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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 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

Prior to #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 : J → C) : Type (max (max uC uC') w)

• 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) → s.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 ⟶ s.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 (CategoryTheory.Limits.Bicone.ι s j) (CategoryTheory.Limits.Bicone.π s j') = if h : j = j' then CategoryTheory.eqToHom (_ : F j = F j') 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 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.

@[simp]

theorem CategoryTheory.Limits.bicone_ι_π_self_assoc {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : J) {Z : C} (h : F j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι B j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.π B j) h) = h

@[simp]

theorem CategoryTheory.Limits.bicone_ι_π_self {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι B j) (CategoryTheory.Limits.Bicone.π B j) = CategoryTheory.CategoryStruct.id (F j)

@[simp]

theorem CategoryTheory.Limits.bicone_ι_π_ne_assoc {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) {j : J} {j' : J} (h : j ≠ j') {Z : C} (h : F j' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι B j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.π B j') h) = CategoryTheory.CategoryStruct.comp 0 h

@[simp]

theorem CategoryTheory.Limits.bicone_ι_π_ne {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) {j : J} {j' : J} (h : j ≠ j') : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι B j) (CategoryTheory.Limits.Bicone.π B j') = 0

structure CategoryTheory.Limits.BiconeMorphism {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (A : CategoryTheory.Limits.Bicone F) (B : CategoryTheory.Limits.Bicone F) : Type uC'

• hom : A.pt ⟶ B.pt

  A morphism between the two vertex objects of the bicones

• wπ : ∀ (j : J), CategoryTheory.CategoryStruct.comp s.hom (CategoryTheory.Limits.Bicone.π B j) = CategoryTheory.Limits.Bicone.π A j

  The triangle consisting of the two natural transformations and hom commutes

• wι : ∀ (j : J), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι A j) s.hom = CategoryTheory.Limits.Bicone.ι B j

  The triangle consisting of the two natural transformations and hom commutes

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.

@[simp]

theorem CategoryTheory.Limits.BiconeMorphism.wι_assoc {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} {A : CategoryTheory.Limits.Bicone F} {B : CategoryTheory.Limits.Bicone F} (self : CategoryTheory.Limits.BiconeMorphism A B) (j : J) {Z : C} (h : B.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι A j) (CategoryTheory.CategoryStruct.comp self.hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι B j) h

@[simp]

theorem CategoryTheory.Limits.BiconeMorphism.wπ_assoc {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} {A : CategoryTheory.Limits.Bicone F} {B : CategoryTheory.Limits.Bicone F} (self : CategoryTheory.Limits.BiconeMorphism A B) (j : J) {Z : C} (h : F j ⟶ Z) : CategoryTheory.CategoryStruct.comp self.hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.π B j) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.π A j) h

@[simp]

theorem CategoryTheory.Limits.Bicone.category_comp_hom {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} : ∀ {X Y Z : CategoryTheory.Limits.Bicone F} (f : X ⟶ Y) (g : Y ⟶ Z), (CategoryTheory.CategoryStruct.comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

@[simp]

theorem CategoryTheory.Limits.Bicone.category_id_hom {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) : (CategoryTheory.CategoryStruct.id B).hom = CategoryTheory.CategoryStruct.id B.pt

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

The category of bicones on a given diagram.

theorem CategoryTheory.Limits.BiconeMorphism.ext {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} {c : CategoryTheory.Limits.Bicone F} {c' : CategoryTheory.Limits.Bicone F} (f : c ⟶ c') (g : c ⟶ c') (w : f.hom = g.hom) : f = g

@[simp]

theorem CategoryTheory.Limits.Bicones.ext_hom_hom {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} {c : CategoryTheory.Limits.Bicone F} {c' : CategoryTheory.Limits.Bicone F} (φ : c.pt ≅ c'.pt) (wι : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι c j) φ.hom = CategoryTheory.Limits.Bicone.ι c' j) _auto✝) (wπ : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp φ.hom (CategoryTheory.Limits.Bicone.π c' j) = CategoryTheory.Limits.Bicone.π c j) _auto✝) : (CategoryTheory.Limits.Bicones.ext φ).hom.hom = φ.hom

@[simp]

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

def CategoryTheory.Limits.Bicones.ext {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} {c : CategoryTheory.Limits.Bicone F} {c' : CategoryTheory.Limits.Bicone F} (φ : c.pt ≅ c'.pt) (wι : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Bicone.ι c j) φ.hom = CategoryTheory.Limits.Bicone.ι c' j) _auto✝) (wπ : autoParam (∀ (j : J), CategoryTheory.CategoryStruct.comp φ.hom (CategoryTheory.Limits.Bicone.π c' j) = CategoryTheory.Limits.Bicone.π 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.

