category_theory.limits.preserves.shapes.binary_products
⟷
Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
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) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
-import Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts
-import Mathbin.CategoryTheory.Limits.Preserves.Basic
+import CategoryTheory.Limits.Shapes.BinaryProducts
+import CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.binary_products from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.limits.preserves.shapes.binary_products
-! 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.BinaryProducts
import Mathbin.CategoryTheory.Limits.Preserves.Basic
+#align_import category_theory.limits.preserves.shapes.binary_products from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
+
/-!
# Preserving binary products
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -43,6 +43,7 @@ section
variable {P X Y Z : C} (f : P ⟶ X) (g : P ⟶ Y)
+#print CategoryTheory.Limits.isLimitMapConeBinaryFanEquiv /-
/--
The map of a binary fan is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `binary_fan.mk` with `functor.map_cone`.
@@ -52,21 +53,27 @@ def isLimitMapConeBinaryFanEquiv :
(IsLimit.postcomposeHomEquiv (diagramIsoPair _) _).symm.trans
(IsLimit.equivIsoLimit (Cones.ext (Iso.refl _) (by rintro (_ | _); tidy)))
#align category_theory.limits.is_limit_map_cone_binary_fan_equiv CategoryTheory.Limits.isLimitMapConeBinaryFanEquiv
+-/
+#print CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimit /-
/-- The property of preserving products expressed in terms of binary fans. -/
def mapIsLimitOfPreservesOfIsLimit [PreservesLimit (pair X Y) G] (l : IsLimit (BinaryFan.mk f g)) :
IsLimit (BinaryFan.mk (G.map f) (G.map g)) :=
isLimitMapConeBinaryFanEquiv G f g (PreservesLimit.preserves l)
#align category_theory.limits.map_is_limit_of_preserves_of_is_limit CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimit
+-/
+#print CategoryTheory.Limits.isLimitOfReflectsOfMapIsLimit /-
/-- The property of reflecting products expressed in terms of binary fans. -/
def isLimitOfReflectsOfMapIsLimit [ReflectsLimit (pair X Y) G]
(l : IsLimit (BinaryFan.mk (G.map f) (G.map g))) : IsLimit (BinaryFan.mk f g) :=
ReflectsLimit.reflects ((isLimitMapConeBinaryFanEquiv G f g).symm l)
#align category_theory.limits.is_limit_of_reflects_of_map_is_limit CategoryTheory.Limits.isLimitOfReflectsOfMapIsLimit
+-/
variable (X Y) [HasBinaryProduct X Y]
+#print CategoryTheory.Limits.isLimitOfHasBinaryProductOfPreservesLimit /-
/-- If `G` preserves binary products and `C` has them, then the binary fan constructed of the mapped
morphisms of the binary product cone is a limit.
-/
@@ -74,9 +81,11 @@ def isLimitOfHasBinaryProductOfPreservesLimit [PreservesLimit (pair X Y) G] :
IsLimit (BinaryFan.mk (G.map (Limits.prod.fst : X ⨯ Y ⟶ X)) (G.map Limits.prod.snd)) :=
mapIsLimitOfPreservesOfIsLimit G _ _ (prodIsProd X Y)
#align category_theory.limits.is_limit_of_has_binary_product_of_preserves_limit CategoryTheory.Limits.isLimitOfHasBinaryProductOfPreservesLimit
+-/
variable [HasBinaryProduct (G.obj X) (G.obj Y)]
+#print CategoryTheory.Limits.PreservesLimitPair.ofIsoProdComparison /-
/-- If the product comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`.
-/
@@ -88,20 +97,25 @@ def PreservesLimitPair.ofIsoProdComparison [i : IsIso (prodComparison G X Y)] :
apply is_limit.of_point_iso (limit.is_limit (pair (G.obj X) (G.obj Y)))
apply i
#align category_theory.limits.preserves_limit_pair.of_iso_prod_comparison CategoryTheory.Limits.PreservesLimitPair.ofIsoProdComparison
+-/
variable [PreservesLimit (pair X Y) G]
+#print CategoryTheory.Limits.PreservesLimitPair.iso /-
/-- If `G` preserves the product of `(X,Y)`, then the product comparison map for `G` at `(X,Y)` is
an isomorphism.
