category_theory.limits.constructions.epi_mono
⟷
Mathlib.CategoryTheory.Limits.Constructions.EpiMono
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/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -38,7 +38,7 @@ theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit
[Mono f] : Mono (F.map f) :=
by
have := is_limit_pullback_cone_map_of_is_limit F _ (pullback_cone.is_limit_mk_id_id f)
- simp_rw [F.map_id] at this
+ simp_rw [F.map_id] at this
apply pullback_cone.mono_of_is_limit_mk_id_id _ this
#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimit
-/
@@ -56,7 +56,7 @@ theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (c
[Mono (F.map f)] : Mono f :=
by
have := pullback_cone.is_limit_mk_id_id (F.map f)
- simp_rw [← F.map_id] at this
+ simp_rw [← F.map_id] at this
apply pullback_cone.mono_of_is_limit_mk_id_id _ (is_limit_of_is_limit_pullback_cone_map F _ this)
#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimit
-/
@@ -74,7 +74,7 @@ theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColi
[Epi f] : Epi (F.map f) :=
by
have := is_colimit_pushout_cocone_map_of_is_colimit F _ (pushout_cocone.is_colimit_mk_id_id f)
- simp_rw [F.map_id] at this
+ simp_rw [F.map_id] at this
apply pushout_cocone.epi_of_is_colimit_mk_id_id _ this
#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimit
-/
@@ -92,7 +92,7 @@ theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit
[Epi (F.map f)] : Epi f :=
by
have := pushout_cocone.is_colimit_mk_id_id (F.map f)
- simp_rw [← F.map_id] at this
+ simp_rw [← F.map_id] at this
apply
pushout_cocone.epi_of_is_colimit_mk_id_id _
(is_colimit_of_is_colimit_pushout_cocone_map F _ this)
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ Copyright (c) 2021 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.Pullbacks
-import Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts
-import Mathbin.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
+import CategoryTheory.Limits.Shapes.Pullbacks
+import CategoryTheory.Limits.Shapes.BinaryProducts
+import CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
Copyright (c) 2021 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.constructions.epi_mono
-! 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.Pullbacks
import Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathbin.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
+#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
+
/-!
# Relating monomorphisms and epimorphisms to limits and colimits
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -35,6 +35,7 @@ variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
variable (F : C ⥤ D)
+#print CategoryTheory.preserves_mono_of_preservesLimit /-
/-- If `F` preserves pullbacks, then it preserves monomorphisms. -/
theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) :=
@@ -43,6 +44,7 @@ theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit
simp_rw [F.map_id] at this
apply pullback_cone.mono_of_is_limit_mk_id_id _ this
#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimit
+-/
#print CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape /-
instance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape
@@ -51,6 +53,7 @@ instance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape
#align category_theory.preserves_monomorphisms_of_preserves_limits_of_shape CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape
-/
+#print CategoryTheory.reflects_mono_of_reflectsLimit /-
/-- If `F` reflects pullbacks, then it reflects monomorphisms. -/
theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (cospan f f) F]
[Mono (F.map f)] : Mono f :=
@@ -59,6 +62,7 @@ theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (c
simp_rw [← F.map_id] at this
apply pullback_cone.mono_of_is_limit_mk_id_id _ (is_limit_of_is_limit_pullback_cone_map F _ this)
#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimit
+-/
#print CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape /-
instance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape
@@ -67,6 +71,7 @@ instance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape
#align category_theory.reflects_monomorphisms_of_reflects_limits_of_shape CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape
-/
+#print CategoryTheory.preserves_epi_of_preservesColimit /-
/-- If `F` preserves pushouts, then it preserves epimorphisms. -/
theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) :=
@@ -75,6 +80,7 @@ theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColi
simp_rw [F.map_id] at this
apply pushout_cocone.epi_of_is_colimit_mk_id_id _ this
#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimit
+-/
#print CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape /-
instance (priority := 100) preservesEpimorphisms_of_preservesColimitsOfShape
@@ -83,6 +89,7 @@ instance (priority := 100) preservesEpimorphisms_of_preservesColimitsOfShape
#align category_theory.preserves_epimorphisms_of_preserves_colimits_of_shape CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape
-/
+#print CategoryTheory.reflects_epi_of_reflectsColimit /-
/-- If `F` reflects pushouts, then it reflects epimorphisms. -/
theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit (span f f) F]
[Epi (F.map f)] : Epi f :=
@@ -93,6 +100,7 @@ theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit
pushout_cocone.epi_of_is_colimit_mk_id_id _
(is_colimit_of_is_colimit_pushout_cocone_map F _ this)
#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimit
+-/
#print CategoryTheory.reflectsEpimorphisms_of_reflectsColimitsOfShape /-
instance (priority := 100) reflectsEpimorphisms_of_reflectsColimitsOfShape
