category_theory.limits.preserves.shapes.kernels
⟷
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
-import Mathbin.CategoryTheory.Limits.Shapes.Kernels
-import Mathbin.CategoryTheory.Limits.Preserves.Shapes.Zero
+import CategoryTheory.Limits.Shapes.Kernels
+import CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.kernels from "leanprover-community/mathlib"@"10bf4f825ad729c5653adc039dafa3622e7f93c9"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.limits.preserves.shapes.kernels
-! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Shapes.Kernels
import Mathbin.CategoryTheory.Limits.Preserves.Shapes.Zero
+#align_import category_theory.limits.preserves.shapes.kernels from "leanprover-community/mathlib"@"10bf4f825ad729c5653adc039dafa3622e7f93c9"
+
/-!
# Preserving (co)kernels
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -43,6 +43,7 @@ section Kernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)
+#print CategoryTheory.Limits.isLimitMapConeForkEquiv' /-
/-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
This essentially lets us commute `kernel_fork.of_ι` with `functor.map_cone`.
@@ -61,7 +62,9 @@ def isLimitMapConeForkEquiv' :
refine' fork.ext (iso.refl _) _
simp [fork.ι]
#align category_theory.limits.is_limit_map_cone_fork_equiv' CategoryTheory.Limits.isLimitMapConeForkEquiv'
+-/
+#print CategoryTheory.Limits.isLimitForkMapOfIsLimit' /-
/-- The property of preserving kernels expressed in terms of kernel forks.
This is a variant of `is_limit_fork_map_of_is_limit` for equalizers,
@@ -74,9 +77,11 @@ def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G]
Fork (G.map f) 0) :=
isLimitMapConeForkEquiv' G w (PreservesLimit.preserves l)
#align category_theory.limits.is_limit_fork_map_of_is_limit' CategoryTheory.Limits.isLimitForkMapOfIsLimit'
+-/
variable (f) [HasKernel f]
+#print CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit /-
/-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit.
-/
@@ -87,12 +92,14 @@ def isLimitOfHasKernelOfPreservesLimit [PreservesLimit (parallelPair f 0) G] :
Fork (G.map f) 0) :=
isLimitForkMapOfIsLimit' G (kernel.condition f) (kernelIsKernel f)
#align category_theory.limits.is_limit_of_has_kernel_of_preserves_limit CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit
+-/
instance [PreservesLimit (parallelPair f 0) G] : HasKernel (G.map f)
where exists_limit := ⟨⟨_, isLimitOfHasKernelOfPreservesLimit G f⟩⟩
variable [HasKernel (G.map f)]
+#print CategoryTheory.Limits.PreservesKernel.ofIsoComparison /-
/-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
@@ -104,26 +111,32 @@ def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
apply is_limit.of_point_iso (kernel_is_kernel (G.map f))
exact i
#align category_theory.limits.preserves_kernel.of_iso_comparison CategoryTheory.Limits.PreservesKernel.ofIsoComparison
+-/
variable [PreservesLimit (parallelPair f 0) G]
+#print CategoryTheory.Limits.PreservesKernel.iso /-
/-- If `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is
an isomorphism.
-/
def PreservesKernel.iso : G.obj (kernel f) ≅ kernel (G.map f) :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasKernelOfPreservesLimit G f) (kernelIsKernel _)
#align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.iso
+-/
+#print CategoryTheory.Limits.PreservesKernel.iso_hom /-
@[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).Hom = kernelComparison f G :=
rfl
#align category_theory.limits.preserves_kernel.iso_hom CategoryTheory.Limits.PreservesKernel.iso_hom
+-/
instance : IsIso (kernelComparison f G) :=
by
rw [← preserves_kernel.iso_hom]
infer_instance
+#print CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv /-
@[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
@@ -135,6 +148,7 @@ theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [Ha
rw [iso.comp_inv_eq, category.assoc, preserves_kernel.iso_hom, iso.eq_inv_comp]
exact kernel_comparison_comp_kernel_map _ _ _ _ _ _
#align category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv
+-/
end Kernels
@@ -142,6 +156,7 @@ section Cokernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)
+#print CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv' /-
/-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cokernel_cofork.of_π` with `functor.map_cocone`.
@@ -162,7 +177,9 @@ def isColimitMapCoconeCoforkEquiv' :
parallel_pair.ext_hom_app, functor.map_cocone_ι_app, cofork.of_π_ι_app]
apply category.comp_id
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv' CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv'
+-/
+#print CategoryTheory.Limits.isColimitCoforkMapOfIsColimit' /-
/-- The property of preserving cokernels expressed in terms of cokernel coforks.
