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 https://github.com/leanprover-community/mathlib3/pull/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} [Category.{uC', uC} C] [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) : CategoryStruct.comp (self.ι j) (self.π j') = if h : j = j' then 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
    @[simp]
    theorem CategoryTheory.Limits.bicone_ι_π_self {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : J) :
    @[simp]
    theorem CategoryTheory.Limits.bicone_ι_π_self_assoc {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : J) {Z : C} (h : F j Z) :
    @[simp]
    theorem CategoryTheory.Limits.bicone_ι_π_ne {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) {j j' : J} (h : j j') :
    CategoryStruct.comp (B.ι j) (B.π j') = 0
    @[simp]
    theorem CategoryTheory.Limits.bicone_ι_π_ne_assoc {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) {j j' : J} (h : j j') {Z : C} (h✝ : F j' Z) :
    structure CategoryTheory.Limits.BiconeMorphism {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (A B : Bicone F) :
    Type uC'

    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.

    • hom : A.pt B.pt

      A morphism between the two vertex objects of the bicones

    • (j : J) : CategoryStruct.comp self.hom (B.π j) = A.π j

      The triangle consisting of the two natural transformations and hom commutes

    • (j : J) : CategoryStruct.comp (A.ι j) self.hom = B.ι j

      The triangle consisting of the two natural transformations and hom commutes

    Instances For
      @[simp]
      theorem CategoryTheory.Limits.BiconeMorphism.wι_assoc {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {A B : Bicone F} (self : BiconeMorphism A B) (j : J) {Z : C} (h : B.pt Z) :
      @[simp]
      theorem CategoryTheory.Limits.BiconeMorphism.wπ_assoc {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {A B : Bicone F} (self : BiconeMorphism A B) (j : J) {Z : C} (h : F j Z) :
      @[simp]
      theorem CategoryTheory.Limits.Bicone.category_comp_hom {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {X✝ Y✝ Z✝ : Bicone F} (f : X✝ Y✝) (g : Y✝ Z✝) :
      theorem CategoryTheory.Limits.BiconeMorphism.ext {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {c c' : Bicone F} (f g : c c') (w : f.hom = g.hom) :
      f = g
      def CategoryTheory.Limits.Bicones.ext {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {c c' : Bicone F} (φ : c.pt c'.pt) (wι : ∀ (j : J), CategoryStruct.comp (c.ι j) φ.hom = c'.ι j := by aesop_cat) (wπ : ∀ (j : J), CategoryStruct.comp φ.hom (c'.π j) = c.π j := by aesop_cat) :
      c c'

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

      Equations
      Instances For
        @[simp]
        theorem CategoryTheory.Limits.Bicones.ext_hom_hom {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {c c' : Bicone F} (φ : c.pt c'.pt) (wι : ∀ (j : J), CategoryStruct.comp (c.ι j) φ.hom = c'.ι j := by aesop_cat) (wπ : ∀ (j : J), CategoryStruct.comp φ.hom (c'.π j) = c.π j := by aesop_cat) :
        (ext φ ).hom.hom = φ.hom
        @[simp]
        theorem CategoryTheory.Limits.Bicones.ext_inv_hom {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} {c c' : Bicone F} (φ : c.pt c'.pt) (wι : ∀ (j : J), CategoryStruct.comp (c.ι j) φ.hom = c'.ι j := by aesop_cat) (wπ : ∀ (j : J), CategoryStruct.comp φ.hom (c'.π j) = c.π j := by aesop_cat) :
        (ext φ ).inv.hom = φ.inv

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

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          @[simp]
          theorem CategoryTheory.Limits.Bicones.functoriality_map_hom {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {D : Type uD} [Category.{uD', uD} D] [HasZeroMorphisms D] (F : JC) (G : Functor C D) [G.PreservesZeroMorphisms] {X✝ Y✝ : Bicone F} (f : X✝ Y✝) :
          ((functoriality F G).map f).hom = G.map f.hom
          @[simp]
          theorem CategoryTheory.Limits.Bicones.functoriality_obj_ι {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {D : Type uD} [Category.{uD', uD} D] [HasZeroMorphisms D] (F : JC) (G : Functor C D) [G.PreservesZeroMorphisms] (A : Bicone F) (j : J) :
          ((functoriality F G).obj A).ι j = G.map (A.ι j)
          @[simp]
          theorem CategoryTheory.Limits.Bicones.functoriality_obj_π {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {D : Type uD} [Category.{uD', uD} D] [HasZeroMorphisms D] (F : JC) (G : Functor C D) [G.PreservesZeroMorphisms] (A : Bicone F) (j : J) :
          ((functoriality F G).obj A).π j = G.map (A.π j)

          Extract the cone from a bicone.

