category_theory.monad.equiv_mon
⟷
Mathlib.CategoryTheory.Monad.EquivMon
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
-import Mathbin.CategoryTheory.Monad.Basic
-import Mathbin.CategoryTheory.Monoidal.End
-import Mathbin.CategoryTheory.Monoidal.Mon_
+import CategoryTheory.Monad.Basic
+import CategoryTheory.Monoidal.End
+import CategoryTheory.Monoidal.Mon_
#align_import category_theory.monad.equiv_mon from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
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 9240e8be927a0955b9a82c6c85ef499ee3a626b8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Monad.Basic
import Mathbin.CategoryTheory.Monoidal.End
import Mathbin.CategoryTheory.Monoidal.Mon_
+#align_import category_theory.monad.equiv_mon from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
+
/-!
# The equivalence between `Monad C` and `Mon_ (C ⥤ C)`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8
@@ -4,7 +4,7 @@ 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
+! leanprover-community/mathlib commit 9240e8be927a0955b9a82c6c85ef499ee3a626b8
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -16,6 +16,9 @@ import Mathbin.CategoryTheory.Monoidal.Mon_
# The equivalence between `Monad C` and `Mon_ (C ⥤ C)`.
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
A monad "is just" a monoid in the category of endofunctors.
# Definitions/Theorems
mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9
@@ -41,6 +41,7 @@ namespace Monad
attribute [local instance, local reducible] endofunctor_monoidal_category
+#print CategoryTheory.Monad.toMon /-
/-- To every `Monad C` we associated a monoid object in `C ⥤ C`.-/
@[simps]
def toMon : Monad C → Mon_ (C ⥤ C) := fun M =>
@@ -53,9 +54,11 @@ def toMon : Monad C → Mon_ (C ⥤ C) := fun M =>
-- `obviously` provides this, but slowly
mul_assoc' := by ext; dsimp; simp [M.assoc] }
#align category_theory.Monad.to_Mon CategoryTheory.Monad.toMon
+-/
variable (C)
+#print CategoryTheory.Monad.monadToMon /-
/-- Passing from `Monad C` to `Mon_ (C ⥤ C)` is functorial. -/
@[simps]
def monadToMon : Monad C ⥤ Mon_ (C ⥤ C) where
@@ -65,9 +68,11 @@ def monadToMon : Monad C ⥤ Mon_ (C ⥤ C) where
-- `obviously` provides this, but slowly
map_comp' := by intro X Y Z f g; rfl
#align category_theory.Monad.Monad_to_Mon CategoryTheory.Monad.monadToMon
+-/
variable {C}
+#print CategoryTheory.Monad.ofMon /-
/-- To every monoid object in `C ⥤ C` we associate a `Monad C`. -/
@[simps]
def ofMon : Mon_ (C ⥤ C) → Monad C := fun M =>
@@ -78,9 +83,11 @@ def ofMon : Mon_ (C ⥤ C) → Monad C := fun M =>
right_unit' := fun X => by rw [← M.one.hcomp_id_app, ← nat_trans.comp_app, M.one_mul]; rfl
assoc' := fun X => by rw [← nat_trans.hcomp_id_app, ← nat_trans.comp_app]; simp }
#align category_theory.Monad.of_Mon CategoryTheory.Monad.ofMon
+-/
variable (C)
+#print CategoryTheory.Monad.monToMonad /-
/-- Passing from `Mon_ (C ⥤ C)` to `Monad C` is functorial. -/
@[simps]
def monToMonad : Mon_ (C ⥤ C) ⥤ Monad C where
@@ -98,6 +105,7 @@ def monToMonad : Mon_ (C ⥤ C) ⥤ Monad C where
simpa only [nat_trans.naturality, nat_trans.hcomp_app, assoc, nat_trans.comp_app,
of_Mon_μ] }
#align category_theory.Monad.Mon_to_Monad CategoryTheory.Monad.monToMonad
+-/
namespace MonadMonEquiv
@@ -112,17 +120,17 @@ def counitIso : monToMonad C ⋙ monadToMon C ≅ 𝟭 _
hom_inv_id' := by ext; simp
-- `obviously` provides these, but slowly
inv_hom_id' := by ext; simp
