Documentation

Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Biproducts and binary biproducts #

We introduce the notion of (finite) biproducts and binary biproducts.

These are slightly unusual relative to the other shapes in the library, as they are simultaneously limits and colimits. (Zero objects are similar; they are "biterminal".)

For results about biproducts in preadditive categories see CategoryTheory.Preadditive.Biproducts.

In a category with zero morphisms, we model the (binary) biproduct of P Q : C using a BinaryBicone, which has a cone point X, and morphisms fst : X ⟶ P, snd : X ⟶ Q, inl : P ⟶ X and inr : X ⟶ Q, such that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q. Such a BinaryBicone is a biproduct if the cone is a limit cone, and the cocone is a colimit cocone.

For biproducts indexed by a Fintype J, a bicone again consists of a cone point X and morphisms π j : X ⟶ F j and ι j : F j ⟶ X for each j, such that ι j ≫ π j' is the identity when j = j' and zero otherwise.

Notation #

As is already taken for the sum of types, we introduce the notation X ⊞ Y for a binary biproduct. We introduce ⨁ f for the indexed biproduct.

Implementation notes #

Prior to leanprover-community/mathlib#14046, HasFiniteBiproducts required a DecidableEq instance on the indexing type. As this had no pay-off (everything about limits is non-constructive in mathlib), and occasional cost (constructing decidability instances appropriate for constructions involving the indexing type), we made everything classical.

structure CategoryTheory.Limits.Bicone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : JC) :
Type (max (max uC uC') w)

A c : Bicone F is:

  • an object c.pt and
  • morphisms π j : pt ⟶ F j and ι j : F j ⟶ pt for each j,
  • such that ι j ≫ π j' is the identity when j = j' and zero otherwise.
  • pt : C

    A c : Bicone F is:

    • an object c.pt and
    • morphisms π j : pt ⟶ F j and ι j : F j ⟶ pt for each j,
    • such that ι j ≫ π j' is the identity when j = j' and zero otherwise.
  • π : (j : J) → self.pt F j

    A c : Bicone F is:

    • an object c.pt and
    • morphisms π j : pt ⟶ F j and ι j : F j ⟶ pt for each j,
    • such that ι j ≫ π j' is the identity when j = j' and zero otherwise.
  • ι : (j : J) → F j self.pt

    A c : Bicone F is:

    • an object c.pt and
    • morphisms π j : pt ⟶ F j and ι j : F j ⟶ pt for each j,
    • such that ι j ≫ π j' is the identity when j = j' and zero otherwise.
  • ι_π : ∀ (j j' : J), CategoryTheory.CategoryStruct.comp (self j) (self j') = if h : j = j' then CategoryTheory.eqToHom else 0

    A c : Bicone F is:

    • an object c.pt and
    • morphisms π j : pt ⟶ F j and ι j : F j ⟶ pt for each j,
    • such that ι j ≫ π j' is the identity when j = j' and zero otherwise.
Instances For
    theorem CategoryTheory.Limits.Bicone.ι_π {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : JC} (self : CategoryTheory.Limits.Bicone F) (j : J) (j' : J) :
    CategoryTheory.CategoryStruct.comp (self j) (self j') = if h : j = j' then CategoryTheory.eqToHom else 0

    A c : Bicone F is:

    • an object c.pt and
    • morphisms π j : pt ⟶ F j and ι j : F j ⟶ pt for each j,
    • such that ι j ≫ π j' is the identity when j = j' and zero otherwise.

    A bicone morphism between two bicones for the same diagram is a morphism of the bicone points which commutes with the cone and cocone legs.

    Instances For
      @[simp]

      The triangle consisting of the two natural transformations and hom commutes

      @[simp]

      The triangle consisting of the two natural transformations and hom commutes

      The category of bicones on a given diagram.