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

            A shorthand for toConeFunctor.obj

            Equations
            Instances For
              @[simp]
              theorem CategoryTheory.Limits.Bicone.toCone_pt {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) :
              @[simp]
              theorem CategoryTheory.Limits.Bicone.toCone_π_app {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : Discrete J) :
              B.toCone.π.app j = B.π j.as
              theorem CategoryTheory.Limits.Bicone.toCone_π_app_mk {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : J) :
              B.toCone.π.app { as := j } = B.π j
              @[simp]
              theorem CategoryTheory.Limits.Bicone.toCone_proj {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : J) :

              Extract the cocone from a bicone.

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

                A shorthand for toCoconeFunctor.obj

                Equations
                Instances For
                  @[simp]
                  theorem CategoryTheory.Limits.Bicone.toCocone_pt {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) :
                  @[simp]
                  theorem CategoryTheory.Limits.Bicone.toCocone_ι_app {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : Discrete J) :
                  B.toCocone.ι.app j = B.ι j.as
                  @[simp]
                  theorem CategoryTheory.Limits.Bicone.toCocone_inj {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : J) :
                  theorem CategoryTheory.Limits.Bicone.toCocone_ι_app_mk {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) (j : J) :
                  B.toCocone.ι.app { as := j } = B.ι j

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

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For
                    @[simp]
                    @[simp]
                    theorem CategoryTheory.Limits.Bicone.ofLimitCone_π {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {f : JC} {t : Cone (Discrete.functor f)} (ht : IsLimit t) (j : J) :
                    (ofLimitCone ht).π j = t.π.app { as := j }
                    @[simp]
                    theorem CategoryTheory.Limits.Bicone.ofLimitCone_ι {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {f : JC} {t : Cone (Discrete.functor f)} (ht : IsLimit t) (j : J) :
                    (ofLimitCone ht).ι j = ht.lift (Fan.mk (f j) fun (j' : J) => if h : j = j' then eqToHom else 0)
                    theorem CategoryTheory.Limits.Bicone.ι_of_isLimit {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {f : JC} {t : Bicone f} (ht : IsLimit t.toCone) (j : J) :
                    t.ι j = ht.lift (Fan.mk (f j) fun (j' : J) => if h : j = j' then eqToHom else 0)

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

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      @[simp]
                      theorem CategoryTheory.Limits.Bicone.ofColimitCocone_ι {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {f : JC} {t : Cocone (Discrete.functor f)} (ht : IsColimit t) (j : J) :
                      (ofColimitCocone ht).ι j = t.ι.app { as := j }
                      @[simp]
                      theorem CategoryTheory.Limits.Bicone.ofColimitCocone_π {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {f : JC} {t : Cocone (Discrete.functor f)} (ht : IsColimit t) (j : J) :
                      (ofColimitCocone ht).π j = ht.desc (Cofan.mk (f j) fun (j' : J) => if h : j' = j then eqToHom else 0)
                      theorem CategoryTheory.Limits.Bicone.π_of_isColimit {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {f : JC} {t : Bicone f} (ht : IsColimit t.toCocone) (j : J) :
                      t.π j = ht.desc (Cofan.mk (f j) fun (j' : J) => if h : j' = j then eqToHom else 0)
                      structure CategoryTheory.Limits.Bicone.IsBilimit {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {F : JC} (B : Bicone F) :
                      Type (max (max uC uC') w)

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

                      • isLimit : IsLimit B.toCone

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

                      • isColimit : IsColimit B.toCocone

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

                      Instances For
                        theorem CategoryTheory.Limits.Bicone.IsBilimit.ext {J : Type w} {C : Type uC} {inst✝ : Category.{uC', uC} C} {inst✝¹ : HasZeroMorphisms C} {F : JC} {B : Bicone F} {x y : B.IsBilimit} (isLimit : x.isLimit = y.isLimit) (isColimit : x.isColimit = y.isColimit) :
                        x = y
                        def CategoryTheory.Limits.Bicone.whisker {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {K : Type w'} {f : JC} (c : Bicone f) (g : K J) :
                        Bicone (f g)

                        Whisker a bicone with an equivalence between the indexing types.