-/
def PreservesLimitPair.iso : G.obj (X ⨯ Y) ≅ G.obj X ⨯ G.obj Y :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasBinaryProductOfPreservesLimit G X Y) (limit.isLimit _)
#align category_theory.limits.preserves_limit_pair.iso CategoryTheory.Limits.PreservesLimitPair.iso
+-/
+#print CategoryTheory.Limits.PreservesLimitPair.iso_hom /-
@[simp]
theorem PreservesLimitPair.iso_hom : (PreservesLimitPair.iso G X Y).Hom = prodComparison G X Y :=
rfl
#align category_theory.limits.preserves_limit_pair.iso_hom CategoryTheory.Limits.PreservesLimitPair.iso_hom
+-/
instance : IsIso (prodComparison G X Y) :=
by
@@ -114,6 +128,7 @@ section
variable {P X Y Z : C} (f : X ⟶ P) (g : Y ⟶ P)
+#print CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquiv /-
/-- The map of a binary cofan is a colimit iff
the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `binary_cofan.mk` with `functor.map_cocone`.
@@ -123,21 +138,27 @@ def isColimitMapCoconeBinaryCofanEquiv :
(IsColimit.precomposeHomEquiv (diagramIsoPair _).symm _).symm.trans
(IsColimit.equivIsoColimit (Cocones.ext (Iso.refl _) (by rintro (_ | _); tidy)))
#align category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquiv
+-/
+#print CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimit /-
/-- The property of preserving coproducts expressed in terms of binary cofans. -/
def mapIsColimitOfPreservesOfIsColimit [PreservesColimit (pair X Y) G]
(l : IsColimit (BinaryCofan.mk f g)) : IsColimit (BinaryCofan.mk (G.map f) (G.map g)) :=
isColimitMapCoconeBinaryCofanEquiv G f g (PreservesColimit.preserves l)
#align category_theory.limits.map_is_colimit_of_preserves_of_is_colimit CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimit
+-/
+#print CategoryTheory.Limits.isColimitOfReflectsOfMapIsColimit /-
/-- The property of reflecting coproducts expressed in terms of binary cofans. -/
def isColimitOfReflectsOfMapIsColimit [ReflectsColimit (pair X Y) G]
(l : IsColimit (BinaryCofan.mk (G.map f) (G.map g))) : IsColimit (BinaryCofan.mk f g) :=
ReflectsColimit.reflects ((isColimitMapCoconeBinaryCofanEquiv G f g).symm l)
#align category_theory.limits.is_colimit_of_reflects_of_map_is_colimit CategoryTheory.Limits.isColimitOfReflectsOfMapIsColimit
+-/
variable (X Y) [HasBinaryCoproduct X Y]
+#print CategoryTheory.Limits.isColimitOfHasBinaryCoproductOfPreservesColimit /-
/--
If `G` preserves binary coproducts and `C` has them, then the binary cofan constructed of the mapped
morphisms of the binary product cocone is a colimit.
@@ -146,9 +167,11 @@ def isColimitOfHasBinaryCoproductOfPreservesColimit [PreservesColimit (pair X Y)
IsColimit (BinaryCofan.mk (G.map (Limits.coprod.inl : X ⟶ X ⨿ Y)) (G.map Limits.coprod.inr)) :=
mapIsColimitOfPreservesOfIsColimit G _ _ (coprodIsCoprod X Y)
#align category_theory.limits.is_colimit_of_has_binary_coproduct_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasBinaryCoproductOfPreservesColimit
+-/
variable [HasBinaryCoproduct (G.obj X) (G.obj Y)]
+#print CategoryTheory.Limits.PreservesColimitPair.ofIsoCoprodComparison /-
/-- If the coproduct comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`.
-/
@@ -160,9 +183,11 @@ def PreservesColimitPair.ofIsoCoprodComparison [i : IsIso (coprodComparison G X
apply is_colimit.of_point_iso (colimit.is_colimit (pair (G.obj X) (G.obj Y)))
apply i
#align category_theory.limits.preserves_colimit_pair.of_iso_coprod_comparison CategoryTheory.Limits.PreservesColimitPair.ofIsoCoprodComparison
+-/
variable [PreservesColimit (pair X Y) G]
+#print CategoryTheory.Limits.PreservesColimitPair.iso /-
/--
If `G` preserves the coproduct of `(X,Y)`, then the coproduct comparison map for `G` at `(X,Y)` is
an isomorphism.