@[simp]

theorem CategoryTheory.Limits.Bicones.functoriality_obj_π {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {D : Type uD} [CategoryTheory.Category.{uD', uD} D] [CategoryTheory.Limits.HasZeroMorphisms D] (F : J → C) (G : CategoryTheory.Functor C D) [CategoryTheory.Functor.PreservesZeroMorphisms G] (A : CategoryTheory.Limits.Bicone F) (j : J) : CategoryTheory.Limits.Bicone.π ((CategoryTheory.Limits.Bicones.functoriality F G).obj A) j = G.map (CategoryTheory.Limits.Bicone.π A j)

@[simp]

theorem CategoryTheory.Limits.Bicones.functoriality_obj_pt {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {D : Type uD} [CategoryTheory.Category.{uD', uD} D] [CategoryTheory.Limits.HasZeroMorphisms D] (F : J → C) (G : CategoryTheory.Functor C D) [CategoryTheory.Functor.PreservesZeroMorphisms G] (A : CategoryTheory.Limits.Bicone F) : ((CategoryTheory.Limits.Bicones.functoriality F G).obj A).pt = G.obj A.pt

@[simp]

theorem CategoryTheory.Limits.Bicones.functoriality_map_hom {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {D : Type uD} [CategoryTheory.Category.{uD', uD} D] [CategoryTheory.Limits.HasZeroMorphisms D] (F : J → C) (G : CategoryTheory.Functor C D) [CategoryTheory.Functor.PreservesZeroMorphisms G] : ∀ {X Y : CategoryTheory.Limits.Bicone F} (f : X ⟶ Y), ((CategoryTheory.Limits.Bicones.functoriality F G).map f).hom = G.map f.hom

@[simp]

theorem CategoryTheory.Limits.Bicones.functoriality_obj_ι {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {D : Type uD} [CategoryTheory.Category.{uD', uD} D] [CategoryTheory.Limits.HasZeroMorphisms D] (F : J → C) (G : CategoryTheory.Functor C D) [CategoryTheory.Functor.PreservesZeroMorphisms G] (A : CategoryTheory.Limits.Bicone F) (j : J) : CategoryTheory.Limits.Bicone.ι ((CategoryTheory.Limits.Bicones.functoriality F G).obj A) j = G.map (CategoryTheory.Limits.Bicone.ι A j)

def CategoryTheory.Limits.Bicones.functoriality {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {D : Type uD} [CategoryTheory.Category.{uD', uD} D] [CategoryTheory.Limits.HasZeroMorphisms D] (F : J → C) (G : CategoryTheory.Functor C D) [CategoryTheory.Functor.PreservesZeroMorphisms G] : CategoryTheory.Functor (CategoryTheory.Limits.Bicone F) (CategoryTheory.Limits.Bicone (G.obj ∘ F))

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

def CategoryTheory.Limits.Bicone.toConeFunctor {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} : CategoryTheory.Functor (CategoryTheory.Limits.Bicone F) (CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor F))

Extract the cone from a bicone.

@[inline, reducible]

abbrev CategoryTheory.Limits.Bicone.toCone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) : CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor F)

A shorthand for toConeFunctor.obj

@[simp]

theorem CategoryTheory.Limits.Bicone.toCone_pt {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) : (CategoryTheory.Limits.Bicone.toCone B).pt = B.pt

@[simp]

theorem CategoryTheory.Limits.Bicone.toCone_π_app {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : CategoryTheory.Discrete J) : (CategoryTheory.Limits.Bicone.toCone B).π.app j = CategoryTheory.Limits.Bicone.π B j.as

theorem CategoryTheory.Limits.Bicone.toCone_π_app_mk {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : J) : (CategoryTheory.Limits.Bicone.toCone B).π.app { as := j } = CategoryTheory.Limits.Bicone.π B j

@[simp]

theorem CategoryTheory.Limits.Bicone.toCone_proj {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : J) : CategoryTheory.Limits.Fan.proj (CategoryTheory.Limits.Bicone.toCone B) j = CategoryTheory.Limits.Bicone.π B j

def CategoryTheory.Limits.Bicone.toCoconeFunctor {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} : CategoryTheory.Functor (CategoryTheory.Limits.Bicone F) (CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor F))

Extract the cocone from a bicone.