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -40,7 +40,7 @@ theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit
[Mono f] : Mono (F.map f) :=
by
have := is_limit_pullback_cone_map_of_is_limit F _ (pullback_cone.is_limit_mk_id_id f)
- simp_rw [F.map_id] at this
+ simp_rw [F.map_id] at this
apply pullback_cone.mono_of_is_limit_mk_id_id _ this
#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimit
@@ -56,7 +56,7 @@ theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (c
[Mono (F.map f)] : Mono f :=
by
have := pullback_cone.is_limit_mk_id_id (F.map f)
- simp_rw [← F.map_id] at this
+ simp_rw [← F.map_id] at this
apply pullback_cone.mono_of_is_limit_mk_id_id _ (is_limit_of_is_limit_pullback_cone_map F _ this)
#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimit
@@ -72,7 +72,7 @@ theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColi
[Epi f] : Epi (F.map f) :=
by
have := is_colimit_pushout_cocone_map_of_is_colimit F _ (pushout_cocone.is_colimit_mk_id_id f)
- simp_rw [F.map_id] at this
+ simp_rw [F.map_id] at this
apply pushout_cocone.epi_of_is_colimit_mk_id_id _ this
#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimit
@@ -88,7 +88,7 @@ theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit
[Epi (F.map f)] : Epi f :=
by
have := pushout_cocone.is_colimit_mk_id_id (F.map f)
- simp_rw [← F.map_id] at this
+ simp_rw [← F.map_id] at this
apply
pushout_cocone.epi_of_is_colimit_mk_id_id _
(is_colimit_of_is_colimit_pushout_cocone_map F _ this)
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -35,12 +35,6 @@ variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
variable (F : C ⥤ D)
-/- warning: category_theory.preserves_mono_of_preserves_limit -> CategoryTheory.preserves_mono_of_preservesLimit is a dubious translation:
-lean 3 declaration is
- forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f], CategoryTheory.Mono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)
-but is expected to have type
- forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f], CategoryTheory.Mono.{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 F) 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 F) 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 F) X Y f)
-Case conversion may be inaccurate. Consider using '#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimitₓ'. -/
/-- If `F` preserves pullbacks, then it preserves monomorphisms. -/
theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) :=
@@ -57,12 +51,6 @@ instance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape
#align category_theory.preserves_monomorphisms_of_preserves_limits_of_shape CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape
-/
-/- warning: category_theory.reflects_mono_of_reflects_limit -> CategoryTheory.reflects_mono_of_reflectsLimit is a dubious translation:
-lean 3 declaration is
- forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)], CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f
-but is expected to have type
- forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{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 F) 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 F) 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 F) X Y f)], CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f
-Case conversion may be inaccurate. Consider using '#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimitₓ'. -/
/-- If `F` reflects pullbacks, then it reflects monomorphisms. -/
theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (cospan f f) F]
[Mono (F.map f)] : Mono f :=
@@ -79,12 +67,6 @@ instance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape
#align category_theory.reflects_monomorphisms_of_reflects_limits_of_shape CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape
-/
-/- warning: category_theory.preserves_epi_of_preserves_colimit -> CategoryTheory.preserves_epi_of_preservesColimit is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
- forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f], 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 F) 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 F) 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 F) X Y f)
-Case conversion may be inaccurate. Consider using '#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimitₓ'. -/
/-- If `F` preserves pushouts, then it preserves epimorphisms. -/
theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) :=
@@ -101,12 +83,6 @@ instance (priority := 100) preservesEpimorphisms_of_preservesColimitsOfShape
#align category_theory.preserves_epimorphisms_of_preserves_colimits_of_shape CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape
-/
-/- warning: category_theory.reflects_epi_of_reflects_colimit -> CategoryTheory.reflects_epi_of_reflectsColimit is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
- forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : 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 F) 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 F) 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 F) X Y f)], CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f
-Case conversion may be inaccurate. Consider using '#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimitₓ'. -/
/-- If `F` reflects pushouts, then it reflects epimorphisms. -/
theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit (span f f) F]
[Epi (F.map f)] : Epi f :=
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.constructions.epi_mono
-! leanprover-community/mathlib commit f7baecbb54bd0f24f228576f97b1752fc3c9b318
+! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
/-!
