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feat(category_theory): preservation of (co)limits, (co)limits in functor categories
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‎category_theory/limits/functor_category.lean

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-- Copyright (c) 2018 Scott Morrison. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Scott Morrison
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import category_theory.products
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import category_theory.limits.preserves
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open category_theory category_theory.category
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namespace category_theory.limits
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universes u v
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variables {C : Type u} [𝒞 : category.{u v} C]
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include 𝒞
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variables {J K : Type v} [small_category J] [small_category K]
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@[simp] lemma cone.functor_w {F : J ⥤ (K ⥤ C)} (c : cone F) {j j' : J} (f : j ⟶ j') (k : K) :
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(c.π.app j).app k ≫ (F.map f).app k = (c.π.app j').app k :=
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by convert ←nat_trans.congr_app (c.π.naturality f).symm k; apply id_comp
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@[simp] lemma cocone.functor_w {F : J ⥤ (K ⥤ C)} (c : cocone F) {j j' : J} (f : j ⟶ j') (k : K) :
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(F.map f).app k ≫ (c.ι.app j').app k = (c.ι.app j).app k :=
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by convert ←nat_trans.congr_app (c.ι.naturality f) k; apply comp_id
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@[simp] def functor_category_limit_cone [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
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cone F :=
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{ X := F.flip ⋙ lim,
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π :=
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{ app := λ j,
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{ app := λ k, limit.π (F.flip.obj k) j },
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naturality' := λ j j' f,
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by ext k; convert (limit.w (F.flip.obj k) _).symm using 1; apply id_comp } }
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@[simp] def functor_category_colimit_cocone [has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) :
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cocone F :=
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{ X := F.flip ⋙ colim,
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ι :=
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{ app := λ j,
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{ app := λ k, colimit.ι (F.flip.obj k) j },
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naturality' := λ j j' f,
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by ext k; convert (colimit.w (F.flip.obj k) _) using 1; apply comp_id } }
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@[simp] def evaluate_functor_category_limit_cone
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[has_limits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
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((evaluation K C).obj k).map_cone (functor_category_limit_cone F) ≅
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limit.cone (F.flip.obj k) :=
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cones.ext (iso.refl _) (by tidy)
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@[simp] def evaluate_functor_category_colimit_cocone
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[has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
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((evaluation K C).obj k).map_cocone (functor_category_colimit_cocone F) ≅
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colimit.cocone (F.flip.obj k) :=
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cocones.ext (iso.refl _) (by tidy)
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def functor_category_is_limit_cone [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
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is_limit (functor_category_limit_cone F) :=
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{ lift := λ s,
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{ app := λ k, limit.lift (F.flip.obj k) (((evaluation K C).obj k).map_cone s),
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naturality' := λ k k' f,
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by ext; dsimp; simpa using (s.π.app j).naturality f },
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uniq' := λ s m w,
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begin
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ext1 k,
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exact is_limit.uniq _
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(((evaluation K C).obj k).map_cone s) (m.app k) (λ j, nat_trans.congr_app (w j) k)
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end }
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def functor_category_is_colimit_cocone [has_colimits_of_shape.{u v} J C] (F : J ⥤ K ⥤ C) :
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is_colimit (functor_category_colimit_cocone F) :=
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{ desc := λ s,
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{ app := λ k, colimit.desc (F.flip.obj k) (((evaluation K C).obj k).map_cocone s),
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naturality' := λ k k' f,
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begin
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ext,
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rw [←assoc, ←assoc],
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dsimp [functor.flip],
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simpa using (s.ι.app j).naturality f
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end },
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uniq' := λ s m w,
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begin
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ext1 k,
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exact is_colimit.uniq _
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(((evaluation K C).obj k).map_cocone s) (m.app k) (λ j, nat_trans.congr_app (w j) k)
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end }
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instance functor_category_has_limits_of_shape
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[has_limits_of_shape J C] : has_limits_of_shape J (K ⥤ C) :=
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λ F,
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{ cone := functor_category_limit_cone F,
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is_limit := functor_category_is_limit_cone F }
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instance functor_category_has_colimits_of_shape
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[has_colimits_of_shape J C] : has_colimits_of_shape J (K ⥤ C) :=
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λ F,
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{ cocone := functor_category_colimit_cocone F,
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is_colimit := functor_category_is_colimit_cocone F }
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instance functor_category_has_limits [has_limits C] : has_limits (K ⥤ C) :=
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λ J 𝒥, by resetI; apply_instance
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instance functor_category_has_colimits [has_colimits C] : has_colimits (K ⥤ C) :=
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λ J 𝒥, by resetI; apply_instance
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instance evaluation_preserves_limits_of_shape [has_limits_of_shape J C] (k : K) :
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preserves_limits_of_shape J ((evaluation K C).obj k) :=
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λ F, preserves_limit_of_preserves_limit_cone (limit.is_limit _) $
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is_limit.of_iso_limit (limit.is_limit _)
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(evaluate_functor_category_limit_cone F k).symm
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instance evaluation_preserves_colimits_of_shape [has_colimits_of_shape J C] (k : K) :
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preserves_colimits_of_shape J ((evaluation K C).obj k) :=
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λ F, preserves_colimit_of_preserves_colimit_cocone (colimit.is_colimit _) $
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is_colimit.of_iso_colimit (colimit.is_colimit _)
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(evaluate_functor_category_colimit_cocone F k).symm
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instance evaluation_preserves_limits [has_limits C] (k : K) :
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preserves_limits ((evaluation K C).obj k) :=
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λ J 𝒥, by resetI; apply_instance
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instance evaluation_preserves_colimits [has_colimits C] (k : K) :
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preserves_colimits ((evaluation K C).obj k) :=
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λ J 𝒥, by resetI; apply_instance
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end category_theory.limits