@[inline, reducible]

abbrev CategoryTheory.Limits.Bicone.toCocone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) : CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor F)

A shorthand for toCoconeFunctor.obj

@[simp]

theorem CategoryTheory.Limits.Bicone.toCocone_pt {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) : (CategoryTheory.Limits.Bicone.toCocone B).pt = B.pt

@[simp]

theorem CategoryTheory.Limits.Bicone.toCocone_ι_app {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : CategoryTheory.Discrete J) : (CategoryTheory.Limits.Bicone.toCocone B).ι.app j = CategoryTheory.Limits.Bicone.ι B j.as

@[simp]

theorem CategoryTheory.Limits.Bicone.toCocone_proj {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : J) : CategoryTheory.Limits.Cofan.proj (CategoryTheory.Limits.Bicone.toCocone B) j = CategoryTheory.Limits.Bicone.ι B j

theorem CategoryTheory.Limits.Bicone.toCocone_ι_app_mk {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (B : CategoryTheory.Limits.Bicone F) (j : J) : (CategoryTheory.Limits.Bicone.toCocone B).ι.app { as := j } = CategoryTheory.Limits.Bicone.ι B j

@[simp]

theorem CategoryTheory.Limits.Bicone.ofLimitCone_ι {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsLimit t) (j : J) : CategoryTheory.Limits.Bicone.ι (CategoryTheory.Limits.Bicone.ofLimitCone ht) j = CategoryTheory.Limits.IsLimit.lift ht (CategoryTheory.Limits.Fan.mk (f j) fun j' => if h : j = j' then CategoryTheory.eqToHom (_ : f j = f j') else 0)

@[simp]

theorem CategoryTheory.Limits.Bicone.ofLimitCone_π {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsLimit t) (j : J) : CategoryTheory.Limits.Bicone.π (CategoryTheory.Limits.Bicone.ofLimitCone ht) j = t.π.app { as := j }

@[simp]

theorem CategoryTheory.Limits.Bicone.ofLimitCone_pt {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsLimit t) : (CategoryTheory.Limits.Bicone.ofLimitCone ht).pt = t.pt

def CategoryTheory.Limits.Bicone.ofLimitCone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsLimit t) : CategoryTheory.Limits.Bicone f

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

theorem CategoryTheory.Limits.Bicone.ι_of_isLimit {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Bicone f} (ht : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.Bicone.toCone t)) (j : J) : CategoryTheory.Limits.Bicone.ι t j = CategoryTheory.Limits.IsLimit.lift ht (CategoryTheory.Limits.Fan.mk (f j) fun j' => if h : j = j' then CategoryTheory.eqToHom (_ : f j = f j') else 0)

@[simp]

theorem CategoryTheory.Limits.Bicone.ofColimitCocone_ι {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsColimit t) (j : J) : CategoryTheory.Limits.Bicone.ι (CategoryTheory.Limits.Bicone.ofColimitCocone ht) j = t.ι.app { as := j }

@[simp]

theorem CategoryTheory.Limits.Bicone.ofColimitCocone_π {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsColimit t) (j : J) : CategoryTheory.Limits.Bicone.π (CategoryTheory.Limits.Bicone.ofColimitCocone ht) j = CategoryTheory.Limits.IsColimit.desc ht (CategoryTheory.Limits.Cofan.mk (f j) fun j' => if h : j' = j then CategoryTheory.eqToHom (_ : f j' = f j) else 0)

@[simp]

theorem CategoryTheory.Limits.Bicone.ofColimitCocone_pt {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsColimit t) : (CategoryTheory.Limits.Bicone.ofColimitCocone ht).pt = t.pt

def CategoryTheory.Limits.Bicone.ofColimitCocone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor f)} (ht : CategoryTheory.Limits.IsColimit t) : CategoryTheory.Limits.Bicone f

