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feat: port CategoryTheory.Monad.EquivMon (#5086)
Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>
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| /- | ||
| Copyright (c) 2020 Adam Topaz. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Adam Topaz | ||
| ! This file was ported from Lean 3 source module category_theory.monad.equiv_mon | ||
| ! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a | ||
| ! Please do not edit these lines, except to modify the commit id | ||
| ! if you have ported upstream changes. | ||
| -/ | ||
| import Mathlib.CategoryTheory.Monad.Basic | ||
| import Mathlib.CategoryTheory.Monoidal.End | ||
| import Mathlib.CategoryTheory.Monoidal.Mon_ | ||
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| /-! | ||
| # The equivalence between `Monad C` and `Mon_ (C ⥤ C)`. | ||
| A monad "is just" a monoid in the category of endofunctors. | ||
| # Definitions/Theorems | ||
| 1. `toMon` associates a monoid object in `C ⥤ C` to any monad on `C`. | ||
| 2. `monadToMon` is the functorial version of `toMon`. | ||
| 3. `ofMon` associates a monad on `C` to any monoid object in `C ⥤ C`. | ||
| 4. `monadMonEquiv` is the equivalence between `Monad C` and `Mon_ (C ⥤ C)`. | ||
| -/ | ||
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| set_option linter.uppercaseLean3 false | ||
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| namespace CategoryTheory | ||
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| open Category | ||
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| universe v u -- morphism levels before object levels. See note [category_theory universes]. | ||
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| variable {C : Type u} [Category.{v} C] | ||
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| namespace Monad | ||
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| attribute [local instance] endofunctorMonoidalCategory | ||
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| /-- To every `Monad C` we associated a monoid object in `C ⥤ C`.-/ | ||
| @[simps] | ||
| def toMon (M : Monad C) : Mon_ (C ⥤ C) where | ||
| X := (M : C ⥤ C) | ||
| one := M.η | ||
| mul := M.μ | ||
| mul_assoc := by ext; simp [M.assoc] | ||
| #align category_theory.Monad.to_Mon CategoryTheory.Monad.toMon | ||
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| variable (C) | ||
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| /-- Passing from `Monad C` to `Mon_ (C ⥤ C)` is functorial. -/ | ||
| @[simps] | ||
| def monadToMon : Monad C ⥤ Mon_ (C ⥤ C) where | ||
| obj := toMon | ||
| map f := { hom := f.toNatTrans } | ||
| #align category_theory.Monad.Monad_to_Mon CategoryTheory.Monad.monadToMon | ||
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| variable {C} | ||
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| /-- To every monoid object in `C ⥤ C` we associate a `Monad C`. -/ | ||
| @[simps η μ] | ||
| def ofMon (M : Mon_ (C ⥤ C)) : Monad C where | ||
| toFunctor := M.X | ||
| η' := M.one | ||
| μ' := M.mul | ||
| left_unit' := fun X => by | ||
| -- Porting note: now using `erw` | ||
| erw [← NatTrans.id_hcomp_app M.one, ← NatTrans.comp_app, M.mul_one] | ||
| rfl | ||
| right_unit' := fun X => by | ||
| -- Porting note: now using `erw` | ||
| erw [← NatTrans.hcomp_id_app M.one, ← NatTrans.comp_app, M.one_mul] | ||
| rfl | ||
| assoc' := fun X => by | ||
| rw [← NatTrans.hcomp_id_app, ← NatTrans.comp_app] | ||
| -- Porting note: had to add this step: | ||
| erw [M.mul_assoc] | ||
| simp | ||
| #align category_theory.Monad.of_Mon CategoryTheory.Monad.ofMon | ||
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| -- Porting note: `@[simps]` fails to generate `ofMon_obj`: | ||
| @[simp] lemma ofMon_obj (M : Mon_ (C ⥤ C)) (X : C) : (ofMon M).obj X = M.X.obj X := rfl | ||
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| variable (C) | ||
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| /-- Passing from `Mon_ (C ⥤ C)` to `Monad C` is functorial. -/ | ||
| @[simps] | ||
| def monToMonad : Mon_ (C ⥤ C) ⥤ Monad C where | ||
| obj := ofMon | ||
| map {X Y} f := | ||
| { f.hom with | ||
| app_η := by | ||
| intro X | ||
| erw [← NatTrans.comp_app, f.one_hom] | ||
| rfl | ||
| app_μ := by | ||
| intro Z | ||
| erw [← NatTrans.comp_app, f.mul_hom] | ||
| dsimp | ||
| simp only [NatTrans.naturality, NatTrans.hcomp_app, assoc, NatTrans.comp_app, | ||
| ofMon_μ] } | ||
| #align category_theory.Monad.Mon_to_Monad CategoryTheory.Monad.monToMonad | ||
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| /-- Oh, monads are just monoids in the category of endofunctors (equivalence of categories). -/ | ||
| @[simps] | ||
| def monadMonEquiv : Monad C ≌ Mon_ (C ⥤ C) where | ||
| functor := monadToMon _ | ||
| inverse := monToMonad _ | ||
| unitIso := | ||
| { hom := { app := fun _ => { app := fun _ => 𝟙 _ } } | ||
| inv := { app := fun _ => { app := fun _ => 𝟙 _ } } } | ||
| counitIso := | ||
| { hom := { app := fun _ => { hom := 𝟙 _ } } | ||
| inv := { app := fun _ => { hom := 𝟙 _ } } } | ||
| #align category_theory.Monad.Monad_Mon_equiv CategoryTheory.Monad.monadMonEquiv | ||
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| -- Sanity check | ||
| example (A : Monad C) {X : C} : ((monadMonEquiv C).unitIso.app A).hom.app X = 𝟙 _ := | ||
| rfl | ||
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| end Monad | ||
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| end CategoryTheory |
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