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Mathlib.CategoryTheory.Filtered.Basic

Filtered categories #

A category is filtered if every finite diagram admits a cocone. We give a simple characterisation of this condition as

  1. for every pair of objects there exists another object "to the right",
  2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal, and
  3. there exists some object.

Filtered colimits are often better behaved than arbitrary colimits. See CategoryTheory/Limits/Types for some details.

Filtered categories are nice because colimits indexed by filtered categories tend to be easier to describe than general colimits (and more often preserved by functors).

In this file we show that any functor from a finite category to a filtered category admits a cocone:

We also prove the converse of cocone_nonempty as of_cocone_nonempty.

Furthermore, we give special support for two diagram categories: The bowtie and the tulip. This is because these shapes show up in the proofs that forgetful functors of algebraic categories (e.g. MonCat, CommRingCat, ...) preserve filtered colimits.

All of the above API, except for the bowtie and the tulip, is also provided for cofiltered categories.

See also #

In CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit we show that filtered colimits commute with finite limits.

There is another characterization of filtered categories, namely that whenever F : J ⥤ C is a functor from a finite category, there is X : C such that Nonempty (limit (F.op ⋙ yoneda.obj X)). This is shown in CategoryTheory.Limits.Filtered.

A category IsFilteredOrEmpty if

  1. for every pair of objects there exists another object "to the right", and
  2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal.
Instances
    theorem CategoryTheory.IsFilteredOrEmpty.cocone_objs {C : Type u} [CategoryTheory.Category.{v, u} C] [self : CategoryTheory.IsFilteredOrEmpty C] (X : C) (Y : C) :
    ∃ (Z : C) (x : X Z) (x : Y Z), True

    for every pair of objects there exists another object "to the right"

    for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal

    A category IsFiltered if

    1. for every pair of objects there exists another object "to the right",
    2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal, and
    3. there exists some object.

    See . (They also define a diagram being filtered.)

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      a filtered category must be non empty

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      max j j' is an arbitrary choice of object to the right of both j and j', whose existence is ensured by IsFiltered.

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        leftToMax j j' is an arbitrary choice of morphism from j to max j j', whose existence is ensured by IsFiltered.

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          rightToMax j j' is an arbitrary choice of morphism from j' to max j j', whose existence is ensured by IsFiltered.

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            noncomputable def CategoryTheory.IsFiltered.coeq {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] {j : C} {j' : C} (f : j j') (f' : j j') :
            C

            coeq f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of object which admits a morphism coeqHom f f' : j' ⟶ coeq f f' such that coeq_condition : f ≫ coeqHom f f' = f' ≫ coeqHom f f'. Its existence is ensured by IsFiltered.

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              coeqHom f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of morphism coeqHom f f' : j' ⟶ coeq f f' such that coeq_condition : f ≫ coeqHom f f' = f' ≫ coeqHom f f'. Its existence is ensured by IsFiltered.

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                If C is filtered or emtpy, and we have a functor R : C ⥤ D with a left adjoint, then D is filtered or empty.

                If C is filtered or empty, and we have a right adjoint functor R : C ⥤ D, then D is filtered or empty.

                theorem CategoryTheory.IsFiltered.sup_objs_exists {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFiltered C] (O : Finset C) :
                ∃ (S : C), ∀ {X : C}, X ONonempty (X S)

                Any finite collection of objects in a filtered category has an object "to the right".

                theorem CategoryTheory.IsFiltered.sup_exists {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) :
                ∃ (S : C) (T : {X : C} → X O(X S)), ∀ {X Y : C} (mX : X O) (mY : Y O) {f : X Y}, X, Y, mX, mY, f HCategoryTheory.CategoryStruct.comp f (T mY) = T mX

                Given any Finset of objects {X, ...} and indexed collection of Finsets of morphisms {f, ...} in C, there exists an object S, with a morphism T X : X ⟶ S from each X, such that the triangles commute: f ≫ T Y = T X, for f : X ⟶ Y in the Finset.

                noncomputable def CategoryTheory.IsFiltered.sup {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) :
                C

                An arbitrary choice of object "to the right" of a finite collection of objects O and morphisms H, making all the triangles commute.

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                  noncomputable def CategoryTheory.IsFiltered.toSup {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) {X : C} (m : X O) :

                  The morphisms to sup O H.

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                    theorem CategoryTheory.IsFiltered.toSup_commutes {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) {X : C} {Y : C} (mX : X O) (mY : Y O) {f : X Y} (mf : X, Y, mX, mY, f H) :

                    The triangles of consisting of a morphism in H and the maps to sup O H commute.

                    If we have IsFiltered C, then for any functor F : J ⥤ C with FinCategory J, there exists a cocone over F.