                        Equations
                        • c.whisker g = { pt := c.pt, π := fun (k : K) => c.π (g k), ι := fun (k : K) => c.ι (g k), ι_π := }
                        Instances For
                          @[simp]
                          theorem CategoryTheory.Limits.Bicone.whisker_ι {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {K : Type w'} {f : JC} (c : Bicone f) (g : K J) (k : K) :
                          (c.whisker g).ι k = c.ι (g k)
                          @[simp]
                          theorem CategoryTheory.Limits.Bicone.whisker_π {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {K : Type w'} {f : JC} (c : Bicone f) (g : K J) (k : K) :
                          (c.whisker g).π k = c.π (g k)
                          @[simp]
                          theorem CategoryTheory.Limits.Bicone.whisker_pt {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] {K : Type w'} {f : JC} (c : Bicone f) (g : K J) :
                          (c.whisker g).pt = c.pt

                          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.

                          Equations
                          Instances For

                            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.

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

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

                              Equations
                              • One or more equations did not get rendered due to their size.
                              Instances For
                                structure CategoryTheory.Limits.LimitBicone {J : Type w} {C : Type uC} [Category.{uC', uC} C] [HasZeroMorphisms C] (F : JC) :
                                Type (max (max uC uC') w)

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

                                • bicone : Bicone F

                                  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.

                                  • mk' :: (
                                    • exists_biproduct : Nonempty (LimitBicone F)

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

                                  • )
                                  Instances

                                    Use the axiom of choice to extract explicit BiproductData F from HasBiproduct F.

                                    Equations
                                    Instances For

                                      A bicone for F which is both a limit cone and a colimit cocone.

                                      Equations
                                      Instances For

                                        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.

                                            Equations
                                            • One or more equations did not get rendered due to their size.
                                            Instances For
                                              @[reducible, inline]
                                              abbrev CategoryTheory.Limits.biproduct {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] :
                                              C

                                              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.)

                                              Equations
                                              Instances For

                                                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.)

                                                Equations
                                                Instances For
                                                  @[reducible, inline]
                                                  abbrev CategoryTheory.Limits.biproduct.π {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (b : J) :
                                                  f f b

                                                  The projection onto a summand of a biproduct.

                                                  Equations
                                                  Instances For
                                                    @[simp]
                                                    theorem CategoryTheory.Limits.biproduct.bicone_π {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (b : J) :
                                                    (bicone f).π b = π f b
                                                    @[reducible, inline]
                                                    abbrev CategoryTheory.Limits.biproduct.ι {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (b : J) :
                                                    f b f

                                                    The inclusion into a summand of a biproduct.

                                                    Equations
                                                    Instances For
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.bicone_ι {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (b : J) :
                                                      (bicone f).ι b = ι f b
                                                      theorem CategoryTheory.Limits.biproduct.ι_π {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [DecidableEq J] (f : JC) [HasBiproduct f] (j j' : J) :
                                                      CategoryStruct.comp (ι f j) (π f j') = if h : j = j' then eqToHom else 0

                                                      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.

