category_theory.monad.coequalizer
⟷
Mathlib.CategoryTheory.Monad.Coequalizer
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 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
-import Mathbin.CategoryTheory.Limits.Shapes.Reflexive
-import Mathbin.CategoryTheory.Limits.Shapes.SplitCoequalizer
-import Mathbin.CategoryTheory.Monad.Algebra
+import CategoryTheory.Limits.Shapes.Reflexive
+import CategoryTheory.Limits.Shapes.SplitCoequalizer
+import CategoryTheory.Monad.Algebra
#align_import category_theory.monad.coequalizer from "leanprover-community/mathlib"@"ef55335933293309ff8c0b1d20ffffeecbe5c39f"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.monad.coequalizer
-! leanprover-community/mathlib commit ef55335933293309ff8c0b1d20ffffeecbe5c39f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Shapes.Reflexive
import Mathbin.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathbin.CategoryTheory.Monad.Algebra
+#align_import category_theory.monad.coequalizer from "leanprover-community/mathlib"@"ef55335933293309ff8c0b1d20ffffeecbe5c39f"
+
/-!
# Special coequalizers associated to a monad
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -46,12 +46,15 @@ Show that any algebra is a coequalizer of free algebras.
-/
+#print CategoryTheory.Monad.FreeCoequalizer.topMap /-
/-- The top map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.topMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T).obj X.A :=
(Monad.free T).map X.a
#align category_theory.monad.free_coequalizer.top_map CategoryTheory.Monad.FreeCoequalizer.topMap
+-/
+#print CategoryTheory.Monad.FreeCoequalizer.bottomMap /-
/-- The bottom map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.bottomMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T).obj X.A
@@ -59,7 +62,9 @@ def FreeCoequalizer.bottomMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T
f := T.μ.app X.A
h' := T.and_assoc X.A
#align category_theory.monad.free_coequalizer.bottom_map CategoryTheory.Monad.FreeCoequalizer.bottomMap
+-/
+#print CategoryTheory.Monad.FreeCoequalizer.π /-
/-- The cofork map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.π : (Monad.free T).obj X.A ⟶ X
@@ -67,12 +72,15 @@ def FreeCoequalizer.π : (Monad.free T).obj X.A ⟶ X
f := X.a
h' := X.and_assoc.symm
#align category_theory.monad.free_coequalizer.π CategoryTheory.Monad.FreeCoequalizer.π
+-/
+#print CategoryTheory.Monad.FreeCoequalizer.condition /-
theorem FreeCoequalizer.condition :
FreeCoequalizer.topMap X ≫ FreeCoequalizer.π X =
FreeCoequalizer.bottomMap X ≫ FreeCoequalizer.π X :=
Algebra.Hom.ext _ _ X.and_assoc.symm
#align category_theory.monad.free_coequalizer.condition CategoryTheory.Monad.FreeCoequalizer.condition
+-/
instance : IsReflexivePair (FreeCoequalizer.topMap X) (FreeCoequalizer.bottomMap X) :=
by
@@ -84,6 +92,7 @@ instance : IsReflexivePair (FreeCoequalizer.topMap X) (FreeCoequalizer.bottomMap
· ext
apply monad.right_unit
+#print CategoryTheory.Monad.beckAlgebraCofork /-
/-- Construct the Beck cofork in the category of algebras. This cofork is reflexive as well as a
coequalizer.
-/
@@ -91,7 +100,9 @@ coequalizer.
def beckAlgebraCofork : Cofork (FreeCoequalizer.topMap X) (FreeCoequalizer.bottomMap X) :=
Cofork.ofπ _ (FreeCoequalizer.condition X)
#align category_theory.monad.beck_algebra_cofork CategoryTheory.Monad.beckAlgebraCofork
+-/
+#print CategoryTheory.Monad.beckAlgebraCoequalizer /-
/-- The cofork constructed is a colimit. This shows that any algebra is a (reflexive) coequalizer of
free algebras.