@@ -171,12 +196,15 @@ def PreservesColimitPair.iso : G.obj X ⨿ G.obj Y ≅ G.obj (X ⨿ Y) :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit _)
(isColimitOfHasBinaryCoproductOfPreservesColimit G X Y)
#align category_theory.limits.preserves_colimit_pair.iso CategoryTheory.Limits.PreservesColimitPair.iso
+-/
+#print CategoryTheory.Limits.PreservesColimitPair.iso_hom /-
@[simp]
theorem PreservesColimitPair.iso_hom :
(PreservesColimitPair.iso G X Y).Hom = coprodComparison G X Y :=
rfl
#align category_theory.limits.preserves_colimit_pair.iso_hom CategoryTheory.Limits.PreservesColimitPair.iso_hom
+-/
instance : IsIso (coprodComparison G X Y) :=
by
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -43,12 +43,6 @@ section
variable {P X Y Z : C} (f : P ⟶ X) (g : P ⟶ Y)
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/--
The map of a binary fan is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `binary_fan.mk` with `functor.map_cone`.
@@ -59,24 +53,12 @@ def isLimitMapConeBinaryFanEquiv :
(IsLimit.equivIsoLimit (Cones.ext (Iso.refl _) (by rintro (_ | _); tidy)))
#align category_theory.limits.is_limit_map_cone_binary_fan_equiv CategoryTheory.Limits.isLimitMapConeBinaryFanEquiv
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/-- The property of preserving products expressed in terms of binary fans. -/
def mapIsLimitOfPreservesOfIsLimit [PreservesLimit (pair X Y) G] (l : IsLimit (BinaryFan.mk f g)) :
IsLimit (BinaryFan.mk (G.map f) (G.map g)) :=
isLimitMapConeBinaryFanEquiv G f g (PreservesLimit.preserves l)
#align category_theory.limits.map_is_limit_of_preserves_of_is_limit CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimit
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/-- The property of reflecting products expressed in terms of binary fans. -/
def isLimitOfReflectsOfMapIsLimit [ReflectsLimit (pair X Y) G]
(l : IsLimit (BinaryFan.mk (G.map f) (G.map g))) : IsLimit (BinaryFan.mk f g) :=
@@ -85,12 +67,6 @@ def isLimitOfReflectsOfMapIsLimit [ReflectsLimit (pair X Y) G]
variable (X Y) [HasBinaryProduct X Y]
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/-- If `G` preserves binary products and `C` has them, then the binary fan constructed of the mapped
morphisms of the binary product cone is a limit.
-/
@@ -101,12 +77,6 @@ def isLimitOfHasBinaryProductOfPreservesLimit [PreservesLimit (pair X Y) G] :
variable [HasBinaryProduct (G.obj X) (G.obj Y)]
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/-- If the product comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`.
-/
@@ -121,12 +91,6 @@ def PreservesLimitPair.ofIsoProdComparison [i : IsIso (prodComparison G X Y)] :
variable [PreservesLimit (pair X Y) G]
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/-- If `G` preserves the product of `(X,Y)`, then the product comparison map for `G` at `(X,Y)` is
an isomorphism.
-/
@@ -134,12 +98,6 @@ def PreservesLimitPair.iso : G.obj (X ⨯ Y) ≅ G.obj X ⨯ G.obj Y :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasBinaryProductOfPreservesLimit G X Y) (limit.isLimit _)
#align category_theory.limits.preserves_limit_pair.iso CategoryTheory.Limits.PreservesLimitPair.iso
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@[simp]
theorem PreservesLimitPair.iso_hom : (PreservesLimitPair.iso G X Y).Hom = prodComparison G X Y :=
rfl
@@ -156,12 +114,6 @@ section
variable {P X Y Z : C} (f : X ⟶ P) (g : Y ⟶ P)
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/-- The map of a binary cofan is a colimit iff
the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `binary_cofan.mk` with `functor.map_cocone`.