                                                      theorem CategoryTheory.Limits.biproduct.ι_π_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [DecidableEq J] (f : JC) [HasBiproduct f] (j j' : J) {Z : C} (h : f j' Z) :
                                                      CategoryStruct.comp (ι f j) (CategoryStruct.comp (π f j') h) = CategoryStruct.comp (if h : j = j' then eqToHom else 0) h
                                                      theorem CategoryTheory.Limits.biproduct.ι_π_self_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (j : J) {Z : C} (h : f j Z) :
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.ι_π_ne {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] {j j' : J} (h : j j') :
                                                      CategoryStruct.comp (ι f j) (π f j') = 0
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.ι_π_ne_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] {j j' : J} (h : j j') {Z : C} (h✝ : f j' Z) :
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.eqToHom_comp_ι {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] {j j' : J} (w : j = j') :
                                                      CategoryStruct.comp (eqToHom ) (ι f j') = ι f j
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.eqToHom_comp_ι_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] {j j' : J} (w : j = j') {Z : C} (h : f Z) :
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.π_comp_eqToHom {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] {j j' : J} (w : j = j') :
                                                      CategoryStruct.comp (π f j) (eqToHom ) = π f j'
                                                      @[simp]
                                                      theorem CategoryTheory.Limits.biproduct.π_comp_eqToHom_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] {j j' : J} (w : j = j') {Z : C} (h : f j' Z) :
                                                      @[reducible, inline]
                                                      abbrev CategoryTheory.Limits.biproduct.lift {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {P : C} (p : (b : J) → P f b) :
                                                      P f

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

                                                      Equations
                                                      Instances For
                                                        @[reducible, inline]
                                                        abbrev CategoryTheory.Limits.biproduct.desc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {P : C} (p : (b : J) → f b P) :
                                                        f P

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

                                                        Equations
                                                        Instances For
                                                          @[simp]
                                                          theorem CategoryTheory.Limits.biproduct.lift_π {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {P : C} (p : (b : J) → P f b) (j : J) :
                                                          @[simp]
                                                          theorem CategoryTheory.Limits.biproduct.lift_π_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {P : C} (p : (b : J) → P f b) (j : J) {Z : C} (h : f j Z) :
                                                          @[simp]
                                                          theorem CategoryTheory.Limits.biproduct.ι_desc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {P : C} (p : (b : J) → f b P) (j : J) :
                                                          @[simp]
                                                          theorem CategoryTheory.Limits.biproduct.ι_desc_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {P : C} (p : (b : J) → f b P) (j : J) {Z : C} (h : P Z) :
                                                          @[reducible, inline]
                                                          abbrev CategoryTheory.Limits.biproduct.map {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (b : J) → f b g b) :
                                                          f g

                                                          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.

                                                          Equations
                                                          • One or more equations did not get rendered due to their size.
                                                          Instances For
                                                            @[reducible, inline]
                                                            abbrev CategoryTheory.Limits.biproduct.map' {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (b : J) → f b g b) :
                                                            f g

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

                                                            Equations
                                                            • One or more equations did not get rendered due to their size.
                                                            Instances For
                                                              theorem CategoryTheory.Limits.biproduct.hom_ext {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {Z : C} (g h : Z f) (w : ∀ (j : J), CategoryStruct.comp g (π f j) = CategoryStruct.comp h (π f j)) :
                                                              g = h
                                                              theorem CategoryTheory.Limits.biproduct.hom_ext' {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} [HasBiproduct f] {Z : C} (g h : f Z) (w : ∀ (j : J), CategoryStruct.comp (ι f j) g = CategoryStruct.comp (ι f j) h) :
                                                              g = h

                                                              The canonical isomorphism between the chosen biproduct and the chosen coproduct.

                                                              Equations
                                                              • One or more equations did not get rendered due to their size.
                                                              Instances For

                                                                If a category has biproducts of a shape J, its colim and lim functor on diagrams over J are isomorphic.

                                                                Equations
                                                                • One or more equations did not get rendered due to their size.
                                                                Instances For
                                                                  theorem CategoryTheory.Limits.biproduct.map_eq_map' {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (b : J) → f b g b) :
                                                                  map p = map' p
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.map_π {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) (j : J) :
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.map_π_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) (j : J) {Z : C} (h : g j Z) :
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.ι_map {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) (j : J) :
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.ι_map_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) (j : J) {Z : C} (h : g Z) :
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.map_desc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) {P : C} (k : (j : J) → g j P) :
                                                                  CategoryStruct.comp (map p) (desc k) = desc fun (j : J) => CategoryStruct.comp (p j) (k j)
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.map_desc_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) {P : C} (k : (j : J) → g j P) {Z : C} (h : P Z) :
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.lift_map {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] {P : C} (k : (j : J) → P f j) (p : (j : J) → f j g j) :
                                                                  CategoryStruct.comp (lift k) (map p) = lift fun (j : J) => CategoryStruct.comp (k j) (p j)
                                                                  @[simp]
                                                                  theorem CategoryTheory.Limits.biproduct.lift_map_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] {P : C} (k : (j : J) → P f j) (p : (j : J) → f j g j) {Z : C} (h : g Z) :
                                                                  def CategoryTheory.Limits.biproduct.mapIso {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (b : J) → f b g b) :
                                                                  f g