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

theorem CategoryTheory.Limits.Bicone.π_of_isColimit {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {t : CategoryTheory.Limits.Bicone f} (ht : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Bicone.toCocone t)) (j : J) : CategoryTheory.Limits.Bicone.π t j = CategoryTheory.Limits.IsColimit.desc ht (CategoryTheory.Limits.Cofan.mk (f j) fun j' => if h : j' = j then CategoryTheory.eqToHom (_ : f j' = f j) else 0)

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

• isLimit : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.Bicone.toCone B)

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

• isColimit : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Bicone.toCocone B)

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

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

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 : J → C} {B : CategoryTheory.Limits.Bicone F} (x y : CategoryTheory.Limits.Bicone.IsBilimit B), 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 : J → C} {B : CategoryTheory.Limits.Bicone F} (x y : CategoryTheory.Limits.Bicone.IsBilimit B), x.isLimit = y.isLimit → x.isColimit = y.isColimit → x = y

instance CategoryTheory.Limits.Bicone.subsingleton_isBilimit {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {c : CategoryTheory.Limits.Bicone f} : Subsingleton (CategoryTheory.Limits.Bicone.IsBilimit c)

@[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 : J → C} (c : CategoryTheory.Limits.Bicone f) (g : K ≃ J) (k : K) : CategoryTheory.Limits.Bicone.ι (CategoryTheory.Limits.Bicone.whisker c g) k = CategoryTheory.Limits.Bicone.ι c (↑g k)

@[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 : J → C} (c : CategoryTheory.Limits.Bicone f) (g : K ≃ J) (k : K) : CategoryTheory.Limits.Bicone.π (CategoryTheory.Limits.Bicone.whisker c g) k = CategoryTheory.Limits.Bicone.π 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 : J → C} (c : CategoryTheory.Limits.Bicone f) (g : K ≃ J) : (CategoryTheory.Limits.Bicone.whisker c g).pt = c.pt

def CategoryTheory.Limits.Bicone.whisker {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : J → C} (c : CategoryTheory.Limits.Bicone f) (g : K ≃ J) : CategoryTheory.Limits.Bicone (f ∘ ↑g)

Whisker a bicone with an equivalence between the indexing types.

def CategoryTheory.Limits.Bicone.whiskerToCone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : J → C} (c : CategoryTheory.Limits.Bicone f) (g : K ≃ J) : CategoryTheory.Limits.Bicone.toCone (CategoryTheory.Limits.Bicone.whisker c g) ≅ (CategoryTheory.Limits.Cones.postcompose (CategoryTheory.Discrete.functorComp f ↑g).inv).obj (CategoryTheory.Limits.Cone.whisker (CategoryTheory.Discrete.functor (CategoryTheory.Discrete.mk ∘ ↑g)) (CategoryTheory.Limits.Bicone.toCone c))

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.

def CategoryTheory.Limits.Bicone.whiskerToCocone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} {f : J → C} (c : CategoryTheory.Limits.Bicone f) (g : K ≃ J) : CategoryTheory.Limits.Bicone.toCocone (CategoryTheory.Limits.Bicone.whisker c g) ≅ (CategoryTheory.Limits.Cocones.precompose (CategoryTheory.Discrete.functorComp f ↑g).hom).obj (CategoryTheory.Limits.Cocone.whisker (CategoryTheory.Discrete.functor (CategoryTheory.Discrete.mk ∘ ↑g)) (CategoryTheory.Limits.Bicone.toCocone c))

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.

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

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

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

• bicone : CategoryTheory.Limits.Bicone F

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

• isBilimit : CategoryTheory.Limits.Bicone.IsBilimit s.bicone

  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.

class CategoryTheory.Limits.HasBiproduct {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) : Prop

• mk' :: (
• exists_biproduct : Nonempty (CategoryTheory.Limits.LimitBicone F)

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

• )

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

theorem CategoryTheory.Limits.HasBiproduct.mk {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} (d : CategoryTheory.Limits.LimitBicone F) : CategoryTheory.Limits.HasBiproduct F

def CategoryTheory.Limits.getBiproductData {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.LimitBicone F

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

def CategoryTheory.Limits.biproduct.bicone {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.Bicone F

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

def CategoryTheory.Limits.biproduct.isBilimit {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.Bicone.IsBilimit (CategoryTheory.Limits.biproduct.bicone F)

biproduct.bicone F is a bilimit bicone.