‎category_theory/limits/preserves.lean

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-- Copyright (c) 2018 Scott Morrison. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Scott Morrison, Reid Barton
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-- Preservation and reflection of (co)limits.
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import category_theory.whiskering
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import category_theory.limits.limits
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open category_theory
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namespace category_theory.limits
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universes u₁ u₂ u₃ v
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variables {C : Type u₁} [𝒞 : category.{u₁ v} C]
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variables {D : Type u₂} [𝒟 : category.{u₂ v} D]
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include 𝒞 𝒟
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variables {J : Type v} [small_category J] {K : J ⥤ C}
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/- Note on "preservation of (co)limits"
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There are various distinct notions of "preserving limits". The one we
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aim to capture here is: A functor F : C → D "preserves limits" if it
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sends every limit cone in C to a limit cone in D. Informally, F
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preserves all the limits which exist in C.
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Note that:
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* Of course, we do not want to require F to *strictly* take chosen
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limit cones of C to chosen limit cones of D. Indeed, the above
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definition makes no reference to a choice of limit cones so it makes
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sense without any conditions on C or D.
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* Some diagrams in C may have no limit. In this case, there is no
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condition on the behavior of F on such diagrams. There are other
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notions (such as "flat functor") which impose conditions also on
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diagrams in C with no limits, but these are not considered here.
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In order to be able to express the property of preserving limits of a
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certain form, we say that a functor F preserves the limit of a
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diagram K if F sends every limit cone on K to a limit cone. This is
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vacuously satisfied when K does not admit a limit, which is consistent
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with the above definition of "preserves limits".
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-/
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class preserves_limit (K : J ⥤ C) (F : C ⥤ D) :=
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(preserves : Π {c : cone K}, is_limit c → is_limit (F.map_cone c))
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class preserves_colimit (K : J ⥤ C) (F : C ⥤ D) :=
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(preserves : Π {c : cocone K}, is_colimit c → is_colimit (F.map_cocone c))
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@[class] def preserves_limits_of_shape (J : Type v) [small_category J] (F : C ⥤ D) :=
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Π {K : J ⥤ C}, preserves_limit K F
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@[class] def preserves_colimits_of_shape (J : Type v) [small_category J] (F : C ⥤ D) :=
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Π {K : J ⥤ C}, preserves_colimit K F
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@[class] def preserves_limits (F : C ⥤ D) :=
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Π {J : Type v} {𝒥 : small_category J}, by exactI preserves_limits_of_shape J F
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@[class] def preserves_colimits (F : C ⥤ D) :=
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Π {J : Type v} {𝒥 : small_category J}, by exactI preserves_colimits_of_shape J F
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instance preserves_limit_of_preserves_limits_of_shape (F : C ⥤ D)
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[H : preserves_limits_of_shape J F] : preserves_limit K F :=
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H
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instance preserves_colimit_of_preserves_colimits_of_shape (F : C ⥤ D)
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[H : preserves_colimits_of_shape J F] : preserves_colimit K F :=
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H
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instance preserves_limits_of_shape_of_preserves_limits (F : C ⥤ D)
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[H : preserves_limits F] : preserves_limits_of_shape J F :=
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@H J _
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instance preserves_colimits_of_shape_of_preserves_colimits (F : C ⥤ D)
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[H : preserves_colimits F] : preserves_colimits_of_shape J F :=
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@H J _
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instance id_preserves_limits : preserves_limits (functor.id C) :=
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λ J 𝒥 K, by exactI ⟨λ c h,
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⟨λ s, h.lift ⟨s.X, λ j, s.π.app j, λ j j' f, s.π.naturality f⟩,
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by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
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by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩; exact h.uniq _ m w⟩⟩
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instance id_preserves_colimits : preserves_colimits (functor.id C) :=
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λ J 𝒥 K, by exactI ⟨λ c h,
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⟨λ s, h.desc ⟨s.X, λ j, s.ι.app j, λ j j' f, s.ι.naturality f⟩,
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by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
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by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩; exact h.uniq _ m w⟩⟩
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section
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variables {E : Type u₃} [ℰ : category.{u₃ v} E]
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variables (F : C ⥤ D) (G : D ⥤ E)
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local attribute [elab_simple] preserves_limit.preserves preserves_colimit.preserves
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instance comp_preserves_limit [preserves_limit K F] [preserves_limit (K ⋙ F) G] :
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preserves_limit K (F ⋙ G) :=