                                                                  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.

                                                                  Equations
                                                                  • One or more equations did not get rendered due to their size.
                                                                  Instances For
                                                                    @[simp]
                                                                    theorem CategoryTheory.Limits.biproduct.mapIso_hom {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (b : J) → f b g b) :
                                                                    (mapIso p).hom = map fun (b : J) => (p b).hom
                                                                    @[simp]
                                                                    theorem CategoryTheory.Limits.biproduct.mapIso_inv {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (b : J) → f b g b) :
                                                                    (mapIso p).inv = map fun (b : J) => (p b).inv
                                                                    instance CategoryTheory.Limits.biproduct.map_epi {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) [∀ (j : J), Epi (p j)] :
                                                                    Epi (map p)
                                                                    instance CategoryTheory.Limits.Pi.map_epi {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) [∀ (j : J), Epi (p j)] :
                                                                    Epi (map p)
                                                                    instance CategoryTheory.Limits.biproduct.map_mono {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) [∀ (j : J), Mono (p j)] :
                                                                    Mono (map p)
                                                                    instance CategoryTheory.Limits.Sigma.map_mono {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f g : JC} [HasBiproduct f] [HasBiproduct g] (p : (j : J) → f j g j) [∀ (j : J), Mono (p j)] :
                                                                    Mono (map p)
                                                                    def CategoryTheory.Limits.biproduct.whiskerEquiv {J : Type w} {K : Type u_1} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) [HasBiproduct f] [HasBiproduct g] :
                                                                    f g

                                                                    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.

                                                                    Equations
                                                                    • One or more equations did not get rendered due to their size.
                                                                    Instances For
                                                                      @[simp]
                                                                      theorem CategoryTheory.Limits.biproduct.whiskerEquiv_inv {J : Type w} {K : Type u_1} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) [HasBiproduct f] [HasBiproduct g] :
                                                                      (whiskerEquiv e w).inv = desc fun (k : K) => CategoryStruct.comp (eqToHom ) (CategoryStruct.comp (w (e.symm k)).hom (ι f (e.symm k)))
                                                                      @[simp]
                                                                      theorem CategoryTheory.Limits.biproduct.whiskerEquiv_hom {J : Type w} {K : Type u_1} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) [HasBiproduct f] [HasBiproduct g] :
                                                                      (whiskerEquiv e w).hom = desc fun (j : J) => CategoryStruct.comp (w j).inv (ι g (e j))
                                                                      theorem CategoryTheory.Limits.biproduct.whiskerEquiv_hom_eq_lift {J : Type w} {K : Type u_1} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) [HasBiproduct f] [HasBiproduct g] :
                                                                      (whiskerEquiv e w).hom = lift fun (k : K) => CategoryStruct.comp (π f (e.symm k)) (CategoryStruct.comp (w (e.symm k)).inv (eqToHom ))
                                                                      theorem CategoryTheory.Limits.biproduct.whiskerEquiv_inv_eq_lift {J : Type w} {K : Type u_1} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) [HasBiproduct f] [HasBiproduct g] :
                                                                      (whiskerEquiv e w).inv = lift fun (j : J) => CategoryStruct.comp (π g (e j)) (w j).hom
                                                                      instance CategoryTheory.Limits.instHasBiproductSigmaFstSndOfBiproduct {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), HasBiproduct (g i)] [HasBiproduct fun (i : ι) => g i] :
                                                                      HasBiproduct fun (p : (i : ι) × f i) => g p.fst p.snd
                                                                      def CategoryTheory.Limits.biproductBiproductIso {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), HasBiproduct (g i)] [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.