def CategoryTheory.Limits.biproduct.isLimit {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.Bicone.toCone (CategoryTheory.Limits.biproduct.bicone F))

biproduct.bicone F is a limit cone.

def CategoryTheory.Limits.biproduct.isColimit {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Bicone.toCocone (CategoryTheory.Limits.biproduct.bicone F))

biproduct.bicone F is a colimit cocone.

instance CategoryTheory.Limits.hasProduct_of_hasBiproduct {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.HasProduct F

instance CategoryTheory.Limits.hasCoproduct_of_hasBiproduct {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {F : J → C} [CategoryTheory.Limits.HasBiproduct F] : CategoryTheory.Limits.HasCoproduct F

class CategoryTheory.Limits.HasBiproductsOfShape (J : Type w) (C : Type uC) [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] : Prop

• has_biproduct : ∀ (F : J → C), CategoryTheory.Limits.HasBiproduct 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.

class CategoryTheory.Limits.HasFiniteBiproducts (C : Type uC) [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] : Prop

• out : ∀ (n : ℕ), CategoryTheory.Limits.HasBiproductsOfShape (Fin n) C

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

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

theorem CategoryTheory.Limits.hasBiproductsOfShape_of_equiv {J : Type w} (C : Type uC) [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type w'} [CategoryTheory.Limits.HasBiproductsOfShape K C] (e : J ≃ K) : CategoryTheory.Limits.HasBiproductsOfShape J C

instance CategoryTheory.Limits.hasBiproductsOfShape_finite {J : Type w} (C : Type uC) [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] [Finite J] : CategoryTheory.Limits.HasBiproductsOfShape J C

instance CategoryTheory.Limits.hasFiniteProducts_of_hasFiniteBiproducts (C : Type uC) [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] : CategoryTheory.Limits.HasFiniteProducts C

instance CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteBiproducts (C : Type uC) [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] : CategoryTheory.Limits.HasFiniteCoproducts C

def CategoryTheory.Limits.biproductIso {J : Type w} {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (F : J → C) [CategoryTheory.Limits.HasBiproduct F] : ∏ F ≅ ∐ F

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

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.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.)

def CategoryTheory.Limits.«term⨁_» : Lean.ParserDescr

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.)

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct.π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (b : J) : ⨁ f ⟶ f b

The projection onto a summand of a biproduct.

@[simp]

theorem CategoryTheory.Limits.biproduct.bicone_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (b : J) : CategoryTheory.Limits.Bicone.π (CategoryTheory.Limits.biproduct.bicone f) b = CategoryTheory.Limits.biproduct.π f b

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct.ι {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (b : J) : f b ⟶ ⨁ f

The inclusion into a summand of a biproduct.

@[simp]

theorem CategoryTheory.Limits.biproduct.bicone_ι {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (b : J) : CategoryTheory.Limits.Bicone.ι (CategoryTheory.Limits.biproduct.bicone f) b = CategoryTheory.Limits.biproduct.ι f b

theorem CategoryTheory.Limits.biproduct.ι_π_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [DecidableEq J] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (j : J) (j' : J) {Z : C} (h : f j' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j') h) = CategoryTheory.CategoryStruct.comp (if h : j = j' then CategoryTheory.eqToHom (_ : f j = f j') else 0) h

theorem CategoryTheory.Limits.biproduct.ι_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [DecidableEq J] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (j : J) (j' : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.π f j') = if h : j = j' then CategoryTheory.eqToHom (_ : f j = f j') 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 Classical.

theorem CategoryTheory.Limits.biproduct.ι_π_self_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (j : J) {Z : C} (h : f j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) h) = h

theorem CategoryTheory.Limits.biproduct.ι_π_self {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.π f j) = CategoryTheory.CategoryStruct.id (f j)

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_π_ne_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {j : J} {j' : J} (h : j ≠ j') {Z : C} (h : f j' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j') h) = CategoryTheory.CategoryStruct.comp 0 h

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_π_ne {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {j : J} {j' : J} (h : j ≠ j') : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.π f j') = 0

@[simp]

theorem CategoryTheory.Limits.biproduct.eqToHom_comp_ι_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {j : J} {j' : J} (w : j = j') {Z : C} (h : ⨁ f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom (_ : f j = f j')) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j') h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) h