                    An arbitrary choice of cocone over F : J ⥤ C, for FinCategory J and IsFiltered C.

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                      If C is filtered, and we have a functor R : C ⥤ D with a left adjoint, then D is filtered.

                      If C is filtered, and we have a right adjoint functor R : C ⥤ D, then D is filtered.

                      Being filtered is preserved by equivalence of categories.

                      If every finite diagram in C admits a cocone, then C is filtered. It is sufficient to verify this for diagrams whose shape lives in any one fixed universe.

                      For every universe w, C is filtered if and only if every finite diagram in C with shape in w admits a cocone.

                      noncomputable def CategoryTheory.IsFiltered.max₃ {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] (j₁ : C) (j₂ : C) (j₃ : C) :
                      C

                      max₃ j₁ j₂ j₃ is an arbitrary choice of object to the right of j₁, j₂ and j₃, whose existence is ensured by IsFiltered.

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                        firstToMax₃ j₁ j₂ j₃ is an arbitrary choice of morphism from j₁ to max₃ j₁ j₂ j₃, whose existence is ensured by IsFiltered.

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                          secondToMax₃ j₁ j₂ j₃ is an arbitrary choice of morphism from j₂ to max₃ j₁ j₂ j₃, whose existence is ensured by IsFiltered.

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                            thirdToMax₃ j₁ j₂ j₃ is an arbitrary choice of morphism from j₃ to max₃ j₁ j₂ j₃, whose existence is ensured by IsFiltered.

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                              noncomputable def CategoryTheory.IsFiltered.coeq₃ {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] {j₁ : C} {j₂ : C} (f : j₁ j₂) (g : j₁ j₂) (h : j₁ j₂) :
                              C

                              coeq₃ f g h, for morphisms f g h : j₁ ⟶ j₂, is an arbitrary choice of object which admits a morphism coeq₃Hom f g h : j₂ ⟶ coeq₃ f g h such that coeq₃_condition₁, coeq₃_condition₂ and coeq₃_condition₃ are satisfied. Its existence is ensured by IsFiltered.

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                                noncomputable def CategoryTheory.IsFiltered.coeq₃Hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] {j₁ : C} {j₂ : C} (f : j₁ j₂) (g : j₁ j₂) (h : j₁ j₂) :

                                coeq₃Hom f g h, for morphisms f g h : j₁ ⟶ j₂, is an arbitrary choice of morphism j₂ ⟶ coeq₃ f g h such that coeq₃_condition₁, coeq₃_condition₂ and coeq₃_condition₃ are satisfied. Its existence is ensured by IsFiltered.

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                                  theorem CategoryTheory.IsFiltered.span {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] {i : C} {j : C} {j' : C} (f : i j) (f' : i j') :

                                  For every span j ⟵ i ⟶ j', there exists a cocone j ⟶ k ⟵ j' such that the square commutes.

                                  theorem CategoryTheory.IsFiltered.bowtie {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] {j₁ : C} {j₂ : C} {k₁ : C} {k₂ : C} (f₁ : j₁ k₁) (g₁ : j₁ k₂) (f₂ : j₂ k₁) (g₂ : j₂ k₂) :

                                  Given a "bowtie" of morphisms

                                   j₁   j₂
                                   |\  /|
                                   | \/ |
                                   | /\ |
                                   |/  \∣
                                   vv  vv
                                   k₁  k₂
                                  

                                  in a filtered category, we can construct an object s and two morphisms from k₁ and k₂ to s, making the resulting squares commute.

                                  theorem CategoryTheory.IsFiltered.tulip {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsFilteredOrEmpty C] {j₁ : C} {j₂ : C} {j₃ : C} {k₁ : C} {k₂ : C} {l : C} (f₁ : j₁ k₁) (f₂ : j₂ k₁) (f₃ : j₂ k₂) (f₄ : j₃ k₂) (g₁ : j₁ l) (g₂ : j₃ l) :

                                  Given a "tulip" of morphisms

                                   j₁    j₂    j₃
                                   |\   / \   / |
                                   | \ /   \ /  |
                                   |  vv    vv  |
                                   \  k₁    k₂ /
                                    \         /
                                     \       /
                                      \     /
                                       \   /
                                        v v
                                         l
                                  

                                  in a filtered category, we can construct an object s and three morphisms from k₁, k₂ and l to s, making the resulting squares commute.