-/
@@ -113,33 +124,44 @@ def beckAlgebraCoequalizer : IsColimit (beckAlgebraCofork X) :=
rw [← hm]
apply (X.unit_assoc _).symm
#align category_theory.monad.beck_algebra_coequalizer CategoryTheory.Monad.beckAlgebraCoequalizer
+-/
+#print CategoryTheory.Monad.beckSplitCoequalizer /-
/-- The Beck cofork is a split coequalizer. -/
def beckSplitCoequalizer : IsSplitCoequalizer (T.map X.a) (T.μ.app _) X.a :=
⟨T.η.app _, T.η.app _, X.and_assoc.symm, X.Unit, T.left_unit _, (T.η.naturality _).symm⟩
#align category_theory.monad.beck_split_coequalizer CategoryTheory.Monad.beckSplitCoequalizer
+-/
+#print CategoryTheory.Monad.beckCofork /-
/-- This is the Beck cofork. It is a split coequalizer, in particular a coequalizer. -/
@[simps pt]
def beckCofork : Cofork (T.map X.a) (T.μ.app _) :=
(beckSplitCoequalizer X).asCofork
#align category_theory.monad.beck_cofork CategoryTheory.Monad.beckCofork
+-/
+#print CategoryTheory.Monad.beckCofork_π /-
@[simp]
theorem beckCofork_π : (beckCofork X).π = X.a :=
rfl
#align category_theory.monad.beck_cofork_π CategoryTheory.Monad.beckCofork_π
+-/
+#print CategoryTheory.Monad.beckCoequalizer /-
/-- The Beck cofork is a coequalizer. -/
def beckCoequalizer : IsColimit (beckCofork X) :=
(beckSplitCoequalizer X).isCoequalizer
#align category_theory.monad.beck_coequalizer CategoryTheory.Monad.beckCoequalizer
+-/
+#print CategoryTheory.Monad.beckCoequalizer_desc /-
@[simp]
theorem beckCoequalizer_desc (s : Cofork (T.toFunctor.map X.a) (T.μ.app X.A)) :
(beckCoequalizer X).desc s = T.η.app _ ≫ s.π :=
rfl
#align category_theory.monad.beck_coequalizer_desc CategoryTheory.Monad.beckCoequalizer_desc
+-/
end Monad
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -46,24 +46,12 @@ Show that any algebra is a coequalizer of free algebras.
-/
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/-- The top map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.topMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T).obj X.A :=
(Monad.free T).map X.a
#align category_theory.monad.free_coequalizer.top_map CategoryTheory.Monad.FreeCoequalizer.topMap
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/-- The bottom map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.bottomMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T).obj X.A
@@ -72,12 +60,6 @@ def FreeCoequalizer.bottomMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T
h' := T.and_assoc X.A
#align category_theory.monad.free_coequalizer.bottom_map CategoryTheory.Monad.FreeCoequalizer.bottomMap
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/-- The cofork map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.π : (Monad.free T).obj X.A ⟶ X
@@ -86,12 +68,6 @@ def FreeCoequalizer.π : (Monad.free T).obj X.A ⟶ X
h' := X.and_assoc.symm
#align category_theory.monad.free_coequalizer.π CategoryTheory.Monad.FreeCoequalizer.π
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theorem FreeCoequalizer.condition :
FreeCoequalizer.topMap X ≫ FreeCoequalizer.π X =
FreeCoequalizer.bottomMap X ≫ FreeCoequalizer.π X :=
@@ -108,12 +84,6 @@ instance : IsReflexivePair (FreeCoequalizer.topMap X) (FreeCoequalizer.bottomMap
· ext
apply monad.right_unit
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/-- Construct the Beck cofork in the category of algebras. This cofork is reflexive as well as a
coequalizer.
-/
@@ -122,12 +92,6 @@ def beckAlgebraCofork : Cofork (FreeCoequalizer.topMap X) (FreeCoequalizer.botto
Cofork.ofπ _ (FreeCoequalizer.condition X)
#align category_theory.monad.beck_algebra_cofork CategoryTheory.Monad.beckAlgebraCofork
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/-- The cofork constructed is a colimit. This shows that any algebra is a (reflexive) coequalizer of
free algebras.