This is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,
@@ -175,9 +192,11 @@ def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G]
Cofork (G.map f) 0) :=
isColimitMapCoconeCoforkEquiv' G w (PreservesColimit.preserves l)
#align category_theory.limits.is_colimit_cofork_map_of_is_colimit' CategoryTheory.Limits.isColimitCoforkMapOfIsColimit'
+-/
variable (f) [HasCokernel f]
+#print CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit /-
/--
If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
a cokernel cofork is a colimit.
@@ -189,12 +208,14 @@ def isColimitOfHasCokernelOfPreservesColimit [PreservesColimit (parallelPair f 0
Cofork (G.map f) 0) :=
isColimitCoforkMapOfIsColimit' G (cokernel.condition f) (cokernelIsCokernel f)
#align category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit
+-/
instance [PreservesColimit (parallelPair f 0) G] : HasCokernel (G.map f)
where exists_colimit := ⟨⟨_, isColimitOfHasCokernelOfPreservesColimit G f⟩⟩
variable [HasCokernel (G.map f)]
+#print CategoryTheory.Limits.PreservesCokernel.ofIsoComparison /-
/-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
@@ -206,9 +227,11 @@ def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
apply is_colimit.of_point_iso (cokernel_is_cokernel (G.map f))
exact i
#align category_theory.limits.preserves_cokernel.of_iso_comparison CategoryTheory.Limits.PreservesCokernel.ofIsoComparison
+-/
variable [PreservesColimit (parallelPair f 0) G]
+#print CategoryTheory.Limits.PreservesCokernel.iso /-
/-- If `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is
an isomorphism.
-/
@@ -216,17 +239,21 @@ def PreservesCokernel.iso : G.obj (cokernel f) ≅ cokernel (G.map f) :=
IsColimit.coconePointUniqueUpToIso (isColimitOfHasCokernelOfPreservesColimit G f)
(cokernelIsCokernel _)
#align category_theory.limits.preserves_cokernel.iso CategoryTheory.Limits.PreservesCokernel.iso
+-/
+#print CategoryTheory.Limits.PreservesCokernel.iso_inv /-
@[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G :=
rfl
#align category_theory.limits.preserves_cokernel.iso_inv CategoryTheory.Limits.PreservesCokernel.iso_inv
+-/
instance : IsIso (cokernelComparison f G) :=
by
rw [← preserves_cokernel.iso_inv]
infer_instance
+#print CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map /-
@[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
@@ -239,6 +266,7 @@ theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [Ha
rw [← iso.comp_inv_eq, category.assoc, ← iso.eq_inv_comp]
exact cokernel_map_comp_cokernel_comparison _ _ _ _ _ _
#align category_theory.limits.preserves_cokernel_iso_comp_cokernel_map CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map
+-/
end Cokernels
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -43,9 +43,6 @@ section Kernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)
-/- 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.isLimitMapConeForkEquiv'ₓ'. -/
/-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
This essentially lets us commute `kernel_fork.of_ι` with `functor.map_cone`.
@@ -65,9 +62,6 @@ def isLimitMapConeForkEquiv' :
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.isLimitForkMapOfIsLimit'ₓ'. -/
/-- The property of preserving kernels expressed in terms of kernel forks.
This is a variant of `is_limit_fork_map_of_is_limit` for equalizers,
@@ -83,9 +77,6 @@ def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G]
variable (f) [HasKernel f]
-/- warning: category_theory.limits.is_limit_of_has_kernel_of_preserves_limit -> CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_has_kernel_of_preserves_limit CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimitₓ'. -/
/-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit.
-/
@@ -102,12 +93,6 @@ instance [PreservesLimit (parallelPair f 0) G] : HasKernel (G.map f)
variable [HasKernel (G.map f)]
-/- warning: category_theory.limits.preserves_kernel.of_iso_comparison -> CategoryTheory.Limits.PreservesKernel.ofIsoComparison is a dubious translation:
-lean 3 declaration is
- forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_3 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6)) (CategoryTheory.Limits.kernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f) _inst_7) (CategoryTheory.Limits.kernelComparison.{u1, u2, u3, u4} C _inst_1 _inst_2 X Y f D _inst_3 _inst_4 G _inst_5 _inst_6 _inst_7)], CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Y))))) G
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.of_iso_comparison CategoryTheory.Limits.PreservesKernel.ofIsoComparisonₓ'. -/
/-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
@@ -122,12 +107,6 @@ def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
variable [PreservesLimit (parallelPair f 0) G]
-/- warning: category_theory.limits.preserves_kernel.iso -> CategoryTheory.Limits.PreservesKernel.iso is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.isoₓ'. -/
/-- If `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is
an isomorphism.