@@ -172,24 +124,12 @@ def isColimitMapCoconeBinaryCofanEquiv :
(IsColimit.equivIsoColimit (Cocones.ext (Iso.refl _) (by rintro (_ | _); tidy)))
#align category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquiv
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/-- The property of preserving coproducts expressed in terms of binary cofans. -/
def mapIsColimitOfPreservesOfIsColimit [PreservesColimit (pair X Y) G]
(l : IsColimit (BinaryCofan.mk f g)) : IsColimit (BinaryCofan.mk (G.map f) (G.map g)) :=
isColimitMapCoconeBinaryCofanEquiv G f g (PreservesColimit.preserves l)
#align category_theory.limits.map_is_colimit_of_preserves_of_is_colimit CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimit
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/-- The property of reflecting coproducts expressed in terms of binary cofans. -/
def isColimitOfReflectsOfMapIsColimit [ReflectsColimit (pair X Y) G]
(l : IsColimit (BinaryCofan.mk (G.map f) (G.map g))) : IsColimit (BinaryCofan.mk f g) :=
@@ -198,12 +138,6 @@ def isColimitOfReflectsOfMapIsColimit [ReflectsColimit (pair X Y) G]
variable (X Y) [HasBinaryCoproduct X Y]
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/--
If `G` preserves binary coproducts and `C` has them, then the binary cofan constructed of the mapped
morphisms of the binary product cocone is a colimit.
@@ -215,12 +149,6 @@ def isColimitOfHasBinaryCoproductOfPreservesColimit [PreservesColimit (pair X Y)
variable [HasBinaryCoproduct (G.obj X) (G.obj Y)]
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/-- If the coproduct comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`.
-/
@@ -235,12 +163,6 @@ def PreservesColimitPair.ofIsoCoprodComparison [i : IsIso (coprodComparison G X
variable [PreservesColimit (pair X Y) G]
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/--
If `G` preserves the coproduct of `(X,Y)`, then the coproduct comparison map for `G` at `(X,Y)` is
an isomorphism.
@@ -250,12 +172,6 @@ def PreservesColimitPair.iso : G.obj X ⨿ G.obj Y ≅ G.obj (X ⨿ Y) :=
(isColimitOfHasBinaryCoproductOfPreservesColimit G X Y)
#align category_theory.limits.preserves_colimit_pair.iso CategoryTheory.Limits.PreservesColimitPair.iso
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@[simp]
theorem PreservesColimitPair.iso_hom :
(PreservesColimitPair.iso G X Y).Hom = coprodComparison G X Y :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -56,11 +56,7 @@ essentially lets us commute `binary_fan.mk` with `functor.map_cone`.
def isLimitMapConeBinaryFanEquiv :
IsLimit (G.mapCone (BinaryFan.mk f g)) ≃ IsLimit (BinaryFan.mk (G.map f) (G.map g)) :=
(IsLimit.postcomposeHomEquiv (diagramIsoPair _) _).symm.trans
- (IsLimit.equivIsoLimit
- (Cones.ext (Iso.refl _)
- (by
- rintro (_ | _)
- tidy)))
+ (IsLimit.equivIsoLimit (Cones.ext (Iso.refl _) (by rintro (_ | _); tidy)))
#align category_theory.limits.is_limit_map_cone_binary_fan_equiv CategoryTheory.Limits.isLimitMapConeBinaryFanEquiv
/- warning: category_theory.limits.map_is_limit_of_preserves_of_is_limit -> CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimit is a dubious translation:
@@ -173,11 +169,7 @@ This essentially lets us commute `binary_cofan.mk` with `functor.map_cocone`.
def isColimitMapCoconeBinaryCofanEquiv :
IsColimit (G.mapCocone (BinaryCofan.mk f g)) ≃ IsColimit (BinaryCofan.mk (G.map f) (G.map g)) :=
(IsColimit.precomposeHomEquiv (diagramIsoPair _).symm _).symm.trans
- (IsColimit.equivIsoColimit
- (Cocones.ext (Iso.refl _)
- (by
- rintro (_ | _)
- tidy)))
+ (IsColimit.equivIsoColimit (Cocones.ext (Iso.refl _) (by rintro (_ | _); tidy)))
#align category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquiv
/- warning: category_theory.limits.map_is_colimit_of_preserves_of_is_colimit -> CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimit is a dubious translation:
mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
! This file was ported from Lean 3 source module category_theory.limits.preserves.shapes.binary_products
-! leanprover-community/mathlib commit 024a4231815538ac739f52d08dd20a55da0d6b23
+! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.CategoryTheory.Limits.Preserves.Basic
/-!