                                                                      Equations
                                                                      • One or more equations did not get rendered due to their size.
                                                                      Instances For
                                                                        @[simp]
                                                                        theorem CategoryTheory.Limits.biproductBiproductIso_inv {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), HasBiproduct (g i)] [HasBiproduct fun (i : ι) => g i] :
                                                                        (biproductBiproductIso f g).inv = biproduct.lift fun (i : ι) => biproduct.lift fun (x : f i) => biproduct.π (fun (p : (i : ι) × f i) => g p.fst p.snd) i, x
                                                                        @[simp]
                                                                        theorem CategoryTheory.Limits.biproductBiproductIso_hom {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {ι : Type u_3} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), HasBiproduct (g i)] [HasBiproduct fun (i : ι) => g i] :
                                                                        (biproductBiproductIso f g).hom = biproduct.lift fun (x : (i : ι) × f i) => match x with | i, x => CategoryStruct.comp (biproduct.π (fun (i : ι) => g i) i) (biproduct.π (g i) x)

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

                                                                        Equations
                                                                        Instances For

                                                                          The canonical morphism from a biproduct to the biproduct over a restriction of its index type.

                                                                          Equations
                                                                          Instances For
                                                                            @[simp]
                                                                            theorem CategoryTheory.Limits.biproduct.fromSubtype_π {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (p : JProp) [HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) :
                                                                            CategoryStruct.comp (fromSubtype f p) (π f j) = if h : p j then π (Subtype.restrict p f) j, h else 0
                                                                            @[simp]
                                                                            theorem CategoryTheory.Limits.biproduct.fromSubtype_π_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (p : JProp) [HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) {Z : C} (h : f j Z) :
                                                                            CategoryStruct.comp (fromSubtype f p) (CategoryStruct.comp (π f j) h) = CategoryStruct.comp (if h : p j then π (Subtype.restrict p f) j, h else 0) h
                                                                            theorem CategoryTheory.Limits.biproduct.fromSubtype_eq_lift {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (p : JProp) [HasBiproduct (Subtype.restrict p f)] [DecidablePred p] :
                                                                            fromSubtype f p = lift fun (j : J) => if h : p j then π (Subtype.restrict p f) j, h else 0
                                                                            @[simp]
                                                                            theorem CategoryTheory.Limits.biproduct.ι_toSubtype {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (p : JProp) [HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) :
                                                                            CategoryStruct.comp (ι f j) (toSubtype f p) = if h : p j then ι (Subtype.restrict p f) j, h else 0
                                                                            @[simp]
                                                                            theorem CategoryTheory.Limits.biproduct.ι_toSubtype_assoc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (p : JProp) [HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) {Z : C} (h : Subtype.restrict p f Z) :
                                                                            CategoryStruct.comp (ι f j) (CategoryStruct.comp (toSubtype f p) h) = CategoryStruct.comp (if h : p j then ι (Subtype.restrict p f) j, h else 0) h
                                                                            theorem CategoryTheory.Limits.biproduct.toSubtype_eq_desc {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (p : JProp) [HasBiproduct (Subtype.restrict p f)] [DecidablePred p] :
                                                                            toSubtype f p = desc fun (j : J) => if h : p j then ι (Subtype.restrict p f) j, h else 0
                                                                            def CategoryTheory.Limits.biproduct.isLimitFromSubtype {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) (i : J) [HasBiproduct f] [HasBiproduct (Subtype.restrict (fun (j : J) => j i) f)] :
                                                                            IsLimit (KernelFork.ofι (fromSubtype f fun (j : J) => j i) )

                                                                            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.