-/
@@ -135,9 +114,6 @@ def PreservesKernel.iso : G.obj (kernel f) ≅ kernel (G.map f) :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasKernelOfPreservesLimit G f) (kernelIsKernel _)
#align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.iso
-/- warning: category_theory.limits.preserves_kernel.iso_hom -> CategoryTheory.Limits.PreservesKernel.iso_hom is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.iso_hom CategoryTheory.Limits.PreservesKernel.iso_homₓ'. -/
@[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).Hom = kernelComparison f G :=
rfl
@@ -148,9 +124,6 @@ instance : IsIso (kernelComparison f G) :=
rw [← preserves_kernel.iso_hom]
infer_instance
-/- warning: category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv -> CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_invₓ'. -/
@[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
@@ -169,9 +142,6 @@ section Cokernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)
-/- 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.isColimitMapCoconeCoforkEquiv'ₓ'. -/
/-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cokernel_cofork.of_π` with `functor.map_cocone`.
@@ -193,9 +163,6 @@ def isColimitMapCoconeCoforkEquiv' :
apply category.comp_id
#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.isColimitCoforkMapOfIsColimit'ₓ'. -/
/-- The property of preserving cokernels expressed in terms of cokernel coforks.
This is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,
@@ -211,9 +178,6 @@ def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G]
variable (f) [HasCokernel f]
-/- warning: category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimitₓ'. -/
/--
If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
a cokernel cofork is a colimit.
@@ -231,12 +195,6 @@ instance [PreservesColimit (parallelPair f 0) G] : HasCokernel (G.map f)
variable [HasCokernel (G.map f)]
-/- warning: category_theory.limits.preserves_cokernel.of_iso_comparison -> CategoryTheory.Limits.PreservesCokernel.ofIsoComparison is a dubious translation:
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/-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
@@ -251,12 +209,6 @@ def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
variable [PreservesColimit (parallelPair f 0) G]
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/-- If `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is
an isomorphism.
-/
@@ -265,9 +217,6 @@ def PreservesCokernel.iso : G.obj (cokernel f) ≅ cokernel (G.map f) :=
(cokernelIsCokernel _)
#align category_theory.limits.preserves_cokernel.iso CategoryTheory.Limits.PreservesCokernel.iso
-/- warning: category_theory.limits.preserves_cokernel.iso_inv -> CategoryTheory.Limits.PreservesCokernel.iso_inv is a dubious translation:
-<too large>
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@[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G :=
rfl
@@ -278,9 +227,6 @@ instance : IsIso (cokernelComparison f G) :=
rw [← preserves_cokernel.iso_inv]
infer_instance
-/- warning: category_theory.limits.preserves_cokernel_iso_comp_cokernel_map -> CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map is a dubious translation:
-<too large>
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@[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -44,10 +44,7 @@ section Kernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_map_cone_fork_equiv' CategoryTheory.Limits.isLimitMapConeForkEquiv'ₓ'. -/
/-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
@@ -69,10 +66,7 @@ def isLimitMapConeForkEquiv' :
#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:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_fork_map_of_is_limit' CategoryTheory.Limits.isLimitForkMapOfIsLimit'ₓ'. -/
/-- The property of preserving kernels expressed in terms of kernel forks.
@@ -90,10 +84,7 @@ def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G]
variable (f) [HasKernel f]
/- warning: category_theory.limits.is_limit_of_has_kernel_of_preserves_limit -> CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit 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_has_kernel_of_preserves_limit CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimitₓ'. -/
/-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit.
@@ -145,10 +136,7 @@ def PreservesKernel.iso : G.obj (kernel f) ≅ kernel (G.map f) :=
#align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.iso
/- warning: category_theory.limits.preserves_kernel.iso_hom -> CategoryTheory.Limits.PreservesKernel.iso_hom is a dubious translation:
-lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.iso_hom CategoryTheory.Limits.PreservesKernel.iso_homₓ'. -/
@[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).Hom = kernelComparison f G :=
@@ -161,10 +149,7 @@ instance : IsIso (kernelComparison f G) :=
infer_instance
/- warning: category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv -> CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_invₓ'. -/
@[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
@@ -185,10 +170,7 @@ section Cokernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_map_cocone_cofork_equiv' CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv'ₓ'. -/
/-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
@@ -212,10 +194,7 @@ 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.isColimitCoforkMapOfIsColimit'ₓ'. -/
/-- The property of preserving cokernels expressed in terms of cokernel coforks.