                                  A category IsCofilteredOrEmpty if

                                  1. for every pair of objects there exists another object "to the left", and
                                  2. for every pair of parallel morphisms there exists a morphism to the left so the compositions are equal.
                                  Instances
                                    theorem CategoryTheory.IsCofilteredOrEmpty.cone_objs {C : Type u} [CategoryTheory.Category.{v, u} C] [self : CategoryTheory.IsCofilteredOrEmpty C] (X : C) (Y : C) :
                                    ∃ (W : C) (x : W X) (x : W Y), True

                                    for every pair of objects there exists another object "to the left"

                                    for every pair of parallel morphisms there exists a morphism to the left so the compositions are equal

                                    A category IsCofiltered if

                                    1. for every pair of objects there exists another object "to the left",
                                    2. for every pair of parallel morphisms there exists a morphism to the left so the compositions are equal, and
                                    3. there exists some object.

                                    See .

                                    Instances

                                      a cofiltered category must be non empty

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                                      min j j' is an arbitrary choice of object to the left of both j and j', whose existence is ensured by IsCofiltered.

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                                        minToLeft j j' is an arbitrary choice of morphism from min j j' to j, whose existence is ensured by IsCofiltered.

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                                          minToRight j j' is an arbitrary choice of morphism from min j j' to j', whose existence is ensured by IsCofiltered.

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                                            noncomputable def CategoryTheory.IsCofiltered.eq {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofilteredOrEmpty C] {j : C} {j' : C} (f : j j') (f' : j j') :
                                            C

                                            eq f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of object which admits a morphism eqHom f f' : eq f f' ⟶ j such that eq_condition : eqHom f f' ≫ f = eqHom f f' ≫ f'. Its existence is ensured by IsCofiltered.

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                                              eqHom f f', for morphisms f f' : j ⟶ j', is an arbitrary choice of morphism eqHom f f' : eq f f' ⟶ j such that eq_condition : eqHom f f' ≫ f = eqHom f f' ≫ f'. Its existence is ensured by IsCofiltered.

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                                                theorem CategoryTheory.IsCofiltered.cospan {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofilteredOrEmpty C] {i : C} {j : C} {j' : C} (f : j i) (f' : j' i) :

                                                For every cospan j ⟶ i ⟵ j', there exists a cone j ⟵ k ⟶ j' such that the square commutes.

                                                theorem CategoryTheory.Functor.ranges_directed {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofilteredOrEmpty C] (F : CategoryTheory.Functor C (Type u_1)) (j : C) :
                                                Directed (fun (x x_1 : Set (F.obj j)) => x x_1) fun (f : (i : C) ×' (i j)) => Set.range (F.map f.snd)

                                                If C is cofiltered or empty, and we have a functor L : C ⥤ D with a right adjoint, then D is cofiltered or empty.

                                                If C is cofiltered or empty, and we have a left adjoint functor L : C ⥤ D, then D is cofiltered or empty.

                                                Any finite collection of objects in a cofiltered category has an object "to the left".

                                                theorem CategoryTheory.IsCofiltered.inf_exists {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) :
                                                ∃ (S : C) (T : {X : C} → X O(S X)), ∀ {X Y : C} (mX : X O) (mY : Y O) {f : X Y}, X, Y, mX, mY, f HCategoryTheory.CategoryStruct.comp (T mX) f = T mY

                                                Given any Finset of objects {X, ...} and indexed collection of Finsets of morphisms {f, ...} in C, there exists an object S, with a morphism T X : S ⟶ X from each X, such that the triangles commute: T X ≫ f = T Y, for f : X ⟶ Y in the Finset.

                                                noncomputable def CategoryTheory.IsCofiltered.inf {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) :
                                                C

                                                An arbitrary choice of object "to the left" of a finite collection of objects O and morphisms H, making all the triangles commute.

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                                                  noncomputable def CategoryTheory.IsCofiltered.infTo {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) {X : C} (m : X O) :

                                                  The morphisms from inf O H.

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                                                    theorem CategoryTheory.IsCofiltered.infTo_commutes {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.IsCofiltered C] (O : Finset C) (H : Finset ((X : C) ×' (Y : C) ×' (_ : X O) ×' (_ : Y O) ×' (X Y))) {X : C} {Y : C} (mX : X O) (mY : Y O) {f : X Y} (mf : X, Y, mX, mY, f H) :

                                                    The triangles consisting of a morphism in H and the maps from inf O H commute.

                                                    If we have IsCofiltered C, then for any functor F : J ⥤ C with FinCategory J, there exists a cone over F.

                                                    An arbitrary choice of cone over F : J ⥤ C, for FinCategory J and IsCofiltered C.

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                                                      If C is cofiltered, and we have a functor L : C ⥤ D with a right adjoint, then D is cofiltered.

                                                      If C is cofiltered, and we have a left adjoint functor L : C ⥤ D, then D is cofiltered.

                                                      If every finite diagram in C admits a cone, then C is cofiltered. It is sufficient to verify this for diagrams whose shape lives in any one fixed universe.

                                                      For every universe w, C is filtered if and only if every finite diagram in C with shape in w admits a cocone.