category_theory.limits.preserves.shapes.equalizers
⟷
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
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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(last sync)
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ 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.SplitCoequalizer
-import Mathbin.CategoryTheory.Limits.Preserves.Basic
+import CategoryTheory.Limits.Shapes.SplitCoequalizer
+import CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
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.limits.preserves.shapes.equalizers
-! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathbin.CategoryTheory.Limits.Preserves.Basic
+#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
+
/-!
# Preserving (co)equalizers
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -43,6 +43,7 @@ section Equalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = h ≫ g)
+#print CategoryTheory.Limits.isLimitMapConeForkEquiv /-
/-- The map of a fork is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `fork.of_ι` with `functor.map_cone`.
-/
@@ -52,22 +53,28 @@ def isLimitMapConeForkEquiv :
(IsLimit.postcomposeHomEquiv (diagramIsoParallelPair _) _).symm.trans
(IsLimit.equivIsoLimit (Fork.ext (Iso.refl _) (by simp [fork.ι])))
#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquiv
+-/
+#print CategoryTheory.Limits.isLimitForkMapOfIsLimit /-
/-- The property of preserving equalizers expressed in terms of forks. -/
def isLimitForkMapOfIsLimit [PreservesLimit (parallelPair f g) G] (l : IsLimit (Fork.ofι h w)) :
IsLimit (Fork.ofι (G.map h) (by simp only [← G.map_comp, w]) : Fork (G.map f) (G.map g)) :=
isLimitMapConeForkEquiv G w (PreservesLimit.preserves l)
#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimit
+-/
+#print CategoryTheory.Limits.isLimitOfIsLimitForkMap /-
/-- The property of reflecting equalizers expressed in terms of forks. -/
def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
(l : IsLimit (Fork.ofι (G.map h) (by simp only [← G.map_comp, w]) : Fork (G.map f) (G.map g))) :
IsLimit (Fork.ofι h w) :=
ReflectsLimit.reflects ((isLimitMapConeForkEquiv G w).symm l)
#align category_theory.limits.is_limit_of_is_limit_fork_map CategoryTheory.Limits.isLimitOfIsLimitForkMap
+-/
variable (f g) [HasEqualizer f g]
+#print CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimit /-
/--
If `G` preserves equalizers and `C` has them, then the fork constructed of the mapped morphisms of
a fork is a limit.
@@ -77,9 +84,11 @@ def isLimitOfHasEqualizerOfPreservesLimit [PreservesLimit (parallelPair f g) G]
(Fork.ofι (G.map (equalizer.ι f g)) (by simp only [← G.map_comp, equalizer.condition])) :=
isLimitForkMapOfIsLimit G _ (equalizerIsEqualizer f g)
#align category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimit
+-/
variable [HasEqualizer (G.map f) (G.map g)]
+#print CategoryTheory.Limits.PreservesEqualizer.ofIsoComparison /-
/-- If the equalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
equalizer of `(f,g)`.
-/
@@ -91,9 +100,11 @@ def PreservesEqualizer.ofIsoComparison [i : IsIso (equalizerComparison f g G)] :
apply is_limit.of_point_iso (limit.is_limit (parallel_pair (G.map f) (G.map g)))
apply i
#align category_theory.limits.preserves_equalizer.of_iso_comparison CategoryTheory.Limits.PreservesEqualizer.ofIsoComparison
+-/
variable [PreservesLimit (parallelPair f g) G]
+#print CategoryTheory.Limits.PreservesEqualizer.iso /-
/--
If `G` preserves the equalizer of `(f,g)`, then the equalizer comparison map for `G` at `(f,g)` is
an isomorphism.
@@ -101,12 +112,15 @@ an isomorphism.
def PreservesEqualizer.iso : G.obj (equalizer f g) ≅ equalizer (G.map f) (G.map g) :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasEqualizerOfPreservesLimit G f g) (limit.isLimit _)
#align category_theory.limits.preserves_equalizer.iso CategoryTheory.Limits.PreservesEqualizer.iso
+-/
+#print CategoryTheory.Limits.PreservesEqualizer.iso_hom /-
@[simp]
theorem PreservesEqualizer.iso_hom :
(PreservesEqualizer.iso G f g).Hom = equalizerComparison f g G :=
rfl
#align category_theory.limits.preserves_equalizer.iso_hom CategoryTheory.Limits.PreservesEqualizer.iso_hom
+-/
instance : IsIso (equalizerComparison f g G) :=
by
@@ -119,6 +133,7 @@ section Coequalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h)
+#print CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv /-
/-- The map of a cofork is a colimit iff the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cofork.of_π` with `functor.map_cocone`.