@[simp]

theorem CategoryTheory.Limits.biproduct.eqToHom_comp_ι {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {j : J} {j' : J} (w : j = j') : CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom (_ : f j = f j')) (CategoryTheory.Limits.biproduct.ι f j') = CategoryTheory.Limits.biproduct.ι f j

@[simp]

theorem CategoryTheory.Limits.biproduct.π_comp_eqToHom_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {j : J} {j' : J} (w : j = j') {Z : C} (h : f j' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom (_ : f j = f j')) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j') h

@[simp]

theorem CategoryTheory.Limits.biproduct.π_comp_eqToHom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {j : J} {j' : J} (w : j = j') : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (CategoryTheory.eqToHom (_ : f j = f j')) = CategoryTheory.Limits.biproduct.π f j'

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct.lift {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.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.

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct.desc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.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.

@[simp]

theorem CategoryTheory.Limits.biproduct.lift_π_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] {P : C} (p : (b : J) → P ⟶ f b) (j : J) {Z : C} (h : f j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) h) = CategoryTheory.CategoryStruct.comp (p j) h

@[simp]

theorem CategoryTheory.Limits.biproduct.lift_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] {P : C} (p : (b : J) → P ⟶ f b) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift p) (CategoryTheory.Limits.biproduct.π f j) = p j

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_desc_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] {P : C} (p : (b : J) → f b ⟶ P) (j : J) {Z : C} (h : P ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.desc p) h) = CategoryTheory.CategoryStruct.comp (p j) h

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_desc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] {P : C} (p : (b : J) → f b ⟶ P) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.desc p) = p j

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct.map {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.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.

@[inline, reducible]

abbrev CategoryTheory.Limits.biproduct.map' {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.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.

theorem CategoryTheory.Limits.biproduct.hom_ext {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] {Z : C} (g : Z ⟶ ⨁ f) (h : Z ⟶ ⨁ f) (w : ∀ (j : J), CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.π f j) = CategoryTheory.CategoryStruct.comp h (CategoryTheory.Limits.biproduct.π f j)) : g = h

theorem CategoryTheory.Limits.biproduct.hom_ext' {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] {Z : C} (g : ⨁ f ⟶ Z) (h : ⨁ f ⟶ Z) (w : ∀ (j : J), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) g = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) h) : g = h

def CategoryTheory.Limits.biproduct.isoProduct {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] : ⨁ f ≅ ∏ f

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

@[simp]

theorem CategoryTheory.Limits.biproduct.isoProduct_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] : (CategoryTheory.Limits.biproduct.isoProduct f).hom = CategoryTheory.Limits.Pi.lift (CategoryTheory.Limits.biproduct.π f)

@[simp]

theorem CategoryTheory.Limits.biproduct.isoProduct_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] : (CategoryTheory.Limits.biproduct.isoProduct f).inv = CategoryTheory.Limits.biproduct.lift (CategoryTheory.Limits.Pi.π f)

def CategoryTheory.Limits.biproduct.isoCoproduct {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] : ⨁ f ≅ ∐ f

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

@[simp]

theorem CategoryTheory.Limits.biproduct.isoCoproduct_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] : (CategoryTheory.Limits.biproduct.isoCoproduct f).inv = CategoryTheory.Limits.Sigma.desc (CategoryTheory.Limits.biproduct.ι f)

@[simp]

theorem CategoryTheory.Limits.biproduct.isoCoproduct_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} [CategoryTheory.Limits.HasBiproduct f] : (CategoryTheory.Limits.biproduct.isoCoproduct f).hom = CategoryTheory.Limits.biproduct.desc (CategoryTheory.Limits.Sigma.ι f)

theorem CategoryTheory.Limits.biproduct.map_eq_map' {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ⟶ g b) : CategoryTheory.Limits.biproduct.map p = CategoryTheory.Limits.biproduct.map' p

@[simp]

theorem CategoryTheory.Limits.biproduct.map_π_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) {Z : C} (h : g j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π g j) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (CategoryTheory.CategoryStruct.comp (p j) h)

@[simp]

theorem CategoryTheory.Limits.biproduct.map_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.Limits.biproduct.π g j) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (p j)

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_map_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) {Z : C} (h : (⨁ fun b => g b) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) h) = CategoryTheory.CategoryStruct.comp (p j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι g j) h)

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_map {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.map p) = CategoryTheory.CategoryStruct.comp (p j) (CategoryTheory.Limits.biproduct.ι g j)

