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Mathlib.CategoryTheory.Idempotents.FunctorCategories

Idempotent completeness and functor categories #

In this file we define an instance functor_category_isIdempotentComplete expressing that a functor category J ⥤ C is idempotent complete when the target category C is.

We also provide a fully faithful functor karoubiFunctorCategoryEmbedding : Karoubi (J ⥤ C)) : J ⥤ Karoubi C for all categories J and C.

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@[simp]
theorem CategoryTheory.Idempotents.app_idem {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) (X : J) :
CategoryTheory.CategoryStruct.comp (P.p.app X) (P.p.app X) = P.p.app X
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@[simp]
theorem CategoryTheory.Idempotents.app_idem_assoc {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) (X : J) {Z : C} (h : P.X.obj X ⟶ Z) :
CategoryTheory.CategoryStruct.comp (P.p.app X) (CategoryTheory.CategoryStruct.comp (P.p.app X) h) = CategoryTheory.CategoryStruct.comp (P.p.app X) h
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@[simp]
theorem CategoryTheory.Idempotents.app_p_comp {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (X : J) :
CategoryTheory.CategoryStruct.comp (P.p.app X) (f.f.app X) = f.f.app X
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@[simp]
theorem CategoryTheory.Idempotents.app_p_comp_assoc {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (X : J) {Z : C} (h : Q.X.obj X ⟶ Z) :
CategoryTheory.CategoryStruct.comp (P.p.app X) (CategoryTheory.CategoryStruct.comp (f.f.app X) h) = CategoryTheory.CategoryStruct.comp (f.f.app X) h
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@[simp]
theorem CategoryTheory.Idempotents.app_comp_p {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (X : J) :
CategoryTheory.CategoryStruct.comp (f.f.app X) (Q.p.app X) = f.f.app X
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@[simp]
theorem CategoryTheory.Idempotents.app_comp_p_assoc {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (X : J) {Z : C} (h : Q.X.obj X ⟶ Z) :
CategoryTheory.CategoryStruct.comp (f.f.app X) (CategoryTheory.CategoryStruct.comp (Q.p.app X) h) = CategoryTheory.CategoryStruct.comp (f.f.app X) h
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theorem CategoryTheory.Idempotents.app_p_comm {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (X : J) :
CategoryTheory.CategoryStruct.comp (P.p.app X) (f.f.app X) = CategoryTheory.CategoryStruct.comp (f.f.app X) (Q.p.app X)
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theorem CategoryTheory.Idempotents.app_p_comm_assoc {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (X : J) {Z : C} (h : Q.X.obj X ⟶ Z) :
CategoryTheory.CategoryStruct.comp (P.p.app X) (CategoryTheory.CategoryStruct.comp (f.f.app X) h) = CategoryTheory.CategoryStruct.comp (f.f.app X) (CategoryTheory.CategoryStruct.comp (Q.p.app X) h)
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instance CategoryTheory.Idempotents.functor_category_isIdempotentComplete (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.IsIdempotentComplete C] :
CategoryTheory.IsIdempotentComplete (CategoryTheory.Functor J C)
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def CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) :
CategoryTheory.Functor J (CategoryTheory.Idempotents.Karoubi C)

On objects, the functor which sends a formal direct factor P of a functor F : J ⥤ C to the functor J ⥤ Karoubi C which sends (j : J) to the corresponding direct factor of F.obj j.

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    theorem CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj_obj_p {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) (j : J) :
    ((CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P).obj j).p = P.p.app j
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    theorem CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj_obj_X {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) (j : J) :
    ((CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P).obj j).X = P.X.obj j
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    theorem CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj_map_f {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) {j : J} {j' : J} (φ : j ⟶ j') :
    ((CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P).map φ).f = CategoryTheory.CategoryStruct.comp (P.p.app j) (P.X.map φ)
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    def CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.map {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) :
    CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P ⟶ CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj Q

    Tautological action on maps of the functor Karoubi (J ⥤ C) ⥤ (J ⥤ Karoubi C).

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      theorem CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.map_app_f {J : Type u_1} {C : Type u_2} [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] {P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} {Q : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : P ⟶ Q) (j : J) :
      ((CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.map f).app j).f = f.f.app j
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      def CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] :
      CategoryTheory.Functor (CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) (CategoryTheory.Functor J (CategoryTheory.Idempotents.Karoubi C))

      The tautological fully faithful functor Karoubi (J ⥤ C) ⥤ (J ⥤ Karoubi C).

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        theorem CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding_map (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] :
        ∀ {X Y : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)} (f : X ⟶ Y), (CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding J C).map f = CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.map f
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        theorem CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding_obj (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.Functor J C)) :
        (CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding J C).obj P = CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P
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        instance CategoryTheory.Idempotents.instFullKaroubiFunctorKaroubiFunctorCategoryEmbedding (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] :
        (CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding J C).Full
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        • ⋯ = ⋯
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        instance CategoryTheory.Idempotents.instFaithfulKaroubiFunctorKaroubiFunctorCategoryEmbedding (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_4, u_1} J] [CategoryTheory.Category.{u_3, u_2} C] :
        (CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding J C).Faithful
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        theorem CategoryTheory.Idempotents.toKaroubi_comp_karoubiFunctorCategoryEmbedding (J : Type u_1) (C : Type u_2) [CategoryTheory.Category.{u_3, u_1} J] [CategoryTheory.Category.{u_4, u_2} C] :
        (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.Functor J C)).comp (CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding J C) = (CategoryTheory.whiskeringRight J C (CategoryTheory.Idempotents.Karoubi C)).obj (CategoryTheory.Idempotents.toKaroubi C)

        The composition of (J ⥤ C) ⥤ Karoubi (J ⥤ C) and Karoubi (J ⥤ C) ⥤ (J ⥤ Karoubi C) equals the functor (J ⥤ C) ⥤ (J ⥤ Karoubi C) given by the composition with toKaroubi C : C ⥤ Karoubi C.