-/
@@ -133,7 +148,9 @@ def isColimitMapCoconeCoforkEquiv :
dsimp only [cofork.π, cofork.of_π_ι_app]
dsimp; rw [category.comp_id, category.id_comp]
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv
+-/
+#print CategoryTheory.Limits.isColimitCoforkMapOfIsColimit /-
/-- The property of preserving coequalizers expressed in terms of coforks. -/
def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
(l : IsColimit (Cofork.ofπ h w)) :
@@ -141,7 +158,9 @@ def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
(Cofork.ofπ (G.map h) (by simp only [← G.map_comp, w]) : Cofork (G.map f) (G.map g)) :=
isColimitMapCoconeCoforkEquiv G w (PreservesColimit.preserves l)
#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimit
+-/
+#print CategoryTheory.Limits.isColimitOfIsColimitCoforkMap /-
/-- The property of reflecting coequalizers expressed in terms of coforks. -/
def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
(l :
@@ -150,9 +169,11 @@ def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
IsColimit (Cofork.ofπ h w) :=
ReflectsColimit.reflects ((isColimitMapCoconeCoforkEquiv G w).symm l)
#align category_theory.limits.is_colimit_of_is_colimit_cofork_map CategoryTheory.Limits.isColimitOfIsColimitCoforkMap
+-/
variable (f g) [HasCoequalizer f g]
+#print CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit /-
/--
If `G` preserves coequalizers and `C` has them, then the cofork constructed of the mapped morphisms
of a cofork is a colimit.
@@ -161,9 +182,11 @@ def isColimitOfHasCoequalizerOfPreservesColimit [PreservesColimit (parallelPair
IsColimit (Cofork.ofπ (G.map (coequalizer.π f g)) _) :=
isColimitCoforkMapOfIsColimit G _ (coequalizerIsCoequalizer f g)
#align category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit
+-/
variable [HasCoequalizer (G.map f) (G.map g)]
+#print CategoryTheory.Limits.ofIsoComparison /-
/-- If the coequalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
coequalizer of `(f,g)`.
-/
@@ -175,9 +198,11 @@ def ofIsoComparison [i : IsIso (coequalizerComparison f g G)] :
apply is_colimit.of_point_iso (colimit.is_colimit (parallel_pair (G.map f) (G.map g)))
apply i
#align category_theory.limits.of_iso_comparison CategoryTheory.Limits.ofIsoComparison
+-/
variable [PreservesColimit (parallelPair f g) G]
+#print CategoryTheory.Limits.PreservesCoequalizer.iso /-
/--
If `G` preserves the coequalizer of `(f,g)`, then the coequalizer comparison map for `G` at `(f,g)`
is an isomorphism.
@@ -186,22 +211,28 @@ def PreservesCoequalizer.iso : coequalizer (G.map f) (G.map g) ≅ G.obj (coequa
IsColimit.coconePointUniqueUpToIso (colimit.isColimit _)
(isColimitOfHasCoequalizerOfPreservesColimit G f g)
#align category_theory.limits.preserves_coequalizer.iso CategoryTheory.Limits.PreservesCoequalizer.iso
+-/
+#print CategoryTheory.Limits.PreservesCoequalizer.iso_hom /-
@[simp]
theorem PreservesCoequalizer.iso_hom :
(PreservesCoequalizer.iso G f g).Hom = coequalizerComparison f g G :=
rfl
#align category_theory.limits.preserves_coequalizer.iso_hom CategoryTheory.Limits.PreservesCoequalizer.iso_hom
+-/
instance : IsIso (coequalizerComparison f g G) :=
by
rw [← preserves_coequalizer.iso_hom]
infer_instance
+#print CategoryTheory.Limits.map_π_epi /-
instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
⟨fun W h k => by rw [← ι_comp_coequalizer_comparison]; apply (cancel_epi _).1; apply epi_comp⟩
#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epi
+-/
+#print CategoryTheory.Limits.map_π_preserves_coequalizer_inv /-
@[reassoc]
theorem map_π_preserves_coequalizer_inv :
G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv =
@@ -210,14 +241,18 @@ theorem map_π_preserves_coequalizer_inv :
rw [← ι_comp_coequalizer_comparison_assoc, ← preserves_coequalizer.iso_hom, iso.hom_inv_id,
comp_id]
#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_inv
+-/
+#print CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc /-
@[reassoc]
theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
(wk : G.map f ≫ k = G.map g ≫ k) :
G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv ≫ coequalizer.desc k wk = k :=
by rw [← category.assoc, map_π_preserves_coequalizer_inv, coequalizer.π_desc]
#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc
+-/
+#print CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap /-
@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
@@ -228,7 +263,9 @@ theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y'
q ≫ coequalizer.π f' g' :=
by rw [← category.assoc, map_π_preserves_coequalizer_inv, ι_colim_map, parallel_pair_hom_app_one]
#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap
+-/
+#print CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_desc /-
@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
@@ -241,6 +278,7 @@ theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X'
slice_lhs 1 3 => rw [map_π_preserves_coequalizer_inv_colim_map]
slice_lhs 2 3 => rw [coequalizer.π_desc]
#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_desc
+-/
#print CategoryTheory.Limits.preservesSplitCoequalizers /-
/-- Any functor preserves coequalizers of split pairs. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -43,9 +43,6 @@ section Equalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = h ≫ g)
-/- warning: category_theory.limits.is_limit_map_cone_fork_equiv -> CategoryTheory.Limits.isLimitMapConeForkEquiv is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquivₓ'. -/
/-- The map of a fork is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `fork.of_ι` with `functor.map_cone`.