                                                                            Equations
                                                                            • One or more equations did not get rendered due to their size.
                                                                            Instances For
                                                                              instance CategoryTheory.Limits.instHasKernelπ {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) (i : J) [HasBiproduct f] [HasBiproduct (Subtype.restrict (fun (j : J) => j i) f)] :
                                                                              def CategoryTheory.Limits.kernelBiproductπIso {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) (i : J) [HasBiproduct f] [HasBiproduct (Subtype.restrict (fun (j : J) => j i) f)] :
                                                                              kernel (biproduct.π f i) Subtype.restrict (fun (j : J) => j i) f

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

                                                                              Equations
                                                                              • One or more equations did not get rendered due to their size.
                                                                              Instances For
                                                                                @[simp]
                                                                                def CategoryTheory.Limits.biproduct.isColimitToSubtype {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) (i : J) [HasBiproduct f] [HasBiproduct (Subtype.restrict (fun (j : J) => j i) f)] :
                                                                                IsColimit (CokernelCofork.ofπ (toSubtype f fun (j : J) => j i) )

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

                                                                                Equations
                                                                                • One or more equations did not get rendered due to their size.
                                                                                Instances For
                                                                                  instance CategoryTheory.Limits.instHasCokernelι {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) (i : J) [HasBiproduct f] [HasBiproduct (Subtype.restrict (fun (j : J) => j i) f)] :
                                                                                  def CategoryTheory.Limits.cokernelBiproductιIso {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) (i : J) [HasBiproduct f] [HasBiproduct (Subtype.restrict (fun (j : J) => j i) f)] :
                                                                                  cokernel (biproduct.ι f i) Subtype.restrict (fun (j : J) => j i) f

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

                                                                                  Equations
                                                                                  • One or more equations did not get rendered due to their size.
                                                                                  Instances For

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

                                                                                    Equations
                                                                                    • One or more equations did not get rendered due to their size.
                                                                                    Instances For

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

                                                                                      Equations
                                                                                      • One or more equations did not get rendered due to their size.
                                                                                      Instances For
                                                                                        def CategoryTheory.Limits.biproduct.matrix {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [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.

                                                                                        Equations
                                                                                        Instances For
                                                                                          @[simp]
                                                                                          theorem CategoryTheory.Limits.biproduct.matrix_π {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) (k : K) :
                                                                                          CategoryStruct.comp (matrix m) (π g k) = desc fun (j : J) => m j k
                                                                                          @[simp]
                                                                                          theorem CategoryTheory.Limits.biproduct.matrix_π_assoc {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) (k : K) {Z : C} (h : g k Z) :
                                                                                          @[simp]
                                                                                          theorem CategoryTheory.Limits.biproduct.ι_matrix {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) (j : J) :
                                                                                          CategoryStruct.comp (ι f j) (matrix m) = lift fun (k : K) => m j k
                                                                                          @[simp]
                                                                                          theorem CategoryTheory.Limits.biproduct.ι_matrix_assoc {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) (j : J) {Z : C} (h : g Z) :
                                                                                          def CategoryTheory.Limits.biproduct.components {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : f g) (j : J) (k : K) :
                                                                                          f j g k

                                                                                          Extract the matrix components from a morphism of biproducts.

                                                                                          Equations
                                                                                          • One or more equations did not get rendered due to their size.
                                                                                          Instances For
                                                                                            @[simp]
                                                                                            theorem CategoryTheory.Limits.biproduct.matrix_components {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) (j : J) (k : K) :
                                                                                            components (matrix m) j k = m j k
                                                                                            @[simp]
                                                                                            theorem CategoryTheory.Limits.biproduct.components_matrix {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : f g) :
                                                                                            (matrix fun (j : J) (k : K) => components m j k) = m
                                                                                            def CategoryTheory.Limits.biproduct.matrixEquiv {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} :
                                                                                            ( f g) ((j : J) → (k : K) → f j g k)

                                                                                            Morphisms between direct sums are matrices.

                                                                                            Equations
                                                                                            Instances For
                                                                                              @[simp]
                                                                                              theorem CategoryTheory.Limits.biproduct.matrixEquiv_symm_apply {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : (j : J) → (k : K) → f j g k) :
                                                                                              @[simp]
                                                                                              theorem CategoryTheory.Limits.biproduct.matrixEquiv_apply {J : Type} [Finite J] {K : Type} [Finite K] {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] [HasFiniteBiproducts C] {f : JC} {g : KC} (m : f g) (j : J) (k : K) :
                                                                                              instance CategoryTheory.Limits.biproduct.ι_mono {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (b : J) :
                                                                                              instance CategoryTheory.Limits.biproduct.π_epi {J : Type w} {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (f : JC) [HasBiproduct f] (b : J) :

                                                                                              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.