@@ -233,10 +212,7 @@ def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G]
variable (f) [HasCokernel f]
/- warning: category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimitₓ'. -/
/--
If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
@@ -290,10 +266,7 @@ def PreservesCokernel.iso : G.obj (cokernel f) ≅ cokernel (G.map f) :=
#align category_theory.limits.preserves_cokernel.iso CategoryTheory.Limits.PreservesCokernel.iso
/- warning: category_theory.limits.preserves_cokernel.iso_inv -> CategoryTheory.Limits.PreservesCokernel.iso_inv is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_cokernel.iso_inv CategoryTheory.Limits.PreservesCokernel.iso_invₓ'. -/
@[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G :=
@@ -306,10 +279,7 @@ instance : IsIso (cokernelComparison f G) :=
infer_instance
/- warning: category_theory.limits.preserves_cokernel_iso_comp_cokernel_map -> CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map is a dubious translation:
-lean 3 declaration is
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_cokernel_iso_comp_cokernel_map CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_mapₓ'. -/
@[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -166,7 +166,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X Y)))) G] {X' : C} {Y' : C} (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') [_inst_9 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X' Y' g] [_inst_10 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X' Y' g)] [_inst_11 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X' Y' g (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X' Y')))) G] (p : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X X') (q : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) Y Y') (hpq : Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} 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_inst_6) (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X' Y' g _inst_9) (CategoryTheory.Limits.kernel.map.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6 X' Y' g _inst_9 p q hpq)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_invₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
@@ -311,7 +311,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasCokernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasCokernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X Y)))) G] {X' : C} {Y' : C} (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') [_inst_9 : CategoryTheory.Limits.HasCokernel.{u1, u3} C _inst_1 _inst_2 X' Y' g] [_inst_10 : CategoryTheory.Limits.HasCokernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X' Y' g)] [_inst_11 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X' Y' g (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X' Y')))) G] (p : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X X') (q : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) Y Y') (hpq : Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C 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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_cokernel_iso_comp_cokernel_map CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_mapₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
! This file was ported from Lean 3 source module category_theory.limits.preserves.shapes.kernels
-! leanprover-community/mathlib commit 956af7c76589f444f2e1313911bad16366ea476d
+! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
! 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.Shapes.Zero
/-!
# Preserving (co)kernels
+> 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)kernels and reflecting (co)kernels
to concrete (co)forks.
mathlib commit https://github.com/leanprover-community/mathlib/commit/2af0836443b4cfb5feda0df0051acdb398304931
@@ -40,6 +40,12 @@ section Kernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)
+/- warning: category_theory.limits.is_limit_map_cone_fork_equiv' -> CategoryTheory.Limits.isLimitMapConeForkEquiv' 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_map_cone_fork_equiv' CategoryTheory.Limits.isLimitMapConeForkEquiv'ₓ'. -/
/-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
This essentially lets us commute `kernel_fork.of_ι` with `functor.map_cone`.
@@ -59,6 +65,12 @@ def isLimitMapConeForkEquiv' :
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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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_fork_map_of_is_limit' CategoryTheory.Limits.isLimitForkMapOfIsLimit'ₓ'. -/
/-- The property of preserving kernels expressed in terms of kernel forks.
This is a variant of `is_limit_fork_map_of_is_limit` for equalizers,
@@ -74,6 +86,12 @@ def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G]
variable (f) [HasKernel f]
+/- warning: category_theory.limits.is_limit_of_has_kernel_of_preserves_limit -> CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit 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_has_kernel_of_preserves_limit CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimitₓ'. -/
/-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit.
-/
@@ -90,6 +108,12 @@ instance [PreservesLimit (parallelPair f 0) G] : HasKernel (G.map f)
variable [HasKernel (G.map f)]
+/- warning: category_theory.limits.preserves_kernel.of_iso_comparison -> CategoryTheory.Limits.PreservesKernel.ofIsoComparison is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_3 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6)) (CategoryTheory.Limits.kernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f) _inst_7) (CategoryTheory.Limits.kernelComparison.{u1, u2, u3, u4} C _inst_1 _inst_2 X Y f D _inst_3 _inst_4 G _inst_5 _inst_6 _inst_7)], CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Y))))) G
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.of_iso_comparison CategoryTheory.Limits.PreservesKernel.ofIsoComparisonₓ'. -/
/-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`.