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      @[simp]
      theorem CategoryTheory.Limits.Bicones.ext_inv_hom {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : JC} {c : CategoryTheory.Limits.Bicone F} {c' : CategoryTheory.Limits.Bicone F} (φ : c.pt c'.pt) (wι : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp (c j) φ.hom = c' j) _auto✝) (wπ : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp φ.hom (c' j) = c j) _auto✝) :
      (CategoryTheory.Limits.Bicones.ext φ ).inv.hom = φ.inv
      @[simp]
      theorem CategoryTheory.Limits.Bicones.ext_hom_hom {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : JC} {c : CategoryTheory.Limits.Bicone F} {c' : CategoryTheory.Limits.Bicone F} (φ : c.pt c'.pt) (wι : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp (c j) φ.hom = c' j) _auto✝) (wπ : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp φ.hom (c' j) = c j) _auto✝) :
      (CategoryTheory.Limits.Bicones.ext φ ).hom.hom = φ.hom
      def CategoryTheory.Limits.Bicones.ext {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : JC} {c : CategoryTheory.Limits.Bicone F} {c' : CategoryTheory.Limits.Bicone F} (φ : c.pt c'.pt) (wι : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp (c j) φ.hom = c' j) _auto✝) (wπ : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp φ.hom (c' j) = c j) _auto✝) :
      c c'

      To give an isomorphism between cocones, it suffices to give an isomorphism between their vertices which commutes with the cocone maps.

      Equations
      • CategoryTheory.Limits.Bicones.ext φ = { hom := { hom := φ.hom, := , := }, inv := { hom := φ.inv, := , := }, hom_inv_id := , inv_hom_id := }
      Instances For

        A functor G : C ⥤ D sends bicones over F to bicones over G.obj ∘ F functorially.

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          Extract the cone from a bicone.

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            @[reducible, inline]

            A shorthand for toConeFunctor.obj

            Equations
            • B.toCone = CategoryTheory.Limits.Bicone.toConeFunctor.obj B
            Instances For

              Extract the cocone from a bicone.

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              • One or more equations did not get rendered due to their size.
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                @[reducible, inline]

                A shorthand for toCoconeFunctor.obj

                Equations
                • B.toCocone = CategoryTheory.Limits.Bicone.toCoconeFunctor.obj B
                Instances For

                  We can turn any limit cone over a discrete collection of objects into a bicone.

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                    We can turn any colimit cocone over a discrete collection of objects into a bicone.

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                      Structure witnessing that a bicone is both a limit cone and a colimit cocone.

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                        theorem CategoryTheory.Limits.Bicone.IsBilimit.ext_iff {J : Type w} {C : Type uC} :
                        ∀ {inst : CategoryTheory.Category.{uC', uC} C} {inst_1 : CategoryTheory.Limits.HasZeroMorphisms C} {F : JC} {B : CategoryTheory.Limits.Bicone F} (x y : B.IsBilimit), x = y x.isLimit = y.isLimit x.isColimit = y.isColimit
                        theorem CategoryTheory.Limits.Bicone.IsBilimit.ext {J : Type w} {C : Type uC} :
                        ∀ {inst : CategoryTheory.Category.{uC', uC} C} {inst_1 : CategoryTheory.Limits.HasZeroMorphisms C} {F : JC} {B : CategoryTheory.Limits.Bicone F} (x y : B.IsBilimit), x.isLimit = y.isLimitx.isColimit = y.isColimitx = y
                        @[simp]
                        theorem CategoryTheory.Limits.Bicone.whisker_ι {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : JC} (c : CategoryTheory.Limits.Bicone f) (g : K J) (k : K) :
                        (c.whisker g) k = c (g k)
                        @[simp]
                        theorem CategoryTheory.Limits.Bicone.whisker_pt {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : JC} (c : CategoryTheory.Limits.Bicone f) (g : K J) :
                        (c.whisker g).pt = c.pt
                        @[simp]
                        theorem CategoryTheory.Limits.Bicone.whisker_π {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : JC} (c : CategoryTheory.Limits.Bicone f) (g : K J) (k : K) :
                        (c.whisker g) k = c (g k)

                        Whisker a bicone with an equivalence between the indexing types.

                        Equations
                        • c.whisker g = { pt := c.pt, π := fun (k : K) => c (g k), ι := fun (k : K) => c (g k), ι_π := }
                        Instances For

                          Taking the cone of a whiskered bicone results in a cone isomorphic to one gained by whiskering the cone and postcomposing with a suitable isomorphism.