                                                                                              Equations
                                                                                              Instances For
                                                                                                @[simp]
                                                                                                @[simp]
                                                                                                @[instance 100]

                                                                                                A category with finite biproducts has a zero object.

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

                                                                                                Equations
                                                                                                • One or more equations did not get rendered due to their size.
                                                                                                Instances For
                                                                                                  @[instance 100]

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

                                                                                                  Equations
                                                                                                  Instances For
                                                                                                    structure CategoryTheory.Limits.BinaryBicone {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] (P Q : C) :
                                                                                                    Type (max u v)

                                                                                                    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 : 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 : 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]
                                                                                                      @[simp]

                                                                                                      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]
                                                                                                        theorem CategoryTheory.Limits.BinaryBicone.category_comp_hom {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {P Q : C} {X✝ Y✝ Z✝ : BinaryBicone P Q} (f : X✝ Y✝) (g : Y✝ Z✝) :
                                                                                                        theorem CategoryTheory.Limits.BinaryBiconeMorphism.ext {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {P Q : C} {c c' : BinaryBicone P Q} (f g : c c') (w : f.hom = g.hom) :
                                                                                                        f = g
                                                                                                        def CategoryTheory.Limits.BinaryBicones.ext {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {P Q : C} {c c' : BinaryBicone P Q} (φ : c.pt c'.pt) (winl : CategoryStruct.comp c.inl φ.hom = c'.inl := by aesop_cat) (winr : CategoryStruct.comp c.inr φ.hom = c'.inr := by aesop_cat) (wfst : CategoryStruct.comp φ.hom c'.fst = c.fst := by aesop_cat) (wsnd : CategoryStruct.comp φ.hom c'.snd = c.snd := by aesop_cat) :
                                                                                                        c c'

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

                                                                                                        Equations
                                                                                                        • One or more equations did not get rendered due to their size.
                                                                                                        Instances For
                                                                                                          @[simp]
                                                                                                          theorem CategoryTheory.Limits.BinaryBicones.ext_inv_hom {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {P Q : C} {c c' : BinaryBicone P Q} (φ : c.pt c'.pt) (winl : CategoryStruct.comp c.inl φ.hom = c'.inl := by aesop_cat) (winr : CategoryStruct.comp c.inr φ.hom = c'.inr := by aesop_cat) (wfst : CategoryStruct.comp φ.hom c'.fst = c.fst := by aesop_cat) (wsnd : CategoryStruct.comp φ.hom c'.snd = c.snd := by aesop_cat) :
                                                                                                          (ext φ winl winr wfst wsnd).inv.hom = φ.inv
                                                                                                          @[simp]
                                                                                                          theorem CategoryTheory.Limits.BinaryBicones.ext_hom_hom {C : Type u} [Category.{v, u} C] [HasZeroMorphisms C] {P Q : C} {c c' : BinaryBicone P Q} (φ : c.pt c'.pt) (winl : CategoryStruct.comp c.inl φ.hom = c'.inl := by aesop_cat) (winr : CategoryStruct.comp c.inr φ.hom = c'.inr := by aesop_cat) (wfst : CategoryStruct.comp φ.hom c'.fst = c.fst := by aesop_cat) (wsnd : CategoryStruct.comp φ.hom c'.snd = c.snd := by aesop_cat) :
                                                                                                          (ext φ winl winr wfst wsnd).hom.hom = φ.hom

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

                                                                                                          Equations
                                                                                                          • One or more equations did not get rendered due to their size.
                                                                                                          Instances For
                                                                                                            @[simp]

                                                                                                            Extract the cone from a binary bicone.

                                                                                                            Equations
                                                                                                            Instances For

                                                                                                              Extract the cocone from a binary bicone.

                                                                                                              Equations
                                                                                                              Instances For

                                                                                                                Convert a BinaryBicone into a Bicone over a pair.

                                                                                                                Equations
                                                                                                                • One or more equations did not get rendered due to their size.
                                                                                                                Instances For
                                                                                                                  @[simp]