@[simp]

theorem CategoryTheory.Limits.biproduct.map_desc_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) {P : C} (k : (j : J) → g j ⟶ P) {Z : C} (h : P ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.desc k) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.desc fun j => CategoryTheory.CategoryStruct.comp (p j) (k j)) h

@[simp]

theorem CategoryTheory.Limits.biproduct.map_desc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) {P : C} (k : (j : J) → g j ⟶ P) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.Limits.biproduct.desc k) = CategoryTheory.Limits.biproduct.desc fun j => CategoryTheory.CategoryStruct.comp (p j) (k j)

@[simp]

theorem CategoryTheory.Limits.biproduct.lift_map_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] {P : C} (k : (j : J) → P ⟶ f j) (p : (j : J) → f j ⟶ g j) {Z : C} (h : (⨁ fun b => g b) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift fun j => CategoryTheory.CategoryStruct.comp (k j) (p j)) h

@[simp]

theorem CategoryTheory.Limits.biproduct.lift_map {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] {P : C} (k : (j : J) → P ⟶ f j) (p : (j : J) → f j ⟶ g j) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift k) (CategoryTheory.Limits.biproduct.map p) = CategoryTheory.Limits.biproduct.lift fun j => CategoryTheory.CategoryStruct.comp (k j) (p j)

@[simp]

theorem CategoryTheory.Limits.biproduct.mapIso_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ≅ g b) : (CategoryTheory.Limits.biproduct.mapIso p).inv = CategoryTheory.Limits.biproduct.map fun b => (p b).inv

@[simp]

theorem CategoryTheory.Limits.biproduct.mapIso_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ≅ g b) : (CategoryTheory.Limits.biproduct.mapIso p).hom = CategoryTheory.Limits.biproduct.map fun b => (p b).hom

def CategoryTheory.Limits.biproduct.mapIso {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.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.

@[simp]

theorem CategoryTheory.Limits.biproduct.whiskerEquiv_inv {J : Type w} {K : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : K → C} (e : J ≃ K) (w : (j : J) → g (↑e j) ≅ f j) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] : (CategoryTheory.Limits.biproduct.whiskerEquiv e w).inv = CategoryTheory.Limits.biproduct.desc fun k => CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom (_ : g k = g (↑e (↑e.symm k)))) (CategoryTheory.CategoryStruct.comp (w (↑e.symm k)).hom (CategoryTheory.Limits.biproduct.ι f (↑e.symm k)))

@[simp]

theorem CategoryTheory.Limits.biproduct.whiskerEquiv_hom {J : Type w} {K : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : K → C} (e : J ≃ K) (w : (j : J) → g (↑e j) ≅ f j) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] : (CategoryTheory.Limits.biproduct.whiskerEquiv e w).hom = CategoryTheory.Limits.biproduct.desc fun j => CategoryTheory.CategoryStruct.comp (w j).inv (CategoryTheory.Limits.biproduct.ι g (↑e j))

def CategoryTheory.Limits.biproduct.whiskerEquiv {J : Type w} {K : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : K → C} (e : J ≃ K) (w : (j : J) → g (↑e j) ≅ f j) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.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.

theorem CategoryTheory.Limits.biproduct.whiskerEquiv_hom_eq_lift {J : Type w} {K : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : K → C} (e : J ≃ K) (w : (j : J) → g (↑e j) ≅ f j) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] : (CategoryTheory.Limits.biproduct.whiskerEquiv e w).hom = CategoryTheory.Limits.biproduct.lift fun k => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f (↑e.symm k)) (CategoryTheory.CategoryStruct.comp (w (↑e.symm k)).inv (CategoryTheory.eqToHom (_ : g (↑e (↑e.symm k)) = g k)))

theorem CategoryTheory.Limits.biproduct.whiskerEquiv_inv_eq_lift {J : Type w} {K : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f : J → C} {g : K → C} (e : J ≃ K) (w : (j : J) → g (↑e j) ≅ f j) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] : (CategoryTheory.Limits.biproduct.whiskerEquiv e w).inv = CategoryTheory.Limits.biproduct.lift fun j => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π g (↑e j)) (w j).hom

instance CategoryTheory.Limits.instHasBiproductSigmaFstSnd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ι → Type u_2) (g : (i : ι) → f i → C) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun i => ⨁ g i] : CategoryTheory.Limits.HasBiproduct fun p => g p.fst p.snd