-#align category_theory.Monad.Monad_Mon_equiv.counit_iso CategoryTheory.Monad.MonadMonEquiv.counitIso
+#align category_theory.Monad.Monad_Mon_equiv.counit_iso CategoryTheory.Monad.monadMonEquiv.counitIso
/-- Auxiliary definition for `Monad_Mon_equiv` -/
@[simps]
def unitIsoHom : 𝟭 _ ⟶ monadToMon C ⋙ monToMonad C where app _ := { app := fun _ => 𝟙 _ }
-#align category_theory.Monad.Monad_Mon_equiv.unit_iso_hom CategoryTheory.Monad.MonadMonEquiv.unitIsoHom
+#align category_theory.Monad.Monad_Mon_equiv.unit_iso_hom CategoryTheory.Monad.monadMonEquiv.unitIsoHom
/-- Auxiliary definition for `Monad_Mon_equiv` -/
@[simps]
def unitIsoInv : monadToMon C ⋙ monToMonad C ⟶ 𝟭 _ where app _ := { app := fun _ => 𝟙 _ }
-#align category_theory.Monad.Monad_Mon_equiv.unit_iso_inv CategoryTheory.Monad.MonadMonEquiv.unitIsoInv
+#align category_theory.Monad.Monad_Mon_equiv.unit_iso_inv CategoryTheory.Monad.monadMonEquiv.unitIsoInv
/-- Isomorphism of functors used in `Monad_Mon_equiv` -/
@[simps]
@@ -133,12 +141,13 @@ def unitIso : 𝟭 _ ≅ monadToMon C ⋙ monToMonad C
hom_inv_id' := by ext; simp
-- `obviously` provides these, but slowly
inv_hom_id' := by ext; simp
-#align category_theory.Monad.Monad_Mon_equiv.unit_iso CategoryTheory.Monad.MonadMonEquiv.unitIso
+#align category_theory.Monad.Monad_Mon_equiv.unit_iso CategoryTheory.Monad.monadMonEquiv.unitIso
end MonadMonEquiv
open MonadMonEquiv
+#print CategoryTheory.Monad.monadMonEquiv /-
/-- Oh, monads are just monoids in the category of endofunctors (equivalence of categories). -/
@[simps]
def monadMonEquiv : Monad C ≌ Mon_ (C ⥤ C)
@@ -149,6 +158,7 @@ def monadMonEquiv : Monad C ≌ Mon_ (C ⥤ C)
counitIso := counitIso
functor_unitIso_comp' := by intro X; ext; dsimp; simp
#align category_theory.Monad.Monad_Mon_equiv CategoryTheory.Monad.monadMonEquiv
+-/
-- `obviously`, slowly
-- Sanity check
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -47,18 +47,11 @@ def toMon : Monad C → Mon_ (C ⥤ C) := fun M =>
{ pt := (M : C ⥤ C)
one := M.η
mul := M.μ
- one_mul' := by
- ext
- simp
+ one_mul' := by ext; simp
-- `obviously` provides this, but slowly
- mul_one' := by
- ext
- simp
+ mul_one' := by ext; simp
-- `obviously` provides this, but slowly
- mul_assoc' := by
- ext
- dsimp
- simp [M.assoc] }
+ mul_assoc' := by ext; dsimp; simp [M.assoc] }
#align category_theory.Monad.to_Mon CategoryTheory.Monad.toMon
variable (C)
@@ -68,13 +61,9 @@ variable (C)
def monadToMon : Monad C ⥤ Mon_ (C ⥤ C) where
obj := toMon
map _ _ f := { Hom := f.toNatTrans }
- map_id' := by
- intro X
- rfl
+ map_id' := by intro X; rfl
-- `obviously` provides this, but slowly
- map_comp' := by
- intro X Y Z f g
- rfl
+ map_comp' := by intro X Y Z f g; rfl
#align category_theory.Monad.Monad_to_Mon CategoryTheory.Monad.monadToMon
variable {C}
@@ -85,17 +74,9 @@ def ofMon : Mon_ (C ⥤ C) → Monad C := fun M =>
{ toFunctor := M.pt
η' := M.one
μ' := M.mul
- left_unit' := fun X =>
- by
- rw [← M.one.id_hcomp_app, ← nat_trans.comp_app, M.mul_one]
- rfl
- right_unit' := fun X =>
- by
- rw [← M.one.hcomp_id_app, ← nat_trans.comp_app, M.one_mul]
- rfl
- assoc' := fun X => by
- rw [← nat_trans.hcomp_id_app, ← nat_trans.comp_app]
- simp }
+ left_unit' := fun X => by rw [← M.one.id_hcomp_app, ← nat_trans.comp_app, M.mul_one]; rfl
+ right_unit' := fun X => by rw [← M.one.hcomp_id_app, ← nat_trans.comp_app, M.one_mul]; rfl
+ assoc' := fun X => by rw [← nat_trans.hcomp_id_app, ← nat_trans.comp_app]; simp }