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                            Taking the cocone of a whiskered bicone results in a cone isomorphic to one gained by whiskering the cocone and precomposing with a suitable isomorphism.

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                              def CategoryTheory.Limits.Bicone.whiskerIsBilimitIff {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : JC} (c : CategoryTheory.Limits.Bicone f) (g : K J) :
                              (c.whisker g).IsBilimit c.IsBilimit

                              Whiskering a bicone with an equivalence between types preserves being a bilimit bicone.

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                                A bicone over F : J → C, which is both a limit cone and a colimit cocone.

                                • A bicone over F : J → C, which is both a limit cone and a colimit cocone.

                                • isBilimit : self.bicone.IsBilimit

                                  A bicone over F : J → C, which is both a limit cone and a colimit cocone.

                                Instances For

                                  HasBiproduct F expresses the mere existence of a bicone which is simultaneously a limit and a colimit of the diagram F.

                                  Instances

                                    HasBiproduct F expresses the mere existence of a bicone which is simultaneously a limit and a colimit of the diagram F.

                                    C has biproducts of shape J if we have a limit and a colimit, with the same cone points, of every function F : J → C.

                                    Instances

                                      HasFiniteBiproducts C represents a choice of biproduct for every family of objects in C indexed by a finite type.

                                      Instances

                                        The isomorphism between the specified limit and the specified colimit for a functor with a bilimit.

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                                          @[reducible, inline]

                                          biproduct f computes the biproduct of a family of elements f. (It is defined as an abbreviation for limit (Discrete.functor f), so for most facts about biproduct f, you will just use general facts about limits and colimits.)

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                                            biproduct f computes the biproduct of a family of elements f. (It is defined as an abbreviation for limit (Discrete.functor f), so for most facts about biproduct f, you will just use general facts about limits and colimits.)

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                                              @[reducible, inline]

                                              The projection onto a summand of a biproduct.

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                                                @[reducible, inline]

                                                The inclusion into a summand of a biproduct.

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                                                  Note that as this lemma has an if in the statement, we include a DecidableEq argument. This means you may not be able to simp using this lemma unless you open scoped Classical.

                                                  @[reducible, inline]

                                                  Given a collection of maps into the summands, we obtain a map into the biproduct.

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                                                    @[reducible, inline]

                                                    Given a collection of maps out of the summands, we obtain a map out of the biproduct.

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                                                      @[reducible, inline]

                                                      Given a collection of maps between corresponding summands of a pair of biproducts indexed by the same type, we obtain a map between the biproducts.

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                                                        @[reducible, inline]

                                                        An alternative to biproduct.map constructed via colimits. This construction only exists in order to show it is equal to biproduct.map.

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                                                        • One or more equations did not get rendered due to their size.
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                                                          The canonical isomorphism between the chosen biproduct and the chosen coproduct.

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                                                            Given a collection of isomorphisms between corresponding summands of a pair of biproducts indexed by the same type, we obtain an isomorphism between the biproducts.

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                                                            • One or more equations did not get rendered due to their size.
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                                                              Two biproducts which differ by an equivalence in the indexing type, and up to isomorphism in the factors, are isomorphic.

                                                              Unfortunately there are two natural ways to define each direction of this isomorphism (because it is true for both products and coproducts separately). We give the alternative definitions as lemmas below.

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                                                                instance CategoryTheory.Limits.instHasBiproductSigmaFstSndOfBiproduct {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun (i : ι) => g i] :
                                                                CategoryTheory.Limits.HasBiproduct fun (p : (i : ι) × f i) => g p.fst p.snd
                                                                Equations
                                                                • =
                                                                @[simp]
                                                                theorem CategoryTheory.Limits.biproductBiproductIso_inv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun (i : ι) => g i] :
                                                                (CategoryTheory.Limits.biproductBiproductIso f g).inv = CategoryTheory.Limits.biproduct.lift fun (i : ι) => CategoryTheory.Limits.biproduct.lift fun (x : f i) => CategoryTheory.Limits.biproduct.π (fun (p : (i : ι) × f i) => g p.fst p.snd) i, x
                                                                def CategoryTheory.Limits.biproductBiproductIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun (i : ι) => g i] :
                                                                ( fun (i : ι) => g i) fun (p : (i : ι) × f i) => g p.fst p.snd

                                                                An iterated biproduct is a biproduct over a sigma type.