# Preserving binary products
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
Constructions to relate the notions of preserving binary products and reflecting binary products
to concrete binary fans.
mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25
@@ -44,7 +44,7 @@ variable {P X Y Z : C} (f : P ⟶ X) (g : P ⟶ Y)
lean 3 declaration is
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but is expected to have type
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+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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)) P X) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) P Y), Equiv.{max (succ u4) (succ u2), max (succ u4) (succ u2)} (CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Functor.comp.{0, u1, u2, 0, u3, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 D _inst_2 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G) (CategoryTheory.Functor.mapCone.{0, u1, u2, 0, u3, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 D _inst_2 G (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryFan.mk.{u1, u3} C _inst_1 X Y P f g))) (CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P X f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P Y g)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_map_cone_binary_fan_equiv CategoryTheory.Limits.isLimitMapConeBinaryFanEquivₓ'. -/
/--
The map of a binary fan is a limit iff the fork consisting of the mapped morphisms is a limit. This
@@ -161,7 +161,7 @@ variable {P X Y Z : C} (f : X ⟶ P) (g : Y ⟶ P)
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquivₓ'. -/
/-- The map of a binary cofan is a colimit iff
the cofork consisting of the mapped morphisms is a colimit.
mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25
@@ -40,6 +40,12 @@ section
variable {P X Y Z : C} (f : P ⟶ X) (g : P ⟶ Y)
+/- warning: category_theory.limits.is_limit_map_cone_binary_fan_equiv -> CategoryTheory.Limits.isLimitMapConeBinaryFanEquiv is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_map_cone_binary_fan_equiv CategoryTheory.Limits.isLimitMapConeBinaryFanEquivₓ'. -/
/--
The map of a binary fan is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `binary_fan.mk` with `functor.map_cone`.
@@ -54,12 +60,24 @@ def isLimitMapConeBinaryFanEquiv :
tidy)))
#align category_theory.limits.is_limit_map_cone_binary_fan_equiv CategoryTheory.Limits.isLimitMapConeBinaryFanEquiv
+/- warning: category_theory.limits.map_is_limit_of_preserves_of_is_limit -> CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimit is a dubious translation:
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+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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)) P X) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) P Y) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], (CategoryTheory.Limits.IsLimit.{0, u1, 0, u3} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryFan.mk.{u1, u3} C _inst_1 X Y P f g)) -> (CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P X f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P Y g)))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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)) P X) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) P Y) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], (CategoryTheory.Limits.IsLimit.{0, u1, 0, u3} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryFan.mk.{u1, u3} C _inst_1 X Y P f g)) -> (CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P X f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P Y g)))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_is_limit_of_preserves_of_is_limit CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimitₓ'. -/
/-- The property of preserving products expressed in terms of binary fans. -/
def mapIsLimitOfPreservesOfIsLimit [PreservesLimit (pair X Y) G] (l : IsLimit (BinaryFan.mk f g)) :
IsLimit (BinaryFan.mk (G.map f) (G.map g)) :=
isLimitMapConeBinaryFanEquiv G f g (PreservesLimit.preserves l)
#align category_theory.limits.map_is_limit_of_preserves_of_is_limit CategoryTheory.Limits.mapIsLimitOfPreservesOfIsLimit
+/- warning: category_theory.limits.is_limit_of_reflects_of_map_is_limit -> CategoryTheory.Limits.isLimitOfReflectsOfMapIsLimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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)) P X) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) P Y) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], (CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P X f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P Y g))) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u3} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryFan.mk.{u1, u3} C _inst_1 X Y P f g))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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)) P X) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) P Y) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], (CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P X f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P Y g))) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u3} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryFan.mk.{u1, u3} C _inst_1 X Y P f g))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_reflects_of_map_is_limit CategoryTheory.Limits.isLimitOfReflectsOfMapIsLimitₓ'. -/
/-- The property of reflecting products expressed in terms of binary fans. -/
def isLimitOfReflectsOfMapIsLimit [ReflectsLimit (pair X Y) G]
(l : IsLimit (BinaryFan.mk (G.map f) (G.map g))) : IsLimit (BinaryFan.mk f g) :=
@@ -68,6 +86,12 @@ def isLimitOfReflectsOfMapIsLimit [ReflectsLimit (pair X Y) G]
variable (X Y) [HasBinaryProduct X Y]
+/- warning: category_theory.limits.is_limit_of_has_binary_product_of_preserves_limit -> CategoryTheory.Limits.isLimitOfHasBinaryProductOfPreservesLimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryProduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.prod.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.prod.{u1, u3} C _inst_1 X Y _inst_3) X (CategoryTheory.Limits.prod.fst.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.prod.{u1, u3} C _inst_1 X Y _inst_3) Y (CategoryTheory.Limits.prod.snd.{u1, u3} C _inst_1 X Y _inst_3)))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryProduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], CategoryTheory.Limits.IsLimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (CategoryTheory.Limits.BinaryFan.mk.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.prod.{u1, u3} C _inst_1 X Y _inst_3)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.prod.{u1, u3} C _inst_1 X Y _inst_3) X (CategoryTheory.Limits.prod.fst.{u1, u3} C _inst_1 X Y _inst_3)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.prod.{u1, u3} C _inst_1 X Y _inst_3) Y (CategoryTheory.Limits.prod.snd.{u1, u3} C _inst_1 X Y _inst_3)))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_limit_of_has_binary_product_of_preserves_limit CategoryTheory.Limits.isLimitOfHasBinaryProductOfPreservesLimitₓ'. -/
/-- If `G` preserves binary products and `C` has them, then the binary fan constructed of the mapped
morphisms of the binary product cone is a limit.
