : ∀ (X : S ⥤ D), IsIso (NatTrans.app (adjunction D ι).counit X)
+  · apply NatIso.isIso_of_isIso_app
+  intro F
+  suffices : ∀ (X : S), IsIso (NatTrans.app (NatTrans.app (adjunction D ι).counit F) X)
+  · apply NatIso.isIso_of_isIso_app
+  intro X
+  dsimp [adjunction, equiv]
+  simp only [Category.id_comp]
+  exact
+    IsIso.of_iso
+      ((limit.isLimit _).conePointUniqueUpToIso
+        (limitOfDiagramInitial StructuredArrow.mkIdInitial _))
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Ran.reflective CategoryTheory.Ran.reflective
+
+end Ran
+
+namespace Lan
+
+attribute [local simp] CostructuredArrow.proj
+
+/-- The diagram indexed by `Lan.index ι x` used to define `Lan`. -/
+abbrev diagram (F : S ⥤ D) (x : L) : CostructuredArrow ι x ⥤ D :=
+  CostructuredArrow.proj ι x ⋙ F
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan.diagram CategoryTheory.Lan.diagram
+
+variable {ι}
+
+/-- A cocone over `Lan.diagram ι F x` used to define `Lan`. -/
+@[simp]
+def cocone {F : S ⥤ D} {G : L ⥤ D} (x : L) (f : F ⟶ ι ⋙ G) : Cocone (diagram ι F x)
+    where
+  pt := G.obj x
+  ι :=
+    { app := fun i => f.app i.left ≫ G.map i.hom
+      naturality := by
+        rintro ⟨ir, ⟨il⟩, i⟩ ⟨jl, ⟨jr⟩, j⟩ ⟨fl, ⟨⟨fl⟩⟩, ff⟩
+        dsimp at *
+        simp only [Functor.comp_map, Category.comp_id, NatTrans.naturality_assoc]
+        rw [← G.map_comp, ff]
+        aesop_cat }
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan.cocone CategoryTheory.Lan.cocone
+
+variable (ι)
+
+/-- An auxiliary definition used to define `Lan`. -/
+@[simps]
+def loc (F : S ⥤ D) [I : ∀ x, HasColimit (diagram ι F x)] : L ⥤ D
+    where
+  obj x := colimit (diagram ι F x)
+  map {x y} f :=
+    haveI : HasColimit (CostructuredArrow.map f ⋙ diagram ι F y) := I _
+    colimit.pre (diagram ι F y) (CostructuredArrow.map f)
+  map_id := by
+    intro l
+    dsimp
+    haveI : HasColimit (CostructuredArrow.map (𝟙 l) ⋙ diagram ι F l) := I _
+    ext j
+    erw [colimit.ι_pre, Category.comp_id]
+    congr 1
+    simp
+  map_comp := by
+    intro x y z f g
+    dsimp
+    haveI : HasColimit (CostructuredArrow.map (f ≫ g) ⋙ diagram ι F z) := I _
+    ext j
+    let ff : CostructuredArrow ι _ ⥤ _ := CostructuredArrow.map f
+    let gg : CostructuredArrow ι _ ⥤ _ := CostructuredArrow.map g
+    let dd := diagram ι F z
+    -- Porting note: It seems that even Lean3 had some trouble with instances in this case.
+    -- I don't know why lean can't deduce the following three instances...