-/
@@ -104,6 +128,12 @@ def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] :
variable [PreservesLimit (parallelPair f 0) G]
+/- warning: category_theory.limits.preserves_kernel.iso -> CategoryTheory.Limits.PreservesKernel.iso is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Y))))) G], CategoryTheory.Iso.{u2, u4} D _inst_3 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6)) (CategoryTheory.Limits.kernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f) _inst_7)
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X Y)))) G], CategoryTheory.Iso.{u2, u4} D _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6)) (CategoryTheory.Limits.kernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f) _inst_7)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.isoₓ'. -/
/-- If `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is
an isomorphism.
-/
@@ -111,6 +141,12 @@ def PreservesKernel.iso : G.obj (kernel f) ≅ kernel (G.map f) :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasKernelOfPreservesLimit G f) (kernelIsKernel _)
#align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.iso
+/- warning: category_theory.limits.preserves_kernel.iso_hom -> CategoryTheory.Limits.PreservesKernel.iso_hom is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Y))))) G], Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_3)) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6)) (CategoryTheory.Limits.kernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f) _inst_7)) (CategoryTheory.Iso.hom.{u2, u4} D _inst_3 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G (CategoryTheory.Limits.kernel.{u1, u3} C _inst_1 _inst_2 X Y f _inst_6)) (CategoryTheory.Limits.kernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f) _inst_7) (CategoryTheory.Limits.PreservesKernel.iso.{u1, u2, u3, u4} C _inst_1 _inst_2 D _inst_3 _inst_4 G _inst_5 X Y f _inst_6 _inst_7 _inst_8)) (CategoryTheory.Limits.kernelComparison.{u1, u2, u3, u4} C _inst_1 _inst_2 X Y f D _inst_3 _inst_4 G _inst_5 _inst_6 _inst_7)
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair 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_inst_2 D _inst_3 _inst_4 G _inst_5 X Y f _inst_6 _inst_7 _inst_8)) (CategoryTheory.Limits.kernelComparison.{u1, u2, u3, u4} C _inst_1 _inst_2 X Y f D _inst_3 _inst_4 G _inst_5 _inst_6 _inst_7)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_kernel.iso_hom CategoryTheory.Limits.PreservesKernel.iso_homₓ'. -/
@[simp]
theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).Hom = kernelComparison f G :=
rfl
@@ -121,6 +157,12 @@ instance : IsIso (kernelComparison f G) :=
rw [← preserves_kernel.iso_hom]
infer_instance
+/- warning: category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv -> CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Y))))) G] {X' : C} {Y' : C} (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') [_inst_9 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X' Y' g] [_inst_10 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X') (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y') (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X' Y' g)] [_inst_11 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X' Y' g (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C 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+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X Y)))) G] {X' : C} {Y' : C} (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') [_inst_9 : CategoryTheory.Limits.HasKernel.{u1, u3} C _inst_1 _inst_2 X' Y' g] [_inst_10 : CategoryTheory.Limits.HasKernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X' Y' g)] [_inst_11 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C 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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_invₓ'. -/
@[reassoc.1]
theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g]
[HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
@@ -139,6 +181,12 @@ section Cokernels
variable {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)
+/- 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.isColimitMapCoconeCoforkEquiv'ₓ'. -/
/-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cokernel_cofork.of_π` with `functor.map_cocone`.
@@ -160,6 +208,12 @@ def isColimitMapCoconeCoforkEquiv' :
apply category.comp_id
#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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(CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1) X Y Z f h) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Z) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Z) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Z) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Z))))) [_inst_6 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C 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+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {X : C} {Y : C} {Z : C} {f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y} {h : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) Y Z} (w : Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Z) (CategoryTheory.CategoryStruct.comp.{u1, u3} C 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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_cofork_map_of_is_colimit' CategoryTheory.Limits.isColimitCoforkMapOfIsColimit'ₓ'. -/
/-- The property of preserving cokernels expressed in terms of cokernel coforks.
This is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,
@@ -175,6 +229,12 @@ def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G]
variable (f) [HasCokernel f]
+/- warning: category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit 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_has_cokernel_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimitₓ'. -/
/--
If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
a cokernel cofork is a colimit.