-/
@@ -78,6 +102,12 @@ def isLimitOfHasBinaryProductOfPreservesLimit [PreservesLimit (pair X Y) G] :
variable [HasBinaryProduct (G.obj X) (G.obj Y)]
+/- warning: category_theory.limits.preserves_limit_pair.of_iso_prod_comparison -> CategoryTheory.Limits.PreservesLimitPair.ofIsoProdComparison 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_limit_pair.of_iso_prod_comparison CategoryTheory.Limits.PreservesLimitPair.ofIsoProdComparisonₓ'. -/
/-- If the product comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`.
-/
@@ -92,6 +122,12 @@ def PreservesLimitPair.ofIsoProdComparison [i : IsIso (prodComparison G X Y)] :
variable [PreservesLimit (pair X Y) G]
+/- warning: category_theory.limits.preserves_limit_pair.iso -> CategoryTheory.Limits.PreservesLimitPair.iso is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limit_pair.iso CategoryTheory.Limits.PreservesLimitPair.isoₓ'. -/
/-- If `G` preserves the product of `(X,Y)`, then the product comparison map for `G` at `(X,Y)` is
an isomorphism.
-/
@@ -99,6 +135,12 @@ def PreservesLimitPair.iso : G.obj (X ⨯ Y) ≅ G.obj X ⨯ G.obj Y :=
IsLimit.conePointUniqueUpToIso (isLimitOfHasBinaryProductOfPreservesLimit G X Y) (limit.isLimit _)
#align category_theory.limits.preserves_limit_pair.iso CategoryTheory.Limits.PreservesLimitPair.iso
+/- warning: category_theory.limits.preserves_limit_pair.iso_hom -> CategoryTheory.Limits.PreservesLimitPair.iso_hom 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_limit_pair.iso_hom CategoryTheory.Limits.PreservesLimitPair.iso_homₓ'. -/
@[simp]
theorem PreservesLimitPair.iso_hom : (PreservesLimitPair.iso G X Y).Hom = prodComparison G X Y :=
rfl
@@ -115,6 +157,12 @@ section
variable {P X Y Z : C} (f : X ⟶ P) (g : Y ⟶ P)
+/- warning: category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv -> CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquiv is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquivₓ'. -/
/-- The map of a binary cofan is a colimit iff
the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `binary_cofan.mk` with `functor.map_cocone`.