-/
@@ -150,51 +114,27 @@ def beckAlgebraCoequalizer : IsColimit (beckAlgebraCofork X) :=
apply (X.unit_assoc _).symm
#align category_theory.monad.beck_algebra_coequalizer CategoryTheory.Monad.beckAlgebraCoequalizer
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/-- The Beck cofork is a split coequalizer. -/
def beckSplitCoequalizer : IsSplitCoequalizer (T.map X.a) (T.μ.app _) X.a :=
⟨T.η.app _, T.η.app _, X.and_assoc.symm, X.Unit, T.left_unit _, (T.η.naturality _).symm⟩
#align category_theory.monad.beck_split_coequalizer CategoryTheory.Monad.beckSplitCoequalizer
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/-- This is the Beck cofork. It is a split coequalizer, in particular a coequalizer. -/
@[simps pt]
def beckCofork : Cofork (T.map X.a) (T.μ.app _) :=
(beckSplitCoequalizer X).asCofork
#align category_theory.monad.beck_cofork CategoryTheory.Monad.beckCofork
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@[simp]
theorem beckCofork_π : (beckCofork X).π = X.a :=
rfl
#align category_theory.monad.beck_cofork_π CategoryTheory.Monad.beckCofork_π
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/-- The Beck cofork is a coequalizer. -/
def beckCoequalizer : IsColimit (beckCofork X) :=
(beckSplitCoequalizer X).isCoequalizer
#align category_theory.monad.beck_coequalizer CategoryTheory.Monad.beckCoequalizer
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@[simp]
theorem beckCoequalizer_desc (s : Cofork (T.toFunctor.map X.a) (T.μ.app X.A)) :
(beckCoequalizer X).desc s = T.η.app _ ≫ s.π :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -174,10 +174,7 @@ def beckCofork : Cofork (T.map X.a) (T.μ.app _) :=
#align category_theory.monad.beck_cofork CategoryTheory.Monad.beckCofork
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Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_cofork_π CategoryTheory.Monad.beckCofork_πₓ'. -/
@[simp]
theorem beckCofork_π : (beckCofork X).π = X.a :=
@@ -196,10 +193,7 @@ def beckCoequalizer : IsColimit (beckCofork X) :=
#align category_theory.monad.beck_coequalizer CategoryTheory.Monad.beckCoequalizer
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(CategoryTheory.Functor.comp.{u1, u1, u1, u2, u2, u2} C _inst_1 C _inst_1 C _inst_1 (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T)) (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.μ.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X)) s))
+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_coequalizer_desc CategoryTheory.Monad.beckCoequalizer_descₓ'. -/
@[simp]
theorem beckCoequalizer_desc (s : Cofork (T.toFunctor.map X.a) (T.μ.app X.A)) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
! This file was ported from Lean 3 source module category_theory.monad.coequalizer
-! leanprover-community/mathlib commit 3a061790136d13594ec10c7c90d202335ac5d854
+! leanprover-community/mathlib commit ef55335933293309ff8c0b1d20ffffeecbe5c39f
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.CategoryTheory.Monad.Algebra
/-!
# Special coequalizers associated to a monad
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
Associated to a monad `T : C ⥤ C` we have important coequalizer constructions:
Any algebra is a coequalizer (in the category of algebras) of free algebras. Furthermore, this
coequalizer is reflexive.
mathlib commit https://github.com/leanprover-community/mathlib/commit/a4f99eae998680d3a2c240da4a2b16354c85ee49
@@ -43,12 +43,24 @@ Show that any algebra is a coequalizer of free algebras.