#align category_theory.Monad.of_Mon CategoryTheory.Monad.ofMon
variable (C)
@@ -128,13 +109,9 @@ def counitIso : monToMonad C ⋙ monadToMon C ≅ 𝟭 _
where
Hom := { app := fun _ => { Hom := 𝟙 _ } }
inv := { app := fun _ => { Hom := 𝟙 _ } }
- hom_inv_id' := by
- ext
- simp
+ hom_inv_id' := by ext; simp
-- `obviously` provides these, but slowly
- inv_hom_id' := by
- ext
- simp
+ inv_hom_id' := by ext; simp
#align category_theory.Monad.Monad_Mon_equiv.counit_iso CategoryTheory.Monad.MonadMonEquiv.counitIso
/-- Auxiliary definition for `Monad_Mon_equiv` -/
@@ -153,13 +130,9 @@ def unitIso : 𝟭 _ ≅ monadToMon C ⋙ monToMonad C
where
Hom := unitIsoHom
inv := unitIsoInv
- hom_inv_id' := by
- ext
- simp
+ hom_inv_id' := by ext; simp
-- `obviously` provides these, but slowly
- inv_hom_id' := by
- ext
- simp
+ inv_hom_id' := by ext; simp
#align category_theory.Monad.Monad_Mon_equiv.unit_iso CategoryTheory.Monad.MonadMonEquiv.unitIso
end MonadMonEquiv
@@ -174,11 +147,7 @@ def monadMonEquiv : Monad C ≌ Mon_ (C ⥤ C)
inverse := monToMonad _
unitIso := unitIso
counitIso := counitIso
- functor_unitIso_comp' := by
- intro X
- ext
- dsimp
- simp
+ functor_unitIso_comp' := by intro X; ext; dsimp; simp
#align category_theory.Monad.Monad_Mon_equiv CategoryTheory.Monad.monadMonEquiv
-- `obviously`, slowly
mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211
@@ -44,7 +44,7 @@ attribute [local instance, local reducible] endofunctor_monoidal_category
/-- To every `Monad C` we associated a monoid object in `C ⥤ C`.-/
@[simps]
def toMon : Monad C → Mon_ (C ⥤ C) := fun M =>
- { x := (M : C ⥤ C)
+ { pt := (M : C ⥤ C)
one := M.η
mul := M.μ
one_mul' := by
@@ -82,7 +82,7 @@ variable {C}
/-- To every monoid object in `C ⥤ C` we associate a `Monad C`. -/
@[simps]
def ofMon : Mon_ (C ⥤ C) → Monad C := fun M =>
- { toFunctor := M.x
+ { toFunctor := M.pt
η' := M.one
μ' := M.mul
left_unit' := fun X =>
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.
@@ -38,7 +38,7 @@ namespace Monad
attribute [local instance] endofunctorMonoidalCategory
-/-- To every `Monad C` we associated a monoid object in `C ⥤ C`.-/
+/-- 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)
@@ -66,14 +66,14 @@ def ofMon (M : Mon_ (C ⥤ C)) : Monad C where
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]
+ erw [← whiskerLeft_app, ← 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]
+ erw [← whiskerRight_app, ← NatTrans.comp_app, M.one_mul]
rfl
assoc' := fun X => by
- rw [← NatTrans.hcomp_id_app, ← NatTrans.comp_app]
+ rw [← whiskerLeft_app, ← whiskerRight_app, ← NatTrans.comp_app]
-- Porting note: had to add this step:
erw [M.mul_assoc]
simp
Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>
@@ -98,8 +98,8 @@ def monToMonad : Mon_ (C ⥤ C) ⥤ Monad C where
intro Z
erw [← NatTrans.comp_app, f.mul_hom]
dsimp
- simp only [NatTrans.naturality, NatTrans.hcomp_app, assoc, NatTrans.comp_app,
- ofMon_μ] }
+ simp only [Category.assoc, NatTrans.naturality, ofMon_obj]
+ rfl }
#align category_theory.Monad.Mon_to_Monad CategoryTheory.Monad.monToMonad
/-- Oh, monads are just monoids in the category of endofunctors (equivalence of categories). -/
@@ -2,16 +2,13 @@
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_
+#align_import category_theory.monad.equiv_mon from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
+
/-!
# The equivalence between `Monad C` and `Mon_ (C ⥤ C)`.
The unported dependencies are