148 lines
6.5 KiB
Text
148 lines
6.5 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Jakob von Raumer
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Functor category has (co)limits if the codomain has them
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-/
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import ..colimits
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open functor nat_trans eq is_trunc
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namespace category
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-- preservation of limits
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variables {D C I : Precategory}
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definition limit_functor [constructor]
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[H : has_limits_of_shape D I] (F : I ⇒ D ^c C) : C ⇒ D :=
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begin
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assert lem : Π(c d : carrier C) (f : hom c d) ⦃i j : carrier I⦄ (k : i ⟶ j),
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(constant2_functor F d) k ∘ to_fun_hom (F i) f ∘ limit_morphism (constant2_functor F c) i =
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to_fun_hom (F j) f ∘ limit_morphism (constant2_functor F c) j,
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{ intro c d f i j k, rewrite [-limit_commute _ k,▸*,+assoc,▸*,-naturality (F k) f]},
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fapply functor.mk,
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{ intro c, exact limit_object (constant2_functor F c)},
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{ intro c d f, fapply hom_limit,
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{ intro i, refine to_fun_hom (F i) f ∘ !limit_morphism},
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{ apply lem}},
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{ exact abstract begin intro c, symmetry, apply eq_hom_limit, intro i,
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rewrite [id_right,respect_id,▸*,id_left] end end},
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{ intro a b c g f, symmetry, apply eq_hom_limit, intro i, -- report: adding abstract fails here
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rewrite [respect_comp,assoc,hom_limit_commute,-assoc,hom_limit_commute,assoc]}
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end
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definition limit_functor_cone [constructor]
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[H : has_limits_of_shape D I] (F : I ⇒ D ^c C) : cone_obj F :=
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begin
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fapply cone_obj.mk,
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{ exact limit_functor F},
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{ fapply nat_trans.mk,
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{ intro i, esimp, fapply nat_trans.mk,
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{ intro c, esimp, apply limit_morphism},
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{ intro c d f, rewrite [▸*,hom_limit_commute (constant2_functor F d)]}},
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{ intro i j k, apply nat_trans_eq, intro c,
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rewrite [▸*,id_right,limit_commute (constant2_functor F c)]}}
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end
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variables (D C I)
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definition has_limits_of_shape_functor [instance] [H : has_limits_of_shape D I]
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: has_limits_of_shape (D ^c C) I :=
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begin
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intro F, fapply has_terminal_object.mk,
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{ exact limit_functor_cone F},
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{ intro c, esimp at *, induction c with G η, induction η with η p, esimp at *,
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fapply is_contr.mk,
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{ fapply cone_hom.mk,
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{ fapply nat_trans.mk,
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{ intro c, esimp, fapply hom_limit,
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{ intro i, esimp, exact η i c},
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{ intro i j k, esimp, exact ap010 natural_map (p k) c ⬝ !id_right}},
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{ intro c d f, esimp, fapply @limit_cone_unique,
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{ intro i, esimp, exact to_fun_hom (F i) f ∘ η i c},
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{ intro i j k, rewrite [▸*,assoc,-naturality,-assoc,-compose_def,p k,▸*,id_right]},
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{ intro i, rewrite [assoc, hom_limit_commute (constant2_functor F d),▸*,-assoc,
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hom_limit_commute]},
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{ intro i, rewrite [assoc, hom_limit_commute (constant2_functor F d),naturality]}}},
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{ intro i, apply nat_trans_eq, intro c,
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rewrite [▸*,hom_limit_commute (constant2_functor F c)]}},
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{ intro h, induction h with f q, apply cone_hom_eq,
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apply nat_trans_eq, intro c, esimp at *, symmetry,
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apply eq_hom_limit, intro i, exact ap010 natural_map (q i) c}}
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end
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definition is_complete_functor [instance] [H : is_complete D] : is_complete (D ^c C) :=
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λI, _