@@ -129,12 +177,24 @@ def isColimitMapCoconeBinaryCofanEquiv :
tidy)))
#align category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv CategoryTheory.Limits.isColimitMapCoconeBinaryCofanEquiv
+/- warning: category_theory.limits.map_is_colimit_of_preserves_of_is_colimit -> CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimit is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.limits.map_is_colimit_of_preserves_of_is_colimit CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimitₓ'. -/
/-- The property of preserving coproducts expressed in terms of binary cofans. -/
def mapIsColimitOfPreservesOfIsColimit [PreservesColimit (pair X Y) G]
(l : IsColimit (BinaryCofan.mk f g)) : IsColimit (BinaryCofan.mk (G.map f) (G.map g)) :=
isColimitMapCoconeBinaryCofanEquiv G f g (PreservesColimit.preserves l)
#align category_theory.limits.map_is_colimit_of_preserves_of_is_colimit CategoryTheory.Limits.mapIsColimitOfPreservesOfIsColimit
+/- warning: category_theory.limits.is_colimit_of_reflects_of_map_is_colimit -> CategoryTheory.Limits.isColimitOfReflectsOfMapIsColimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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 P) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) Y P) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], (CategoryTheory.Limits.IsColimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)) (CategoryTheory.Limits.BinaryCofan.mk.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G P) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X P f) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y P g))) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u3} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryCofan.mk.{u1, u3} C _inst_1 X Y P f g))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {P : C} {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 P) (g : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) Y P) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], (CategoryTheory.Limits.IsColimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (CategoryTheory.Limits.BinaryCofan.mk.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) P) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X P f) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y P g))) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u3} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) (CategoryTheory.Limits.BinaryCofan.mk.{u1, u3} C _inst_1 X Y P f g))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_reflects_of_map_is_colimit CategoryTheory.Limits.isColimitOfReflectsOfMapIsColimitₓ'. -/
/-- The property of reflecting coproducts expressed in terms of binary cofans. -/
def isColimitOfReflectsOfMapIsColimit [ReflectsColimit (pair X Y) G]
(l : IsColimit (BinaryCofan.mk (G.map f) (G.map g))) : IsColimit (BinaryCofan.mk f g) :=
@@ -143,6 +203,12 @@ def isColimitOfReflectsOfMapIsColimit [ReflectsColimit (pair X Y) G]
variable (X Y) [HasBinaryCoproduct X Y]
+/- warning: category_theory.limits.is_colimit_of_has_binary_coproduct_of_preserves_colimit -> CategoryTheory.Limits.isColimitOfHasBinaryCoproductOfPreservesColimit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], CategoryTheory.Limits.IsColimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)) (CategoryTheory.Limits.BinaryCofan.mk.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3) (CategoryTheory.Limits.coprod.inl.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3) (CategoryTheory.Limits.coprod.inr.{u1, u3} C _inst_1 X Y _inst_3)))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], CategoryTheory.Limits.IsColimit.{0, u2, 0, u4} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) D _inst_2 (CategoryTheory.Limits.pair.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)) (CategoryTheory.Limits.BinaryCofan.mk.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3) (CategoryTheory.Limits.coprod.inl.{u1, u3} C _inst_1 X Y _inst_3)) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3) (CategoryTheory.Limits.coprod.inr.{u1, u3} C _inst_1 X Y _inst_3)))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.is_colimit_of_has_binary_coproduct_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasBinaryCoproductOfPreservesColimitₓ'. -/
/--
If `G` preserves binary coproducts and `C` has them, then the binary cofan constructed of the mapped
morphisms of the binary product cocone is a colimit.
@@ -154,6 +220,12 @@ def isColimitOfHasBinaryCoproductOfPreservesColimit [PreservesColimit (pair X Y)
variable [HasBinaryCoproduct (G.obj X) (G.obj Y)]
+/- warning: category_theory.limits.preserves_colimit_pair.of_iso_coprod_comparison -> CategoryTheory.Limits.PreservesColimitPair.ofIsoCoprodComparison is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.HasBinaryCoproduct.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) _inst_4) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Limits.coprodComparison.{u1, u3, u4, u2} C _inst_1 D _inst_2 G X Y _inst_3 _inst_4)], CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.HasBinaryCoproduct.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)] [i : CategoryTheory.IsIso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) _inst_4) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Limits.coprodComparison.{u1, u3, u4, u2} C _inst_1 D _inst_2 G X Y _inst_3 _inst_4)], CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimit_pair.of_iso_coprod_comparison CategoryTheory.Limits.PreservesColimitPair.ofIsoCoprodComparisonₓ'. -/
/-- If the coproduct comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`.
-/
@@ -168,6 +240,12 @@ def PreservesColimitPair.ofIsoCoprodComparison [i : IsIso (coprodComparison G X
variable [PreservesColimit (pair X Y) G]
+/- warning: category_theory.limits.preserves_colimit_pair.iso -> CategoryTheory.Limits.PreservesColimitPair.iso is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.HasBinaryCoproduct.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], CategoryTheory.Iso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) _inst_4) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3))
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.HasBinaryCoproduct.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 X Y) G], CategoryTheory.Iso.{u2, u4} D _inst_2 (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) _inst_4) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3))
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimit_pair.iso CategoryTheory.Limits.PreservesColimitPair.isoₓ'. -/
/--
If `G` preserves the coproduct of `(X,Y)`, then the coproduct comparison map for `G` at `(X,Y)` is
an isomorphism.