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                                                                  The canonical morphism from the biproduct over a restricted index type to the biproduct of the full index type.

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                                                                    The kernel of biproduct.π f i is the inclusion from the biproduct which omits i from the index set J into the biproduct over J.

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                                                                      The kernel of biproduct.π f i is Subtype.restrict {i}ᶜ f.

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                                                                        The cokernel of biproduct.ι f i is the projection from the biproduct over the index set J onto the biproduct omitting i.

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                                                                          The cokernel of biproduct.ι f i is Subtype.restrict {i}ᶜ f.

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                                                                            The limit cone exhibiting Subtype.restrict pᶜ f as the kernel of biproduct.toSubtype f p

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                                                                              The colimit cocone exhibiting Subtype.restrict pᶜ f as the cokernel of biproduct.fromSubtype f p

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                                                                                def CategoryTheory.Limits.biproduct.matrix {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) :
                                                                                f g

                                                                                Convert a (dependently typed) matrix to a morphism of biproducts.

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                                                                                  Extract the matrix components from a morphism of biproducts.

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                                                                                    @[simp]
                                                                                    theorem CategoryTheory.Limits.biproduct.matrixEquiv_symm_apply {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) :
                                                                                    CategoryTheory.Limits.biproduct.matrixEquiv.symm m = CategoryTheory.Limits.biproduct.matrix m
                                                                                    @[simp]
                                                                                    theorem CategoryTheory.Limits.biproduct.matrixEquiv_apply {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : JC} {g : KC} (m : f g) (j : J) (k : K) :
                                                                                    CategoryTheory.Limits.biproduct.matrixEquiv m j k = CategoryTheory.Limits.biproduct.components m j k

                                                                                    Morphisms between direct sums are matrices.

                                                                                    Equations
                                                                                    • CategoryTheory.Limits.biproduct.matrixEquiv = { toFun := CategoryTheory.Limits.biproduct.components, invFun := CategoryTheory.Limits.biproduct.matrix, left_inv := , right_inv := }
                                                                                    Instances For

                                                                                      Biproducts are unique up to isomorphism. This already follows because bilimits are limits, but in the case of biproducts we can give an isomorphism with particularly nice definitional properties, namely that biproduct.lift b.π and biproduct.desc b.ι are inverses of each other.

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                                                                                        The limit bicone for the biproduct over an index type with exactly one term.

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                                                                                          A biproduct over an index type with exactly one term is just the object over that term.

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                                                                                            A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • pt : C

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • fst : self.pt P

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • snd : self.pt Q

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • inl : P self.pt

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • inr : Q self.pt

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • inl_snd : CategoryTheory.CategoryStruct.comp self.inl self.snd = 0

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • inr_fst : CategoryTheory.CategoryStruct.comp self.inr self.fst = 0

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            • A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                            Instances For
                                                                                              @[simp]

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                              @[simp]

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                              @[simp]

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                              @[simp]

                                                                                              A binary bicone for a pair of objects P Q : C consists of the cone point X, maps from X to both P and Q, and maps from both P and Q to X, so that inlfst = 𝟙 P, inlsnd = 0, inrfst = 0, and inrsnd = 𝟙 Q

                                                                                              A binary bicone morphism between two binary bicones for the same diagram is a morphism of the binary bicone points which commutes with the cone and cocone legs.

                                                                                              Instances For
                                                                                                @[simp]

                                                                                                The triangle consisting of the two natural transformations and hom commutes

                                                                                                @[simp]

                                                                                                The triangle consisting of the two natural transformations and hom commutes

                                                                                                @[simp]

                                                                                                The triangle consisting of the two natural transformations and hom commutes

                                                                                                @[simp]

                                                                                                The triangle consisting of the two natural transformations and hom commutes

                                                                                                The category of binary bicones on a given diagram.

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                                                                                                To give an isomorphism between cocones, it suffices to give an isomorphism between their vertices which commutes with the cocone maps.

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                                                                                                  A functor F : C ⥤ D sends binary bicones for P and Q to binary bicones for G.obj P and G.obj Q functorially.

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