-/
@@ -56,18 +53,12 @@ def isLimitMapConeForkEquiv :
(IsLimit.equivIsoLimit (Fork.ext (Iso.refl _) (by simp [fork.ι])))
#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquiv
-/- warning: category_theory.limits.is_limit_fork_map_of_is_limit -> CategoryTheory.Limits.isLimitForkMapOfIsLimit is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimitₓ'. -/
/-- The property of preserving equalizers expressed in terms of forks. -/
def isLimitForkMapOfIsLimit [PreservesLimit (parallelPair f g) G] (l : IsLimit (Fork.ofι h w)) :
IsLimit (Fork.ofι (G.map h) (by simp only [← G.map_comp, w]) : Fork (G.map f) (G.map g)) :=
isLimitMapConeForkEquiv G w (PreservesLimit.preserves l)
#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimit
-/- warning: category_theory.limits.is_limit_of_is_limit_fork_map -> CategoryTheory.Limits.isLimitOfIsLimitForkMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_is_limit_fork_map CategoryTheory.Limits.isLimitOfIsLimitForkMapₓ'. -/
/-- The property of reflecting equalizers expressed in terms of forks. -/
def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
(l : IsLimit (Fork.ofι (G.map h) (by simp only [← G.map_comp, w]) : Fork (G.map f) (G.map g))) :
@@ -77,9 +68,6 @@ def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
variable (f g) [HasEqualizer f g]
-/- warning: category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit -> CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimit is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimitₓ'. -/
/--
If `G` preserves equalizers and `C` has them, then the fork constructed of the mapped morphisms of
a fork is a limit.
@@ -92,12 +80,6 @@ def isLimitOfHasEqualizerOfPreservesLimit [PreservesLimit (parallelPair f g) G]
variable [HasEqualizer (G.map f) (G.map g)]
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-Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_equalizer.of_iso_comparison CategoryTheory.Limits.PreservesEqualizer.ofIsoComparisonₓ'. -/
/-- If the equalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
equalizer of `(f,g)`.
-/
@@ -112,12 +94,6 @@ def PreservesEqualizer.ofIsoComparison [i : IsIso (equalizerComparison f g G)] :
variable [PreservesLimit (parallelPair f g) G]
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-Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_equalizer.iso CategoryTheory.Limits.PreservesEqualizer.isoₓ'. -/
/--
If `G` preserves the equalizer of `(f,g)`, then the equalizer comparison map for `G` at `(f,g)` is
an isomorphism.
@@ -126,12 +102,6 @@ def PreservesEqualizer.iso : G.obj (equalizer f g) ≅ equalizer (G.map f) (G.ma
IsLimit.conePointUniqueUpToIso (isLimitOfHasEqualizerOfPreservesLimit G f g) (limit.isLimit _)
#align category_theory.limits.preserves_equalizer.iso CategoryTheory.Limits.PreservesEqualizer.iso
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@[simp]
theorem PreservesEqualizer.iso_hom :
(PreservesEqualizer.iso G f g).Hom = equalizerComparison f g G :=
@@ -149,9 +119,6 @@ section Coequalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h)
-/- warning: category_theory.limits.is_colimit_map_cocone_cofork_equiv -> CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquivₓ'. -/
/-- The map of a cofork is a colimit iff the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cofork.of_π` with `functor.map_cocone`.
-/
@@ -167,9 +134,6 @@ def isColimitMapCoconeCoforkEquiv :
dsimp; rw [category.comp_id, category.id_comp]
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv
-/- warning: category_theory.limits.is_colimit_cofork_map_of_is_colimit -> CategoryTheory.Limits.isColimitCoforkMapOfIsColimit is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimitₓ'. -/
/-- The property of preserving coequalizers expressed in terms of coforks. -/
def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
(l : IsColimit (Cofork.ofπ h w)) :
@@ -178,9 +142,6 @@ def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
isColimitMapCoconeCoforkEquiv G w (PreservesColimit.preserves l)
#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimit
-/- warning: category_theory.limits.is_colimit_of_is_colimit_cofork_map -> CategoryTheory.Limits.isColimitOfIsColimitCoforkMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_is_colimit_cofork_map CategoryTheory.Limits.isColimitOfIsColimitCoforkMapₓ'. -/
/-- The property of reflecting coequalizers expressed in terms of coforks. -/
def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
(l :
@@ -192,9 +153,6 @@ def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
variable (f g) [HasCoequalizer f g]
-/- warning: category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit is a dubious translation:
-<too large>
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/--
If `G` preserves coequalizers and `C` has them, then the cofork constructed of the mapped morphisms
of a cofork is a colimit.
@@ -206,12 +164,6 @@ def isColimitOfHasCoequalizerOfPreservesColimit [PreservesColimit (parallelPair
variable [HasCoequalizer (G.map f) (G.map g)]
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/-- If the coequalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
coequalizer of `(f,g)`.
-/
@@ -226,12 +178,6 @@ def ofIsoComparison [i : IsIso (coequalizerComparison f g G)] :
variable [PreservesColimit (parallelPair f g) G]
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/--
If `G` preserves the coequalizer of `(f,g)`, then the coequalizer comparison map for `G` at `(f,g)`
is an isomorphism.