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⟨λ c h, preserves_limit.preserves G (preserves_limit.preserves F h)⟩
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instance comp_preserves_colimit [preserves_colimit K F] [preserves_colimit (K ⋙ F) G] :
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preserves_colimit K (F ⋙ G) :=
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⟨λ c h, preserves_colimit.preserves G (preserves_colimit.preserves F h)⟩
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end
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/-- If F preserves one limit cone for the diagram K,
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then it preserves any limit cone for K. -/
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def preserves_limit_of_preserves_limit_cone {F : C ⥤ D} {t : cone K}
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(h : is_limit t) (hF : is_limit (F.map_cone t)) : preserves_limit K F :=
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⟨λ t' h', is_limit.of_iso_limit hF (functor.on_iso _ (is_limit.unique h h'))⟩
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/-- If F preserves one colimit cocone for the diagram K,
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then it preserves any colimit cocone for K. -/
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def preserves_colimit_of_preserves_colimit_cocone {F : C ⥤ D} {t : cocone K}
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(h : is_colimit t) (hF : is_colimit (F.map_cocone t)) : preserves_colimit K F :=
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⟨λ t' h', is_colimit.of_iso_colimit hF (functor.on_iso _ (is_colimit.unique h h'))⟩
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/-
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A functor F : C → D reflects limits if whenever the image of a cone
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under F is a limit cone in D, the cone was already a limit cone in C.
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Note that again we do not assume a priori that D actually has any
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limits.
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-/
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class reflects_limit (K : J ⥤ C) (F : C ⥤ D) :=
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(reflects : Π {c : cone K}, is_limit (F.map_cone c) → is_limit c)
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class reflects_colimit (K : J ⥤ C) (F : C ⥤ D) :=
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(reflects : Π {c : cocone K}, is_colimit (F.map_cocone c) → is_colimit c)
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@[class] def reflects_limits_of_shape (J : Type v) [small_category J] (F : C ⥤ D) :=
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Π {K : J ⥤ C}, reflects_limit K F
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@[class] def reflects_colimits_of_shape (J : Type v) [small_category J] (F : C ⥤ D) :=
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Π {K : J ⥤ C}, reflects_colimit K F
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@[class] def reflects_limits (F : C ⥤ D) :=
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Π {J : Type v} {𝒥 : small_category J}, by exactI reflects_limits_of_shape J F
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@[class] def reflects_colimits (F : C ⥤ D) :=
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Π {J : Type v} {𝒥 : small_category J}, by exactI reflects_colimits_of_shape J F
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instance reflects_limit_of_reflects_limits_of_shape (K : J ⥤ C) (F : C ⥤ D)
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[H : reflects_limits_of_shape J F] : reflects_limit K F :=
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H
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instance reflects_colimit_of_reflects_colimits_of_shape (K : J ⥤ C) (F : C ⥤ D)
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[H : reflects_colimits_of_shape J F] : reflects_colimit K F :=
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H
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instance reflects_limits_of_shape_of_reflects_limits (F : C ⥤ D)
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[H : reflects_limits F] : reflects_limits_of_shape J F :=
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@H J _
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instance reflects_colimits_of_shape_of_reflects_colimits (F : C ⥤ D)
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[H : reflects_colimits F] : reflects_colimits_of_shape J F :=
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@H J _
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instance id_reflects_limits : reflects_limits (functor.id C) :=
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λ J 𝒥 K, by exactI ⟨λ c h,
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⟨λ s, h.lift ⟨s.X, λ j, s.π.app j, λ j j' f, s.π.naturality f⟩,
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by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
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by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩; exact h.uniq _ m w⟩⟩
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instance id_reflects_colimits : reflects_colimits (functor.id C) :=
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λ J 𝒥 K, by exactI ⟨λ c h,
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⟨λ s, h.desc ⟨s.X, λ j, s.ι.app j, λ j j' f, s.ι.naturality f⟩,
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by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
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by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩; exact h.uniq _ m w⟩⟩
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section
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variables {E : Type u₃} [ℰ : category.{u₃ v} E]
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variables (F : C ⥤ D) (G : D ⥤ E)
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instance comp_reflects_limit [reflects_limit K F] [reflects_limit (K ⋙ F) G] :
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reflects_limit K (F ⋙ G) :=
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⟨λ c h, reflects_limit.reflects (reflects_limit.reflects h)⟩
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instance comp_reflects_colimit [reflects_colimit K F] [reflects_colimit (K ⋙ F) G] :
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reflects_colimit K (F ⋙ G) :=
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⟨λ c h, reflects_colimit.reflects (reflects_colimit.reflects h)⟩
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end
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end category_theory.limits

‎category_theory/natural_transformation.lean

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subst hc
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end
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lemma congr_app {α β : F ⟹ G} (h : α = β) (X : C) : α.app X = β.app X := by rw h
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/-- `vcomp α β` is the vertical compositions of natural transformations. -/
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def vcomp (α : F ⟹ G) (β : G ⟹ H) : F ⟹ H :=
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{ app := λ X, (α.app X) ≫ (β.app X),

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