-/
+/- warning: category_theory.monad.free_coequalizer.top_map -> CategoryTheory.Monad.FreeCoequalizer.topMap is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {T : CategoryTheory.Monad.{u1, u2} C _inst_1} (X : CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T), Quiver.Hom.{succ u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Category.toCategoryStruct.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T))) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} C _inst_1 C _inst_1 (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X))) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.free_coequalizer.top_map CategoryTheory.Monad.FreeCoequalizer.topMapₓ'. -/
/-- The top map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.topMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T).obj X.A :=
(Monad.free T).map X.a
#align category_theory.monad.free_coequalizer.top_map CategoryTheory.Monad.FreeCoequalizer.topMap
+/- warning: category_theory.monad.free_coequalizer.bottom_map -> CategoryTheory.Monad.FreeCoequalizer.bottomMap is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {T : CategoryTheory.Monad.{u1, u2} C _inst_1} (X : CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T), Quiver.Hom.{succ u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.CategoryTheory.categoryStruct.{u1, u2} C _inst_1 T)) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} C _inst_1 C _inst_1 (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X))) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.free_coequalizer.bottom_map CategoryTheory.Monad.FreeCoequalizer.bottomMapₓ'. -/
/-- The bottom map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.bottomMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T).obj X.A
@@ -57,6 +69,12 @@ def FreeCoequalizer.bottomMap : (Monad.free T).obj (T.obj X.A) ⟶ (Monad.free T
h' := T.and_assoc X.A
#align category_theory.monad.free_coequalizer.bottom_map CategoryTheory.Monad.FreeCoequalizer.bottomMap
+/- warning: category_theory.monad.free_coequalizer.π -> CategoryTheory.Monad.FreeCoequalizer.π is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {T : CategoryTheory.Monad.{u1, u2} C _inst_1} (X : CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T), Quiver.Hom.{succ u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.CategoryTheory.categoryStruct.{u1, u2} C _inst_1 T)) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X)) X
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.free_coequalizer.π CategoryTheory.Monad.FreeCoequalizer.πₓ'. -/
/-- The cofork map in the coequalizer diagram we will construct. -/
@[simps]
def FreeCoequalizer.π : (Monad.free T).obj X.A ⟶ X
@@ -65,6 +83,12 @@ def FreeCoequalizer.π : (Monad.free T).obj X.A ⟶ X
h' := X.and_assoc.symm
#align category_theory.monad.free_coequalizer.π CategoryTheory.Monad.FreeCoequalizer.π
+/- warning: category_theory.monad.free_coequalizer.condition -> CategoryTheory.Monad.FreeCoequalizer.condition is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.free_coequalizer.condition CategoryTheory.Monad.FreeCoequalizer.conditionₓ'. -/
theorem FreeCoequalizer.condition :
FreeCoequalizer.topMap X ≫ FreeCoequalizer.π X =
FreeCoequalizer.bottomMap X ≫ FreeCoequalizer.π X :=
@@ -81,6 +105,12 @@ instance : IsReflexivePair (FreeCoequalizer.topMap X) (FreeCoequalizer.bottomMap
· ext
apply monad.right_unit
+/- warning: category_theory.monad.beck_algebra_cofork -> CategoryTheory.Monad.beckAlgebraCofork is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {T : CategoryTheory.Monad.{u1, u2} C _inst_1} (X : CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T), CategoryTheory.Limits.Cofork.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} C _inst_1 C _inst_1 (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X))) (CategoryTheory.Functor.obj.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X)) (CategoryTheory.Monad.FreeCoequalizer.topMap.{u1, u2} C _inst_1 T X) (CategoryTheory.Monad.FreeCoequalizer.bottomMap.{u1, u2} C _inst_1 T X)
+but is expected to have type
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {T : CategoryTheory.Monad.{u1, u2} C _inst_1} (X : CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T), CategoryTheory.Limits.Cofork.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (Prefunctor.obj.{succ u1, succ u1, u2, max u2 u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Category.toCategoryStruct.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T)) (Prefunctor.obj.{succ u1, succ u1, u2, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} C _inst_1 C _inst_1 (CategoryTheory.Monad.toFunctor.{u1, u2} C _inst_1 T)) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X))) (Prefunctor.obj.{succ u1, succ u1, u2, max u2 u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Category.toCategoryStruct.{u1, max u2 u1} (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Monad.Algebra.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.Algebra.eilenbergMoore.{u1, u2} C _inst_1 T) (CategoryTheory.Monad.free.{u1, u2} C _inst_1 T)) (CategoryTheory.Monad.Algebra.A.{u1, u2} C _inst_1 T X)) (CategoryTheory.Monad.FreeCoequalizer.topMap.{u1, u2} C _inst_1 T X) (CategoryTheory.Monad.FreeCoequalizer.bottomMap.{u1, u2} C _inst_1 T X)
+Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_algebra_cofork CategoryTheory.Monad.beckAlgebraCoforkₓ'. -/
/-- Construct the Beck cofork in the category of algebras. This cofork is reflexive as well as a
coequalizer.