@@ -241,12 +187,6 @@ def PreservesCoequalizer.iso : coequalizer (G.map f) (G.map g) ≅ G.obj (coequa
(isColimitOfHasCoequalizerOfPreservesColimit G f g)
#align category_theory.limits.preserves_coequalizer.iso CategoryTheory.Limits.PreservesCoequalizer.iso
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@[simp]
theorem PreservesCoequalizer.iso_hom :
(PreservesCoequalizer.iso G f g).Hom = coequalizerComparison f g G :=
@@ -258,19 +198,10 @@ instance : IsIso (coequalizerComparison f g G) :=
rw [← preserves_coequalizer.iso_hom]
infer_instance
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-Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epiₓ'. -/
instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
⟨fun W h k => by rw [← ι_comp_coequalizer_comparison]; apply (cancel_epi _).1; apply epi_comp⟩
#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epi
-/- warning: category_theory.limits.map_π_preserves_coequalizer_inv -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_invₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv :
G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv =
@@ -280,9 +211,6 @@ theorem map_π_preserves_coequalizer_inv :
comp_id]
#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_inv
-/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_desc -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_descₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
(wk : G.map f ≫ k = G.map g ≫ k) :
@@ -290,9 +218,6 @@ theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
by rw [← category.assoc, map_π_preserves_coequalizer_inv, coequalizer.π_desc]
#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc
-/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_colim_map -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMapₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
@@ -304,9 +229,6 @@ theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y'
by rw [← category.assoc, map_π_preserves_coequalizer_inv, ι_colim_map, parallel_pair_hom_app_one]
#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap
-/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_desc is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_descₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -265,10 +265,7 @@ but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Epi.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) (CategoryTheory.Limits.coequalizer.π.{u1, u3} C _inst_1 X Y f g _inst_3))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epiₓ'. -/
instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
- ⟨fun W h k => by
- rw [← ι_comp_coequalizer_comparison]
- apply (cancel_epi _).1
- apply epi_comp⟩
+ ⟨fun W h k => by rw [← ι_comp_coequalizer_comparison]; apply (cancel_epi _).1; apply epi_comp⟩
#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epi
/- warning: category_theory.limits.map_π_preserves_coequalizer_inv -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv is a dubious translation:
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -44,10 +44,7 @@ section Equalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = h ≫ g)
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquivₓ'. -/
/-- The map of a fork is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `fork.of_ι` with `functor.map_cone`.
@@ -60,10 +57,7 @@ def isLimitMapConeForkEquiv :
#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquiv
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimitₓ'. -/
/-- The property of preserving equalizers expressed in terms of forks. -/
def isLimitForkMapOfIsLimit [PreservesLimit (parallelPair f g) G] (l : IsLimit (Fork.ofι h w)) :
@@ -72,10 +66,7 @@ def isLimitForkMapOfIsLimit [PreservesLimit (parallelPair f g) G] (l : IsLimit (
#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimit
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_is_limit_fork_map CategoryTheory.Limits.isLimitOfIsLimitForkMapₓ'. -/
/-- The property of reflecting equalizers expressed in terms of forks. -/
def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
@@ -87,10 +78,7 @@ def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
variable (f g) [HasEqualizer f g]
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimitₓ'. -/
/--
If `G` preserves equalizers and `C` has them, then the fork constructed of the mapped morphisms of
@@ -162,10 +150,7 @@ section Coequalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h)
/- warning: category_theory.limits.is_colimit_map_cocone_cofork_equiv -> CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv is a dubious translation:
-lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquivₓ'. -/
/-- The map of a cofork is a colimit iff the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cofork.of_π` with `functor.map_cocone`.
@@ -183,10 +168,7 @@ def isColimitMapCoconeCoforkEquiv :
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimitₓ'. -/
/-- The property of preserving coequalizers expressed in terms of coforks. -/
def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
@@ -197,10 +179,7 @@ def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimit
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_is_colimit_cofork_map CategoryTheory.Limits.isColimitOfIsColimitCoforkMapₓ'. -/
/-- The property of reflecting coequalizers expressed in terms of coforks. -/
def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
@@ -214,10 +193,7 @@ def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
variable (f g) [HasCoequalizer f g]
/- warning: category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimitₓ'. -/
/--
If `G` preserves coequalizers and `C` has them, then the cofork constructed of the mapped morphisms
@@ -296,10 +272,7 @@ instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epi
/- warning: category_theory.limits.map_π_preserves_coequalizer_inv -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_invₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv :
@@ -311,10 +284,7 @@ theorem map_π_preserves_coequalizer_inv :
#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_inv
/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_desc -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_descₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
@@ -324,10 +294,7 @@ theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc
/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_colim_map -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMapₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y')
@@ -341,10 +308,7 @@ theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y'
#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap
/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_desc is a dubious translation:
-lean 3 declaration is
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(CategoryTheory.Limits.coequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D 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_inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)) (CategoryTheory.Limits.parallelPair.{u2, u4} D _inst_2 X' Y' f' g') _inst_4 _inst_6 (CategoryTheory.Limits.parallelPairHom.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) X' Y' (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) f' g' p q wf wg)) (CategoryTheory.Limits.coequalizer.desc.{u2, u4} D _inst_2 X' Y' f' g' _inst_6 Z' h wh)))) (CategoryTheory.CategoryStruct.comp.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) Y' Z' q h)
+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_descₓ'. -/
@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X' ⟶ Y')
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -301,7 +301,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (CategoryTheory.Limits.coequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C 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_inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4)) (CategoryTheory.CategoryStruct.comp.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C 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D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D 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(CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_invₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem map_π_preserves_coequalizer_inv :
G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv =
coequalizer.π (G.map f) (G.map g) :=
@@ -316,7 +316,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G] {W : D} (k : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) W) (wk : Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D 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_inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) W (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) k)), Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, 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(CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) W (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) (CategoryTheory.Limits.coequalizer.π.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.CategoryStruct.comp.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D 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_inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4 W k wk))) k
Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_descₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
(wk : G.map f ≫ k = G.map g ≫ k) :
G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv ≫ coequalizer.desc k wk = k :=
@@ -329,7 +329,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G] {X' : D} {Y' : D} (f' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') (g' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') [_inst_6 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 X' Y' f' g'] (p : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C 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Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMapₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
(wg : G.map g ≫ q = p ≫ g') :
@@ -346,7 +346,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D 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_inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G] {X' : D} {Y' : D} (f' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') (g' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') [_inst_6 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 X' Y' f' g'] (p : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C 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Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_descₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
(wg : G.map g ≫ q = p ≫ g') {Z' : D} (h : Y' ⟶ Z') (wh : f' ≫ h = g' ≫ h) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936
@@ -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.limits.preserves.shapes.equalizers
-! leanprover-community/mathlib commit 4698e35ca56a0d4fa53aa5639c3364e0a77f4eba
+! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.CategoryTheory.Limits.Preserves.Basic
/-!