@@ -192,6 +252,12 @@ instance [PreservesColimit (parallelPair f 0) G] : HasCokernel (G.map f)
variable [HasCokernel (G.map f)]
+/- warning: category_theory.limits.preserves_cokernel.of_iso_comparison -> CategoryTheory.Limits.PreservesCokernel.ofIsoComparison 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.preserves_cokernel.of_iso_comparison CategoryTheory.Limits.PreservesCokernel.ofIsoComparisonₓ'. -/
/-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`.
-/
@@ -206,6 +272,12 @@ def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] :
variable [PreservesColimit (parallelPair f 0) G]
+/- warning: category_theory.limits.preserves_cokernel.iso -> CategoryTheory.Limits.PreservesCokernel.iso 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.preserves_cokernel.iso CategoryTheory.Limits.PreservesCokernel.isoₓ'. -/
/-- If `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is
an isomorphism.
-/
@@ -214,6 +286,12 @@ def PreservesCokernel.iso : G.obj (cokernel f) ≅ cokernel (G.map f) :=
(cokernelIsCokernel _)
#align category_theory.limits.preserves_cokernel.iso CategoryTheory.Limits.PreservesCokernel.iso
+/- warning: category_theory.limits.preserves_cokernel.iso_inv -> CategoryTheory.Limits.PreservesCokernel.iso_inv is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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(CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u3} C _inst_1 _inst_2 X Y)))) G], Eq.{succ u2} (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_3)) (CategoryTheory.Limits.cokernel.{u2, u4} D _inst_3 _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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f) _inst_7) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C 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(CategoryTheory.Limits.PreservesCokernel.iso.{u1, u2, u3, u4} C _inst_1 _inst_2 D _inst_3 _inst_4 G _inst_5 X Y f _inst_6 _inst_7 _inst_8)) (CategoryTheory.Limits.cokernelComparison.{u1, u2, u3, u4} C _inst_1 _inst_2 X Y f D _inst_3 _inst_4 G _inst_5 _inst_6 _inst_7)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_cokernel.iso_inv CategoryTheory.Limits.PreservesCokernel.iso_invₓ'. -/
@[simp]
theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G :=
rfl
@@ -224,6 +302,12 @@ instance : IsIso (cokernelComparison f G) :=
rw [← preserves_cokernel.iso_inv]
infer_instance
+/- warning: category_theory.limits.preserves_cokernel_iso_comp_cokernel_map -> CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u1, u3} C _inst_1] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] [_inst_4 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u4} D _inst_3] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_3) [_inst_5 : CategoryTheory.Functor.PreservesZeroMorphisms.{u1, u2, u3, u4} C _inst_1 D _inst_3 _inst_2 _inst_4 G] {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) [_inst_6 : CategoryTheory.Limits.HasCokernel.{u1, u3} C _inst_1 _inst_2 X Y f] [_inst_7 : CategoryTheory.Limits.HasCokernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X Y f (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u3} C _inst_1 _inst_2 X Y))))) G] {X' : C} {Y' : C} (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') [_inst_9 : CategoryTheory.Limits.HasCokernel.{u1, u3} C _inst_1 _inst_2 X' Y' g] [_inst_10 : CategoryTheory.Limits.HasCokernel.{u2, u4} D _inst_3 _inst_4 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X') (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_3 G Y') (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_3 G X' Y' g)] [_inst_11 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (CategoryTheory.Limits.parallelPair.{u1, u3} C _inst_1 X' Y' g (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X' Y') 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u3} C 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(CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 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_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X Y f)] [_inst_8 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_3 CategoryTheory.Limits.WalkingParallelPair 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(CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_3 G) X' Y' g) _inst_10) (CategoryTheory.Limits.PreservesCokernel.iso.{u1, u2, u3, u4} C _inst_1 _inst_2 D _inst_3 _inst_4 G _inst_5 X' Y' g _inst_9 _inst_10 _inst_11)))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_cokernel_iso_comp_cokernel_map CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_mapₓ'. -/
@[reassoc.1]
theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [HasCokernel g]
[HasCokernel (G.map g)] [PreservesColimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
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)
@@ -26,7 +26,6 @@ universe v₁ v₂ u₁ u₂
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
variable {C : Type u₁} [Category.{v₁} C] [HasZeroMorphisms C]
-
variable {D : Type u₂} [Category.{v₂} D] [HasZeroMorphisms D]
namespace CategoryTheory.Limits
@@ -111,8 +111,8 @@ def isLimitOfHasKernelOfPreservesLimit [PreservesLimit (parallelPair f 0) G] :
isLimitForkMapOfIsLimit' G (kernel.condition f) (kernelIsKernel f)
#align category_theory.limits.is_limit_of_has_kernel_of_preserves_limit CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit
-instance [PreservesLimit (parallelPair f 0) G] : HasKernel (G.map f)
- where exists_limit := ⟨⟨_, isLimitOfHasKernelOfPreservesLimit G f⟩⟩
+instance [PreservesLimit (parallelPair f 0) G] : HasKernel (G.map f) where
+ exists_limit := ⟨⟨_, isLimitOfHasKernelOfPreservesLimit G f⟩⟩
variable [HasKernel (G.map f)]
@@ -239,8 +239,8 @@ def isColimitOfHasCokernelOfPreservesColimit [PreservesColimit (parallelPair f 0
isColimitCoforkMapOfIsColimit' G (cokernel.condition f) (cokernelIsCokernel f)
#align category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit
-instance [PreservesColimit (parallelPair f 0) G] : HasCokernel (G.map f)
- where exists_colimit := ⟨⟨_, isColimitOfHasCokernelOfPreservesColimit G f⟩⟩
+instance [PreservesColimit (parallelPair f 0) G] : HasCokernel (G.map f) where
+ exists_colimit := ⟨⟨_, isColimitOfHasCokernelOfPreservesColimit G f⟩⟩
variable [HasCokernel (G.map f)]
This PR shows basic results about the preservation of (limit) kernels forks by functors which preserve zero morphisms.