# Relating monomorphisms and epimorphisms to limits and colimits
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
If `F` preserves (resp. reflects) pullbacks, then it preserves (resp. reflects) monomorphisms.
We also provide the dual version for epimorphisms.
mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25
@@ -32,6 +32,12 @@ variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
variable (F : C ⥤ D)
+/- warning: category_theory.preserves_mono_of_preserves_limit -> CategoryTheory.preserves_mono_of_preservesLimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f], CategoryTheory.Mono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)
+but is expected to have type
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f], CategoryTheory.Mono.{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 F) 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 F) 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 F) X Y f)
+Case conversion may be inaccurate. Consider using '#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimitₓ'. -/
/-- If `F` preserves pullbacks, then it preserves monomorphisms. -/
theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) :=
@@ -41,11 +47,19 @@ theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit
apply pullback_cone.mono_of_is_limit_mk_id_id _ this
#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimit
+#print CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape /-
instance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape
[PreservesLimitsOfShape WalkingCospan F] : F.PreservesMonomorphisms
where preserves X Y f hf := preserves_mono_of_preserves_limit F f
#align category_theory.preserves_monomorphisms_of_preserves_limits_of_shape CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape
+-/
+/- warning: category_theory.reflects_mono_of_reflects_limit -> CategoryTheory.reflects_mono_of_reflectsLimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)], CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f
+but is expected to have type
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{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 F) 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 F) 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 F) X Y f)], CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f
+Case conversion may be inaccurate. Consider using '#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimitₓ'. -/
/-- If `F` reflects pullbacks, then it reflects monomorphisms. -/
theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (cospan f f) F]
[Mono (F.map f)] : Mono f :=
@@ -55,11 +69,19 @@ theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (c
apply pullback_cone.mono_of_is_limit_mk_id_id _ (is_limit_of_is_limit_pullback_cone_map F _ this)
#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimit
+#print CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape /-
instance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape
[ReflectsLimitsOfShape WalkingCospan F] : F.ReflectsMonomorphisms
where reflects X Y f hf := reflects_mono_of_reflects_limit F f
#align category_theory.reflects_monomorphisms_of_reflects_limits_of_shape CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape
+-/
+/- warning: category_theory.preserves_epi_of_preserves_colimit -> CategoryTheory.preserves_epi_of_preservesColimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f], CategoryTheory.Epi.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)
+but is expected to have type
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f], 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 F) 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 F) 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 F) X Y f)
+Case conversion may be inaccurate. Consider using '#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimitₓ'. -/
/-- If `F` preserves pushouts, then it preserves epimorphisms. -/
theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) :=
@@ -69,11 +91,19 @@ theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColi
apply pushout_cocone.epi_of_is_colimit_mk_id_id _ this
#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimit
+#print CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape /-
instance (priority := 100) preservesEpimorphisms_of_preservesColimitsOfShape
[PreservesColimitsOfShape WalkingSpan F] : F.PreservesEpimorphisms
where preserves X Y f hf := preserves_epi_of_preserves_colimit F f
#align category_theory.preserves_epimorphisms_of_preserves_colimits_of_shape CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape
+-/
+/- warning: category_theory.reflects_epi_of_reflects_colimit -> CategoryTheory.reflects_epi_of_reflectsColimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)], CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f
+but is expected to have type
+ forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : 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) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : 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 F) 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 F) 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 F) X Y f)], CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f
+Case conversion may be inaccurate. Consider using '#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimitₓ'. -/
/-- If `F` reflects pushouts, then it reflects epimorphisms. -/
theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit (span f f) F]
[Epi (F.map f)] : Epi f :=
@@ -85,10 +115,12 @@ theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit
(is_colimit_of_is_colimit_pushout_cocone_map F _ this)
#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimit
+#print CategoryTheory.reflectsEpimorphisms_of_reflectsColimitsOfShape /-
instance (priority := 100) reflectsEpimorphisms_of_reflectsColimitsOfShape
[ReflectsColimitsOfShape WalkingSpan F] : F.ReflectsEpimorphisms
where reflects X Y f hf := reflects_epi_of_reflects_colimit F f
#align category_theory.reflects_epimorphisms_of_reflects_colimits_of_shape CategoryTheory.reflectsEpimorphisms_of_reflectsColimitsOfShape
+-/
end CategoryTheory
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 @@ namespace CategoryTheory
open Category Limits
variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
-
variable (F : C ⥤ D)
/-- If `F` preserves pullbacks, then it preserves monomorphisms. -/
@@ -2,16 +2,13 @@
Copyright (c) 2021 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.constructions.epi_mono
-! leanprover-community/mathlib commit f7baecbb54bd0f24f228576f97b1752fc3c9b318
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
+#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
+
/-!
# Relating monomorphisms and epimorphisms to limits and colimits
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,
@@ -41,7 +41,7 @@ theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit
#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimit
instance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape
- [PreservesLimitsOfShape WalkingCospan F] : F.PreservesMonomorphisms where
+ [PreservesLimitsOfShape WalkingCospan F] : F.PreservesMonomorphisms where
preserves f _ := preserves_mono_of_preservesLimit F f
#align category_theory.preserves_monomorphisms_of_preserves_limits_of_shape CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape
@@ -54,7 +54,7 @@ theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (c
#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimit
instance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape
- [ReflectsLimitsOfShape WalkingCospan F] : F.ReflectsMonomorphisms where
+ [ReflectsLimitsOfShape WalkingCospan F] : F.ReflectsMonomorphisms where
reflects f _ := reflects_mono_of_reflectsLimit F f
#align category_theory.reflects_monomorphisms_of_reflects_limits_of_shape CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape
@@ -67,7 +67,7 @@ theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColi
#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimit
instance (priority := 100) preservesEpimorphisms_of_preservesColimitsOfShape
- [PreservesColimitsOfShape WalkingSpan F] : F.PreservesEpimorphisms where
+ [PreservesColimitsOfShape WalkingSpan F] : F.PreservesEpimorphisms where
preserves f _ := preserves_epi_of_preservesColimit F f
#align category_theory.preserves_epimorphisms_of_preserves_colimits_of_shape CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape
@@ -82,9 +82,8 @@ theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit
#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimit
instance (priority := 100) reflectsEpimorphisms_of_reflectsColimitsOfShape
- [ReflectsColimitsOfShape WalkingSpan F] : F.ReflectsEpimorphisms where
+ [ReflectsColimitsOfShape WalkingSpan F] : F.ReflectsEpimorphisms where
reflects f _ := reflects_epi_of_reflectsColimit F f
#align category_theory.reflects_epimorphisms_of_reflects_colimits_of_shape CategoryTheory.reflectsEpimorphisms_of_reflectsColimitsOfShape
end CategoryTheory
-
All dependencies are ported!