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variables {D C I}
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-- preservation of colimits
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-- definition constant2_functor_op [constructor] (F : I ⇒ (D ^c C)ᵒᵖ) (c : C) : I ⇒ D :=
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-- proof
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-- functor.mk (λi, to_fun_ob (F i) c)
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-- (λi j f, natural_map (F f) c)
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-- abstract (λi, ap010 natural_map !respect_id c ⬝ proof idp qed) end
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-- abstract (λi j k g f, ap010 natural_map !respect_comp c) end
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-- qed
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definition colimit_functor [constructor]
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[H : has_colimits_of_shape D I] (F : Iᵒᵖ ⇒ (D ^c C)ᵒᵖ) : C ⇒ D :=
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begin
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fapply functor.mk,
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{ intro c, exact colimit_object (constant2_functor Fᵒᵖ' c)},
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{ intro c d f, apply colimit_hom_colimit, apply constant2_functor_natural _ f},
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{ exact abstract begin intro c, symmetry, apply eq_colimit_hom, intro i,
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rewrite [id_left,▸*,respect_id,id_right] end end},
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{ intro a b c g f, symmetry, apply eq_colimit_hom, intro i, -- report: adding abstract fails here
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rewrite [▸*,respect_comp,-assoc,colimit_hom_commute,assoc,colimit_hom_commute,-assoc]}
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end
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definition colimit_functor_cone [constructor]
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[H : has_colimits_of_shape D I] (F : Iᵒᵖ ⇒ (D ^c C)ᵒᵖ) : cone_obj F :=
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begin
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fapply cone_obj.mk,
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{ exact colimit_functor F},
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{ fapply nat_trans.mk,
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{ intro i, esimp, fapply nat_trans.mk,
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{ intro c, esimp, apply colimit_morphism},
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{ intro c d f, apply colimit_hom_commute (constant2_functor Fᵒᵖ' c)}},
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{ intro i j k, apply nat_trans_eq, intro c,
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rewrite [▸*,id_left], apply colimit_commute (constant2_functor Fᵒᵖ' c)}}
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end
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variables (D C I)
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definition has_colimits_of_shape_functor [instance] [H : has_colimits_of_shape D I]
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: has_colimits_of_shape (D ^c C) I :=
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begin
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intro F, fapply has_terminal_object.mk,
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{ exact colimit_functor_cone F},
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{ intro c, esimp at *, induction c with G η, induction η with η p, esimp at *,
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fapply is_contr.mk,
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{ fapply cone_hom.mk,
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{ fapply nat_trans.mk,
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{ intro c, esimp, fapply colimit_hom,
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{ intro i, esimp, exact η i c},
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{ intro i j k, esimp, exact ap010 natural_map (p k) c ⬝ !id_left}},
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{ intro c d f, esimp, fapply @colimit_cocone_unique,
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{ intro i, esimp, exact η i d ∘ to_fun_hom (F i) f},
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{ intro i j k, rewrite [▸*,-assoc,naturality,assoc,-compose_def,p k,▸*,id_left]},
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{ intro i, rewrite [-assoc, colimit_hom_commute (constant2_functor Fᵒᵖ' c),
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▸*, naturality]},
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{ intro i, rewrite [-assoc, colimit_hom_commute (constant2_functor Fᵒᵖ' c),▸*,assoc,
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colimit_hom_commute (constant2_functor Fᵒᵖ' d)]}}},
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{ intro i, apply nat_trans_eq, intro c,
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rewrite [▸*,colimit_hom_commute (constant2_functor Fᵒᵖ' c)]}},
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{ intro h, induction h with f q, apply cone_hom_eq,
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apply nat_trans_eq, intro c, esimp at *, symmetry,
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apply eq_colimit_hom, intro i, exact ap010 natural_map (q i) c}}
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end
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local attribute has_limits_of_shape_op_op [instance] [priority 1]
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universe variables u v
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definition is_cocomplete_functor [instance] [H : is_cocomplete.{_ _ u v} D] : is_cocomplete.{_ _ u v} (D ^c C) :=
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λI, _
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end category
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