# Preserving (co)equalizers
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
Constructions to relate the notions of preserving (co)equalizers and reflecting (co)equalizers
to concrete (co)forks.
mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25
@@ -40,6 +40,12 @@ section Equalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = h ≫ g)
+/- warning: category_theory.limits.is_limit_map_cone_fork_equiv -> CategoryTheory.Limits.isLimitMapConeForkEquiv is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquivₓ'. -/
/-- The map of a fork is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `fork.of_ι` with `functor.map_cone`.
-/
@@ -50,12 +56,24 @@ def isLimitMapConeForkEquiv :
(IsLimit.equivIsoLimit (Fork.ext (Iso.refl _) (by simp [fork.ι])))
#align category_theory.limits.is_limit_map_cone_fork_equiv CategoryTheory.Limits.isLimitMapConeForkEquiv
+/- warning: category_theory.limits.is_limit_fork_map_of_is_limit -> CategoryTheory.Limits.isLimitForkMapOfIsLimit is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimitₓ'. -/
/-- The property of preserving equalizers expressed in terms of forks. -/
def isLimitForkMapOfIsLimit [PreservesLimit (parallelPair f g) G] (l : IsLimit (Fork.ofι h w)) :
IsLimit (Fork.ofι (G.map h) (by simp only [← G.map_comp, w]) : Fork (G.map f) (G.map g)) :=
isLimitMapConeForkEquiv G w (PreservesLimit.preserves l)
#align category_theory.limits.is_limit_fork_map_of_is_limit CategoryTheory.Limits.isLimitForkMapOfIsLimit
+/- warning: category_theory.limits.is_limit_of_is_limit_fork_map -> CategoryTheory.Limits.isLimitOfIsLimitForkMap is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_is_limit_fork_map CategoryTheory.Limits.isLimitOfIsLimitForkMapₓ'. -/
/-- The property of reflecting equalizers expressed in terms of forks. -/
def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
(l : IsLimit (Fork.ofι (G.map h) (by simp only [← G.map_comp, w]) : Fork (G.map f) (G.map g))) :
@@ -65,6 +83,12 @@ def isLimitOfIsLimitForkMap [ReflectsLimit (parallelPair f g) G]
variable (f g) [HasEqualizer f g]
+/- warning: category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit -> CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory D _inst_2 (CategoryTheory.Limits.parallelPair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)) (CategoryTheory.Limits.Fork.ofι.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3) X (CategoryTheory.Limits.equalizer.ι.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimit._proof_1.{u3, u4, u1, u2} C _inst_1 D _inst_2 G X Y f g _inst_3))
+but is expected to have type
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C _inst_1 X Y f g _inst_3) X Y (CategoryTheory.Limits.equalizer.ι.{u1, u3} C _inst_1 X Y f g _inst_3) f)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3) Y (CategoryTheory.CategoryStruct.comp.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3) X Y (CategoryTheory.Limits.equalizer.ι.{u1, u3} C _inst_1 X Y f g _inst_3) g)))) (CategoryTheory.CategoryStruct.comp.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3) X Y (CategoryTheory.Limits.equalizer.ι.{u1, u3} C _inst_1 X Y f g _inst_3) g) (CategoryTheory.Limits.equalizer.condition.{u1, u3} C _inst_1 X Y f g _inst_3))) (Eq.refl.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3) Y (CategoryTheory.CategoryStruct.comp.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3) X Y (CategoryTheory.Limits.equalizer.ι.{u1, u3} C _inst_1 X Y f g _inst_3) g))))))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimitₓ'. -/
/--
If `G` preserves equalizers and `C` has them, then the fork constructed of the mapped morphisms of
a fork is a limit.
@@ -77,6 +101,12 @@ def isLimitOfHasEqualizerOfPreservesLimit [PreservesLimit (parallelPair f g) G]
variable [HasEqualizer (G.map f) (G.map g)]
+/- warning: category_theory.limits.preserves_equalizer.of_iso_comparison -> CategoryTheory.Limits.PreservesEqualizer.ofIsoComparison is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasEqualizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) _inst_4) (CategoryTheory.Limits.equalizerComparison.{u1, u2, u3, u4} C _inst_1 X Y f g D _inst_2 G _inst_3 _inst_4)], CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasEqualizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4) (CategoryTheory.Limits.equalizerComparison.{u1, u2, u3, u4} C _inst_1 X Y f g D _inst_2 G _inst_3 _inst_4)], CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_equalizer.of_iso_comparison CategoryTheory.Limits.PreservesEqualizer.ofIsoComparisonₓ'. -/
/-- If the equalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
equalizer of `(f,g)`.