@@ -177,6 +255,12 @@ def PreservesColimitPair.iso : G.obj X ⨿ G.obj Y ≅ G.obj (X ⨿ Y) :=
(isColimitOfHasBinaryCoproductOfPreservesColimit G X Y)
#align category_theory.limits.preserves_colimit_pair.iso CategoryTheory.Limits.PreservesColimitPair.iso
+/- warning: category_theory.limits.preserves_colimit_pair.iso_hom -> CategoryTheory.Limits.PreservesColimitPair.iso_hom is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.HasBinaryCoproduct.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 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_2)) (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) _inst_4) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3))) (CategoryTheory.Iso.hom.{u2, u4} D _inst_2 (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G Y) _inst_4) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 G (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Limits.PreservesColimitPair.iso.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y _inst_3 _inst_4 _inst_5)) (CategoryTheory.Limits.coprodComparison.{u1, u3, u4, u2} C _inst_1 D _inst_2 G X Y _inst_3 _inst_4)
+but is expected to have type
+ forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] (G : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) (X : C) (Y : C) [_inst_3 : CategoryTheory.Limits.HasBinaryCoproduct.{u1, u3} C _inst_1 X Y] [_inst_4 : CategoryTheory.Limits.HasBinaryCoproduct.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y)] [_inst_5 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.pair.{u1, u3} C _inst_1 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_2)) (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) _inst_4) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3))) (CategoryTheory.Iso.hom.{u2, u4} D _inst_2 (CategoryTheory.Limits.coprod.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) Y) _inst_4) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 G) (CategoryTheory.Limits.coprod.{u1, u3} C _inst_1 X Y _inst_3)) (CategoryTheory.Limits.PreservesColimitPair.iso.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X Y _inst_3 _inst_4 _inst_5)) (CategoryTheory.Limits.coprodComparison.{u1, u3, u4, u2} C _inst_1 D _inst_2 G X Y _inst_3 _inst_4)
+Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimit_pair.iso_hom CategoryTheory.Limits.PreservesColimitPair.iso_homₓ'. -/
@[simp]
theorem PreservesColimitPair.iso_hom :
(PreservesColimitPair.iso G X Y).Hom = coprodComparison G X 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)
@@ -26,9 +26,7 @@ universe v₁ v₂ u₁ u₂
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
variable {C : Type u₁} [Category.{v₁} C]
-
variable {D : Type u₂} [Category.{v₂} D]
-
variable (G : C ⥤ D)
namespace CategoryTheory.Limits
@@ -2,15 +2,12 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.limits.preserves.shapes.binary_products
-! leanprover-community/mathlib commit 024a4231815538ac739f52d08dd20a55da0d6b23
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Basic
+#align_import category_theory.limits.preserves.shapes.binary_products from "leanprover-community/mathlib"@"024a4231815538ac739f52d08dd20a55da0d6b23"
+
/-!
# Preserving binary products
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,
@@ -116,8 +116,8 @@ the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `BinaryCofan.mk` with `Functor.mapCocone`.
-/
def isColimitMapCoconeBinaryCofanEquiv :
- IsColimit (Functor.mapCocone G (BinaryCofan.mk f g))
- ≃ IsColimit (BinaryCofan.mk (G.map f) (G.map g)) :=
+ IsColimit (Functor.mapCocone G (BinaryCofan.mk f g))
+ ≃ IsColimit (BinaryCofan.mk (G.map f) (G.map g)) :=
(IsColimit.precomposeHomEquiv (diagramIsoPair _).symm _).symm.trans
(IsColimit.equivIsoColimit
(Cocones.ext (Iso.refl _)
@@ -183,4 +183,3 @@ instance : IsIso (coprodComparison G X Y) := by
end
end CategoryTheory.Limits
-
@@ -45,7 +45,7 @@ The map of a binary fan is a limit iff the fork consisting of the mapped morphis
essentially lets us commute `BinaryFan.mk` with `Functor.mapCone`.
-/
def isLimitMapConeBinaryFanEquiv :
- IsLimit (Functor.mapCone G (BinaryFan.mk f g)) ≃ IsLimit (BinaryFan.mk (G.map f) (G.map g)) :=
+ IsLimit (G.mapCone (BinaryFan.mk f g)) ≃ IsLimit (BinaryFan.mk (G.map f) (G.map g)) :=
(IsLimit.postcomposeHomEquiv (diagramIsoPair _) _).symm.trans
(IsLimit.equivIsoLimit
(Cones.ext (Iso.refl _)
All dependencies are ported!