@@ -29,14 +29,46 @@ variable {C : Type u₁} [Category.{v₁} C] [HasZeroMorphisms C]
variable {D : Type u₂} [Category.{v₂} D] [HasZeroMorphisms D]
-variable (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
-
namespace CategoryTheory.Limits
-section Kernels
+namespace KernelFork
+
+variable {X Y : C} {f : X ⟶ Y} (c : KernelFork f)
+ (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
+
+@[reassoc (attr := simp)]
+lemma map_condition : G.map c.ι ≫ G.map f = 0 := by
+ rw [← G.map_comp, c.condition, G.map_zero]
+
+/-- A kernel fork for `f` is mapped to a kernel fork for `G.map f` if `G` is a functor
+which preserves zero morphisms. -/
+def map : KernelFork (G.map f) :=
+ KernelFork.ofι (G.map c.ι) (c.map_condition G)
+
+@[simp]
+lemma map_ι : (c.map G).ι = G.map c.ι := rfl
+
+/-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
+the mapped kernel fork is limit. -/
+def isLimitMapConeEquiv :
+ IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
+ refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
+ refine' parallelPair.ext (Iso.refl _) (Iso.refl _) _ _ <;> simp
+ exact Cones.ext (Iso.refl _) (by rintro (_|_) <;> aesop_cat)
-variable {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)
+/-- A limit kernel fork is mapped to a limit kernel fork by a functor `G` when this functor
+preserves the corresponding limit. -/
+def mapIsLimit (hc : IsLimit c) (G : C ⥤ D)
+ [Functor.PreservesZeroMorphisms G] [PreservesLimit (parallelPair f 0) G] :
+ IsLimit (c.map G) :=
+ c.isLimitMapConeEquiv G (isLimitOfPreserves G hc)
+end KernelFork
+
+section Kernels
+
+variable (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
+ {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)
/-- The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
@@ -49,11 +81,8 @@ def isLimitMapConeForkEquiv' :
IsLimit (G.mapCone (KernelFork.ofι h w)) ≃
IsLimit
(KernelFork.ofι (G.map h) (by simp only [← G.map_comp, w, Functor.map_zero]) :
- Fork (G.map f) 0) := by
- refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
- refine' parallelPair.ext (Iso.refl _) (Iso.refl _) _ _ <;> simp
- refine' Fork.ext (Iso.refl _) _
- simp [Fork.ι]
+ Fork (G.map f) 0) :=
+ KernelFork.isLimitMapConeEquiv _ _
#align category_theory.limits.is_limit_map_cone_fork_equiv' CategoryTheory.Limits.isLimitMapConeForkEquiv'
/-- The property of preserving kernels expressed in terms of kernel forks.