-/
@@ -91,6 +121,12 @@ def PreservesEqualizer.ofIsoComparison [i : IsIso (equalizerComparison f g G)] :
variable [PreservesLimit (parallelPair f g) G]
+/- warning: category_theory.limits.preserves_equalizer.iso -> CategoryTheory.Limits.PreservesEqualizer.iso is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasEqualizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Iso.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) _inst_4)
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasEqualizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Iso.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_equalizer.iso CategoryTheory.Limits.PreservesEqualizer.isoₓ'. -/
/--
If `G` preserves the equalizer of `(f,g)`, then the equalizer comparison map for `G` at `(f,g)` is
an isomorphism.
@@ -99,6 +135,12 @@ def PreservesEqualizer.iso : G.obj (equalizer f g) ≅ equalizer (G.map f) (G.ma
IsLimit.conePointUniqueUpToIso (isLimitOfHasEqualizerOfPreservesLimit G f g) (limit.isLimit _)
#align category_theory.limits.preserves_equalizer.iso CategoryTheory.Limits.PreservesEqualizer.iso
+/- warning: category_theory.limits.preserves_equalizer.iso_hom -> CategoryTheory.Limits.PreservesEqualizer.iso_hom is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasEqualizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) _inst_4)) (CategoryTheory.Iso.hom.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) _inst_4) (CategoryTheory.Limits.PreservesEqualizer.iso.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f g _inst_3 _inst_4 _inst_5)) (CategoryTheory.Limits.equalizerComparison.{u1, u2, u3, u4} C _inst_1 X Y f g D _inst_2 G _inst_3 _inst_4)
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasEqualizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ 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(CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4)) (CategoryTheory.Iso.hom.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.equalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.equalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4) (CategoryTheory.Limits.PreservesEqualizer.iso.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f g _inst_3 _inst_4 _inst_5)) (CategoryTheory.Limits.equalizerComparison.{u1, u2, u3, u4} C _inst_1 X Y f g D _inst_2 G _inst_3 _inst_4)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_equalizer.iso_hom CategoryTheory.Limits.PreservesEqualizer.iso_homₓ'. -/
@[simp]
theorem PreservesEqualizer.iso_hom :
(PreservesEqualizer.iso G f g).Hom = equalizerComparison f g G :=
@@ -116,6 +158,12 @@ section Coequalizers
variable {X Y Z : C} {f g : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h)
+/- warning: category_theory.limits.is_colimit_map_cocone_cofork_equiv -> CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquivₓ'. -/
/-- The map of a cofork is a colimit iff the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cofork.of_π` with `functor.map_cocone`.
-/
@@ -131,6 +179,12 @@ def isColimitMapCoconeCoforkEquiv :
dsimp; rw [category.comp_id, category.id_comp]
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv
+/- warning: category_theory.limits.is_colimit_cofork_map_of_is_colimit -> CategoryTheory.Limits.isColimitCoforkMapOfIsColimit is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimitₓ'. -/
/-- The property of preserving coequalizers expressed in terms of coforks. -/
def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
(l : IsColimit (Cofork.ofπ h w)) :
@@ -139,6 +193,12 @@ def isColimitCoforkMapOfIsColimit [PreservesColimit (parallelPair f g) G]
isColimitMapCoconeCoforkEquiv G w (PreservesColimit.preserves l)
#align category_theory.limits.is_colimit_cofork_map_of_is_colimit CategoryTheory.Limits.isColimitCoforkMapOfIsColimit
+/- warning: category_theory.limits.is_colimit_of_is_colimit_cofork_map -> CategoryTheory.Limits.isColimitOfIsColimitCoforkMap is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_is_colimit_cofork_map CategoryTheory.Limits.isColimitOfIsColimitCoforkMapₓ'. -/
/-- The property of reflecting coequalizers expressed in terms of coforks. -/
def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
(l :
@@ -150,6 +210,12 @@ def isColimitOfIsColimitCoforkMap [ReflectsColimit (parallelPair f g) G]
variable (f g) [HasCoequalizer f g]
+/- warning: category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Limits.IsColimit.{0, u2, 0, u4} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory D _inst_2 (CategoryTheory.Limits.parallelPair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)) (CategoryTheory.Limits.Cofork.ofπ.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) (CategoryTheory.Limits.coequalizer.π.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit._proof_1.{u3, u4, u1, u2} C _inst_1 D _inst_2 G X Y f g _inst_3))
+but is expected to have type
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(CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3))) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) (CategoryTheory.CategoryStruct.comp.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1) X Y (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) g (CategoryTheory.Limits.coequalizer.π.{u1, u3} C _inst_1 X Y f g _inst_3)))))))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimitₓ'. -/
/--
If `G` preserves coequalizers and `C` has them, then the cofork constructed of the mapped morphisms
of a cofork is a colimit.