-/
@@ -89,6 +119,12 @@ def beckAlgebraCofork : Cofork (FreeCoequalizer.topMap X) (FreeCoequalizer.botto
Cofork.ofπ _ (FreeCoequalizer.condition X)
#align category_theory.monad.beck_algebra_cofork CategoryTheory.Monad.beckAlgebraCofork
+/- warning: category_theory.monad.beck_algebra_coequalizer -> CategoryTheory.Monad.beckAlgebraCoequalizer is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_algebra_coequalizer CategoryTheory.Monad.beckAlgebraCoequalizerₓ'. -/
/-- The cofork constructed is a colimit. This shows that any algebra is a (reflexive) coequalizer of
free algebras.
-/
@@ -111,27 +147,57 @@ def beckAlgebraCoequalizer : IsColimit (beckAlgebraCofork X) :=
apply (X.unit_assoc _).symm
#align category_theory.monad.beck_algebra_coequalizer CategoryTheory.Monad.beckAlgebraCoequalizer
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_split_coequalizer CategoryTheory.Monad.beckSplitCoequalizerₓ'. -/
/-- The Beck cofork is a split coequalizer. -/
def beckSplitCoequalizer : IsSplitCoequalizer (T.map X.a) (T.μ.app _) X.a :=
⟨T.η.app _, T.η.app _, X.and_assoc.symm, X.Unit, T.left_unit _, (T.η.naturality _).symm⟩
#align category_theory.monad.beck_split_coequalizer CategoryTheory.Monad.beckSplitCoequalizer
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_cofork CategoryTheory.Monad.beckCoforkₓ'. -/
/-- This is the Beck cofork. It is a split coequalizer, in particular a coequalizer. -/
@[simps pt]
def beckCofork : Cofork (T.map X.a) (T.μ.app _) :=
(beckSplitCoequalizer X).asCofork
#align category_theory.monad.beck_cofork CategoryTheory.Monad.beckCofork
+/- warning: category_theory.monad.beck_cofork_π -> CategoryTheory.Monad.beckCofork_π is a dubious translation:
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@[simp]
theorem beckCofork_π : (beckCofork X).π = X.a :=
rfl
#align category_theory.monad.beck_cofork_π CategoryTheory.Monad.beckCofork_π
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/-- The Beck cofork is a coequalizer. -/
def beckCoequalizer : IsColimit (beckCofork X) :=
(beckSplitCoequalizer X).isCoequalizer
#align category_theory.monad.beck_coequalizer CategoryTheory.Monad.beckCoequalizer
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+Case conversion may be inaccurate. Consider using '#align category_theory.monad.beck_coequalizer_desc CategoryTheory.Monad.beckCoequalizer_descₓ'. -/
@[simp]
theorem beckCoequalizer_desc (s : Cofork (T.toFunctor.map X.a) (T.μ.app X.A)) :
(beckCoequalizer X).desc s = T.η.app _ ≫ s.π :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211
@@ -117,7 +117,7 @@ def beckSplitCoequalizer : IsSplitCoequalizer (T.map X.a) (T.μ.app _) X.a :=
#align category_theory.monad.beck_split_coequalizer CategoryTheory.Monad.beckSplitCoequalizer
/-- This is the Beck cofork. It is a split coequalizer, in particular a coequalizer. -/
-@[simps x]
+@[simps pt]
def beckCofork : Cofork (T.map X.a) (T.μ.app _) :=
(beckSplitCoequalizer X).asCofork
#align category_theory.monad.beck_cofork CategoryTheory.Monad.beckCofork
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
@@ -30,9 +30,7 @@ namespace Monad
open Limits
variable {C : Type u₁}
-
variable [Category.{v₁} C]
-
variable {T : Monad C} (X : Algebra T)
/-!
@@ -2,16 +2,13 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.monad.coequalizer
-! leanprover-community/mathlib commit 3a061790136d13594ec10c7c90d202335ac5d854
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.Reflexive
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Monad.Algebra
+#align_import category_theory.monad.coequalizer from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
+
/-!
# Special coequalizers associated to a monad
The unported dependencies are