@@ -129,9 +158,44 @@ theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [Ha
end Kernels
+namespace CokernelCofork
+
+variable {X Y : C} {f : X ⟶ Y} (c : CokernelCofork f)
+ (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
+
+@[reassoc (attr := simp)]
+lemma map_condition : G.map f ≫ G.map c.π = 0 := by
+ rw [← G.map_comp, c.condition, G.map_zero]
+
+/-- A cokernel cofork for `f` is mapped to a cokernel cofork for `G.map f` if `G` is a functor
+which preserves zero morphisms. -/
+def map : CokernelCofork (G.map f) :=
+ CokernelCofork.ofπ (G.map c.π) (c.map_condition G)
+
+@[simp]
+lemma map_π : (c.map G).π = G.map c.π := rfl
+
+/-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if
+the mapped cokernel cofork is colimit. -/
+def isColimitMapCoconeEquiv :
+ IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by
+ refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
+ refine' parallelPair.ext (Iso.refl _) (Iso.refl _) _ _ <;> simp
+ exact Cocones.ext (Iso.refl _) (by rintro (_|_) <;> aesop_cat)
+
+/-- A colimit cokernel cofork is mapped to a colimit cokernel cofork by a functor `G`
+when this functor preserves the corresponding colimit. -/
+def mapIsColimit (hc : IsColimit c) (G : C ⥤ D)
+ [Functor.PreservesZeroMorphisms G] [PreservesColimit (parallelPair f 0) G] :
+ IsColimit (c.map G) :=
+ c.isColimitMapCoconeEquiv G (isColimitOfPreserves G hc)
+
+end CokernelCofork
+
section Cokernels
-variable {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)
+variable (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
+ {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)
/-- The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
@@ -144,13 +208,8 @@ def isColimitMapCoconeCoforkEquiv' :
IsColimit (G.mapCocone (CokernelCofork.ofπ h w)) ≃
IsColimit
(CokernelCofork.ofπ (G.map h) (by simp only [← G.map_comp, w, Functor.map_zero]) :
- Cofork (G.map f) 0) := by
- refine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)
- refine' parallelPair.ext (Iso.refl _) (Iso.refl _) _ _ <;> simp
- refine' Cofork.ext (Iso.refl _) _
- simp only [Cofork.π, Iso.refl_hom, id_comp, Cocones.precompose_obj_ι, NatTrans.comp_app,
- parallelPair.ext_hom_app, Functor.mapCocone_ι_app, Cofork.ofπ_ι_app]
- apply Category.comp_id
+ Cofork (G.map f) 0) :=
+ CokernelCofork.isColimitMapCoconeEquiv _ _
#align category_theory.limits.is_colimit_map_cocone_cofork_equiv' CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv'
/-- The property of preserving cokernels expressed in terms of cokernel coforks.
@@ -229,4 +288,36 @@ theorem preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y') [Ha
end Cokernels
+variable (X Y : C) (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
+
+noncomputable instance preservesKernelZero :
+ PreservesLimit (parallelPair (0 : X ⟶ Y) 0) G where
+ preserves {c} hc := by
+ have := KernelFork.IsLimit.isIso_ι c hc rfl
+ refine' (KernelFork.isLimitMapConeEquiv c G).symm _
+ refine' IsLimit.ofIsoLimit (KernelFork.IsLimit.ofId _ (G.map_zero _ _)) _
+ exact (Fork.ext (G.mapIso (asIso (Fork.ι c))).symm (by simp))
+
+noncomputable instance preservesCokernelZero :
+ PreservesColimit (parallelPair (0 : X ⟶ Y) 0) G where
+ preserves {c} hc := by
+ have := CokernelCofork.IsColimit.isIso_π c hc rfl
+ refine' (CokernelCofork.isColimitMapCoconeEquiv c G).symm _
+ refine' IsColimit.ofIsoColimit (CokernelCofork.IsColimit.ofId _ (G.map_zero _ _)) _
+ exact (Cofork.ext (G.mapIso (asIso (Cofork.π c))) (by simp))
+
+variable {X Y}
+
+/-- The kernel of a zero map is preserved by any functor which preserves zero morphisms. -/
+noncomputable def preservesKernelZero' (f : X ⟶ Y) (hf : f = 0) :
+ PreservesLimit (parallelPair f 0) G := by
+ rw [hf]
+ infer_instance
+
+/-- The cokernel of a zero map is preserved by any functor which preserves zero morphisms. -/
+noncomputable def preservesCokernelZero' (f : X ⟶ Y) (hf : f = 0) :
+ PreservesColimit (parallelPair f 0) G := by
+ rw [hf]
+ infer_instance
+
end CategoryTheory.Limits
@@ -2,15 +2,12 @@
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.limits.preserves.shapes.kernels
-! leanprover-community/mathlib commit 956af7c76589f444f2e1313911bad16366ea476d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
+#align_import category_theory.limits.preserves.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
+
/-!
# Preserving (co)kernels
All dependencies are ported!