@@ -161,6 +227,12 @@ def isColimitOfHasCoequalizerOfPreservesColimit [PreservesColimit (parallelPair
variable [HasCoequalizer (G.map f) (G.map g)]
+/- warning: category_theory.limits.of_iso_comparison -> CategoryTheory.Limits.ofIsoComparison is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coequalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) _inst_4) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.coequalizerComparison.{u1, u2, u3, u4} C _inst_1 X Y f g D _inst_2 G _inst_3 _inst_4)], CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Limits.coequalizerComparison.{u1, u2, u3, u4} C _inst_1 X Y f g D _inst_2 G _inst_3 _inst_4)], CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.of_iso_comparison CategoryTheory.Limits.ofIsoComparisonₓ'. -/
/-- If the coequalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
coequalizer of `(f,g)`.
-/
@@ -175,6 +247,12 @@ def ofIsoComparison [i : IsIso (coequalizerComparison f g G)] :
variable [PreservesColimit (parallelPair f g) G]
+/- warning: category_theory.limits.preserves_coequalizer.iso -> CategoryTheory.Limits.PreservesCoequalizer.iso is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Iso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coequalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g) _inst_4) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Iso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g) _inst_4) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_coequalizer.iso CategoryTheory.Limits.PreservesCoequalizer.isoₓ'. -/
/--
If `G` preserves the coequalizer of `(f,g)`, then the coequalizer comparison map for `G` at `(f,g)`
is an isomorphism.
@@ -184,6 +262,12 @@ def PreservesCoequalizer.iso : coequalizer (G.map f) (G.map g) ≅ G.obj (coequa
(isColimitOfHasCoequalizerOfPreservesColimit G f g)
#align category_theory.limits.preserves_coequalizer.iso CategoryTheory.Limits.PreservesCoequalizer.iso
+/- warning: category_theory.limits.preserves_coequalizer.iso_hom -> CategoryTheory.Limits.PreservesCoequalizer.iso_hom is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_coequalizer.iso_hom CategoryTheory.Limits.PreservesCoequalizer.iso_homₓ'. -/
@[simp]
theorem PreservesCoequalizer.iso_hom :
(PreservesCoequalizer.iso G f g).Hom = coequalizerComparison f g G :=
@@ -195,6 +279,12 @@ instance : IsIso (coequalizerComparison f g G) :=
rw [← preserves_coequalizer.iso_hom]
infer_instance
+/- warning: category_theory.limits.map_π_epi -> CategoryTheory.Limits.map_π_epi is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Epi.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) (CategoryTheory.Limits.coequalizer.π.{u1, u3} C _inst_1 X Y f g _inst_3))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G], CategoryTheory.Epi.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y (CategoryTheory.Limits.coequalizer.{u1, u3} C _inst_1 X Y f g _inst_3) (CategoryTheory.Limits.coequalizer.π.{u1, u3} C _inst_1 X Y f g _inst_3))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epiₓ'. -/
instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
⟨fun W h k => by
rw [← ι_comp_coequalizer_comparison]
@@ -202,6 +292,12 @@ instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
apply epi_comp⟩
#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epi
+/- warning: category_theory.limits.map_π_preserves_coequalizer_inv -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_invₓ'. -/
@[reassoc.1]
theorem map_π_preserves_coequalizer_inv :
G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv =
@@ -211,6 +307,12 @@ theorem map_π_preserves_coequalizer_inv :
comp_id]
#align category_theory.limits.map_π_preserves_coequalizer_inv CategoryTheory.Limits.map_π_preserves_coequalizer_inv
+/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_desc -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_descₓ'. -/
@[reassoc.1]
theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
(wk : G.map f ≫ k = G.map g ≫ k) :
@@ -218,6 +320,12 @@ theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
by rw [← category.assoc, map_π_preserves_coequalizer_inv, coequalizer.π_desc]
#align category_theory.limits.map_π_preserves_coequalizer_inv_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_desc
+/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_colim_map -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap is a dubious translation:
+lean 3 declaration is
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CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G] {X' : D} {Y' : D} (f' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') (g' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') [_inst_6 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 X' Y' f' g'] (p : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) X') (q : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, 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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMapₓ'. -/
@[reassoc.1]
theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
@@ -229,6 +337,12 @@ theorem map_π_preserves_coequalizer_inv_colimMap {X' Y' : D} (f' g' : X' ⟶ Y'
by rw [← category.assoc, map_π_preserves_coequalizer_inv, ι_colim_map, parallel_pair_hom_app_one]
#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap
+/- warning: category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc -> CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_desc is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u3} C _inst_1 X Y f g] [_inst_4 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 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_inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X Y g)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f g) G] {X' : D} {Y' : D} (f' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') (g' : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) X' Y') [_inst_6 : CategoryTheory.Limits.HasCoequalizer.{u2, u4} D _inst_2 X' Y' f' g'] (p : Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C 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(CategoryTheory.Limits.coequalizer.desc.{u2, u4} D _inst_2 X' Y' f' g' _inst_6 Z' h wh)))) (CategoryTheory.CategoryStruct.comp.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) Y' Z' q h)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_descₓ'. -/
@[reassoc.1]
theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X' ⟶ Y')
[HasCoequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y') (wf : G.map f ≫ q = p ≫ f')
@@ -242,6 +356,7 @@ theorem map_π_preserves_coequalizer_inv_colimMap_desc {X' Y' : D} (f' g' : X'
slice_lhs 2 3 => rw [coequalizer.π_desc]
#align category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap_desc
+#print CategoryTheory.Limits.preservesSplitCoequalizers /-
/-- Any functor preserves coequalizers of split pairs. -/
instance (priority := 1) preservesSplitCoequalizers (f g : X ⟶ Y) [HasSplitCoequalizer f g] :
PreservesColimit (parallelPair f g) G :=
@@ -253,6 +368,7 @@ instance (priority := 1) preservesSplitCoequalizers (f g : X ⟶ Y) [HasSplitCoe
(is_colimit_map_cocone_cofork_equiv G _).symm
((has_split_coequalizer.is_split_coequalizer f g).map G).isCoequalizer
#align category_theory.limits.preserves_split_coequalizers CategoryTheory.Limits.preservesSplitCoequalizers
+-/
end Coequalizers
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)
@@ -26,9 +26,7 @@ universe w v₁ v₂ u₁ u₂
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
variable {C : Type u₁} [Category.{v₁} C]
-
variable {D : Type u₂} [Category.{v₂} D]
-
variable (G : C ⥤ D)
namespace CategoryTheory.Limits
@@ -2,15 +2,12 @@
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.limits.preserves.shapes.equalizers
-! leanprover-community/mathlib commit 4698e35ca56a0d4fa53aa5639c3364e0a77f4eba
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
+#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
+
/-!