+    haveI : HasColimit (ff ⋙ gg ⋙ dd) := I _
+    haveI : HasColimit ((ff ⋙ gg) ⋙ dd) := I _
+    haveI : HasColimit (gg ⋙ dd) := I _
+    change _ = colimit.ι ((ff ⋙ gg) ⋙ dd) j ≫ _ ≫ _
+    erw [colimit.pre_pre dd gg ff, colimit.ι_pre, colimit.ι_pre]
+    congr 1
+    simp
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan.loc CategoryTheory.Lan.loc
+
+/-- An auxiliary definition used to define `Lan` and `Lan.adjunction`. -/
+@[simps]
+def equiv (F : S ⥤ D) [I : ∀ x, HasColimit (diagram ι F x)] (G : L ⥤ D) :
+    (loc ι F ⟶ G) ≃ (F ⟶ ((whiskeringLeft _ _ _).obj ι).obj G)
+    where
+  toFun f :=
+    { app := fun x => colimit.ι (diagram ι F (ι.obj x)) (CostructuredArrow.mk (𝟙 _)) ≫ f.app _
+      naturality := by
+        intro x y ff
+        dsimp only [whiskeringLeft]
+        simp only [Functor.comp_map, Category.assoc]
+        rw [← f.naturality (ι.map ff), ← Category.assoc, ← Category.assoc]
+        let fff : CostructuredArrow ι _ ⥤ _ := CostructuredArrow.map (ι.map ff)
+        -- same issue :-(
+        haveI : HasColimit (fff ⋙ diagram ι F (ι.obj y)) := I _
+        erw [colimit.ι_pre (diagram ι F (ι.obj y)) fff (CostructuredArrow.mk (𝟙 _))]
+        let xx : CostructuredArrow ι (ι.obj y) := CostructuredArrow.mk (ι.map ff)
+        let yy : CostructuredArrow ι (ι.obj y) := CostructuredArrow.mk (𝟙 _)
+        let fff : xx ⟶ yy :=
+          CostructuredArrow.homMk ff
+            (by
+              simp only [CostructuredArrow.mk_hom_eq_self]
+              erw [Category.comp_id])
+        erw [colimit.w (diagram ι F (ι.obj y)) fff]
+        congr
+        simp }
+  invFun f :=
+    { app := fun x => colimit.desc (diagram ι F x) (cocone _ f)
+      naturality := by
+        intro x y ff
+        apply colimit.hom_ext
+        intros j
+        haveI : HasColimit (CostructuredArrow.map ff ⋙ diagram ι F y) := I _
+        erw [colimit.pre_desc, ← Category.assoc, colimit.ι_desc, colimit.ι_desc]
+        simp }
+  left_inv := by
+    intros x
+    dsimp
+    ext k
+    dsimp
+    apply colimit.hom_ext
+    intros j
+    rw [colimit.ι_desc]
+    dsimp only [cocone]
+    rw [Category.assoc, ← x.naturality j.hom, ← Category.assoc]
+    congr 1
+    dsimp [loc]
+    haveI : HasColimit (CostructuredArrow.map j.hom ⋙ diagram ι F k) := I _
+    erw [colimit.ι_pre (diagram ι F k) (CostructuredArrow.map j.hom)]
+    congr
+    rcases j with ⟨_, ⟨⟩, _⟩
+    simp only [CostructuredArrow.map_mk, Category.id_comp]
+    rfl
+  right_inv := by aesop_cat
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan.equiv CategoryTheory.Lan.equiv
+
+end Lan
+
+/-- The left Kan extension of a functor. -/
+@[simps!]
+def lan [∀ X, HasColimitsOfShape (CostructuredArrow ι X) D] : (S ⥤ D) ⥤ L ⥤ D :=
+  Adjunction.leftAdjointOfEquiv (fun F G => Lan.equiv ι F G) (by {
+    intros X' X Y f g
+    ext
+    simp [Lan.equiv] })
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan CategoryTheory.lan
+
+namespace Lan
+
+variable (D)
+
+/-- The adjunction associated to `Lan`. -/
+def adjunction [∀ X, HasColimitsOfShape (CostructuredArrow ι X) D] :
+    lan ι ⊣ (whiskeringLeft _ _ D).obj ι :=
+  Adjunction.adjunctionOfEquivLeft _ _
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan.adjunction CategoryTheory.Lan.adjunction
+
+theorem coreflective [Full ι] [Faithful ι] [∀ X, HasColimitsOfShape (CostructuredArrow ι X) D] :
+    IsIso (adjunction D ι).unit := by
+  suffices : ∀ (X : S ⥤ D), IsIso (NatTrans.app (adjunction D ι).unit X)
+  · apply NatIso.isIso_of_isIso_app
+  intro F
+  suffices : ∀ (X : S), IsIso (NatTrans.app (NatTrans.app (adjunction D ι).unit F) X)
+  · apply NatIso.isIso_of_isIso_app
+  intro X
+  dsimp [adjunction, equiv]
+  simp only [Category.comp_id]
+  exact
+    IsIso.of_iso
+      ((colimit.isColimit _).coconePointUniqueUpToIso
+          (colimitOfDiagramTerminal CostructuredArrow.mkIdTerminal _)).symm
+set_option linter.uppercaseLean3 false in
+  #align category_theory.Lan.coreflective CategoryTheory.Lan.coreflective
+
+end Lan
+
+end CategoryTheory