# Preserving (co)equalizers
by
s! (#3825)
This PR puts, with one exception, every single remaining by
that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh
. The exception is when the by
begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.
Essentially this is s/\n *by$/ by/g
, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated by
s".
@@ -124,8 +124,7 @@ def isColimitMapCoconeCoforkEquiv :
(Cofork.ofπ (G.map h) (by simp only [← G.map_comp, w]) : Cofork (G.map f) (G.map g)) :=
(IsColimit.precomposeInvEquiv (diagramIsoParallelPair _) _).symm.trans <|
IsColimit.equivIsoColimit <|
- Cofork.ext (Iso.refl _) <|
- by
+ Cofork.ext (Iso.refl _) <| by
dsimp only [Cofork.π, Cofork.ofπ_ι_app]
dsimp; rw [Category.comp_id, Category.id_comp]
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv
vscode is already configured by .vscode/settings.json
to trim these on save. It's not clear how they've managed to stick around.
By doing this all in one PR now, it avoids getting random whitespace diffs in PRs later.
This was done with a regex search in vscode,
@@ -70,8 +70,8 @@ If `G` preserves equalizers and `C` has them, then the fork constructed of the m
a fork is a limit.
-/
def isLimitOfHasEqualizerOfPreservesLimit [PreservesLimit (parallelPair f g) G] :
- IsLimit (Fork.ofι
- (G.map (equalizer.ι f g)) (by simp only [← G.map_comp]; rw [equalizer.condition]) :
+ IsLimit (Fork.ofι
+ (G.map (equalizer.ι f g)) (by simp only [← G.map_comp]; rw [equalizer.condition]) :
Fork (G.map f) (G.map g)) :=
isLimitForkMapOfIsLimit G _ (equalizerIsEqualizer f g)
#align category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit CategoryTheory.Limits.isLimitOfHasEqualizerOfPreservesLimit
@@ -154,7 +154,7 @@ If `G` preserves coequalizers and `C` has them, then the cofork constructed of t
of a cofork is a colimit.
-/
def isColimitOfHasCoequalizerOfPreservesColimit [PreservesColimit (parallelPair f g) G] :
- IsColimit (Cofork.ofπ (G.map (coequalizer.π f g)) (by
+ IsColimit (Cofork.ofπ (G.map (coequalizer.π f g)) (by
simp only [← G.map_comp]; rw [coequalizer.condition]) : Cofork (G.map f) (G.map g)) :=
isColimitCoforkMapOfIsColimit G _ (coequalizerIsCoequalizer f g)
#align category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCoequalizerOfPreservesColimit
@@ -168,7 +168,7 @@ def ofIsoComparison [i : IsIso (coequalizerComparison f g G)] :
PreservesColimit (parallelPair f g) G := by
apply preservesColimitOfPreservesColimitCocone (coequalizerIsCoequalizer f g)
apply (isColimitMapCoconeCoforkEquiv _ _).symm _
- refine
+ refine
@IsColimit.ofPointIso _ _ _ _ _ _ _ (colimit.isColimit (parallelPair (G.map f) (G.map g))) ?_
apply i
#align category_theory.limits.of_iso_comparison CategoryTheory.Limits.ofIsoComparison
@@ -197,9 +197,9 @@ instance : IsIso (coequalizerComparison f g G) := by
instance map_π_epi : Epi (G.map (coequalizer.π f g)) :=
⟨fun {W} h k => by
rw [← ι_comp_coequalizerComparison]
- haveI : Epi (coequalizer.π (G.map f) (G.map g) ≫ coequalizerComparison f g G) := by
+ haveI : Epi (coequalizer.π (G.map f) (G.map g) ≫ coequalizerComparison f g G) := by
apply epi_comp
- apply (cancel_epi _).1⟩
+ apply (cancel_epi _).1⟩
#align category_theory.limits.map_π_epi CategoryTheory.Limits.map_π_epi
@[reassoc]
@@ -254,4 +254,3 @@ instance (priority := 1) preservesSplitCoequalizers (f g : X ⟶ Y) [HasSplitCoe
end Coequalizers
end CategoryTheory.Limits
-
All dependencies are ported!