@[simp]

theorem CategoryTheory.Limits.biproductBiproductIso_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ι → Type u_2) (g : (i : ι) → f i → C) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun i => ⨁ g i] : (CategoryTheory.Limits.biproductBiproductIso f g).hom = CategoryTheory.Limits.biproduct.lift fun x => match x with | { fst := i, snd := x } => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π (fun i => ⨁ g i) i) (CategoryTheory.Limits.biproduct.π (g i) x)

@[simp]

theorem CategoryTheory.Limits.biproductBiproductIso_inv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ι → Type u_2) (g : (i : ι) → f i → C) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun i => ⨁ g i] : (CategoryTheory.Limits.biproductBiproductIso f g).inv = CategoryTheory.Limits.biproduct.lift fun i => CategoryTheory.Limits.biproduct.lift fun x => CategoryTheory.Limits.biproduct.π (fun p => g p.fst p.snd) { fst := i, snd := x }

def CategoryTheory.Limits.biproductBiproductIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_3} (f : ι → Type u_2) (g : (i : ι) → f i → C) [∀ (i : ι), CategoryTheory.Limits.HasBiproduct (g i)] [CategoryTheory.Limits.HasBiproduct fun i => ⨁ g i] : (⨁ fun i => ⨁ g i) ≅ ⨁ fun p => g p.fst p.snd

An iterated biproduct is a biproduct over a sigma type.

def CategoryTheory.Limits.biproduct.fromSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] : ⨁ Subtype.restrict p f ⟶ ⨁ f

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

def CategoryTheory.Limits.biproduct.toSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] : ⨁ f ⟶ ⨁ Subtype.restrict p f

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

@[simp]

theorem CategoryTheory.Limits.biproduct.fromSubtype_π_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) {Z : C} (h : f j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) h) = CategoryTheory.CategoryStruct.comp (if h : p j then CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) { val := j, property := h } else 0) h

@[simp]

theorem CategoryTheory.Limits.biproduct.fromSubtype_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.π f j) = if h : p j then CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) { val := j, property := h } else 0

theorem CategoryTheory.Limits.biproduct.fromSubtype_eq_lift {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] : CategoryTheory.Limits.biproduct.fromSubtype f p = CategoryTheory.Limits.biproduct.lift fun j => if h : p j then CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) { val := j, property := h } else 0

theorem CategoryTheory.Limits.biproduct.fromSubtype_π_subtype_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) {Z : C} (h : f ↑j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f ↑j) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) j) h

theorem CategoryTheory.Limits.biproduct.fromSubtype_π_subtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.π f ↑j) = CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) j

@[simp]

theorem CategoryTheory.Limits.biproduct.toSubtype_π_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) {Z : C} (h : Subtype.restrict p f j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) j) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f ↑j) h

@[simp]

theorem CategoryTheory.Limits.biproduct.toSubtype_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) (CategoryTheory.Limits.biproduct.π (Subtype.restrict p f) j) = CategoryTheory.Limits.biproduct.π f ↑j

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_toSubtype_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) {Z : C} (h : ⨁ Subtype.restrict p f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) h) = CategoryTheory.CategoryStruct.comp (if h : p j then CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) { val := j, property := h } else 0) h

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_toSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.toSubtype f p) = if h : p j then CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) { val := j, property := h } else 0

theorem CategoryTheory.Limits.biproduct.toSubtype_eq_desc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] : CategoryTheory.Limits.biproduct.toSubtype f p = CategoryTheory.Limits.biproduct.desc fun j => if h : p j then CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) { val := j, property := h } else 0

theorem CategoryTheory.Limits.biproduct.ι_toSubtype_subtype_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) {Z : C} (h : ⨁ Subtype.restrict p f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f ↑j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) j) h

theorem CategoryTheory.Limits.biproduct.ι_toSubtype_subtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f ↑j) (CategoryTheory.Limits.biproduct.toSubtype f p) = CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) j

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_fromSubtype_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) {Z : C} (h : ⨁ f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f ↑j) h

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_fromSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] (j : Subtype p) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι (Subtype.restrict p f) j) (CategoryTheory.Limits.biproduct.fromSubtype f p) = CategoryTheory.Limits.biproduct.ι f ↑j

@[simp]

theorem CategoryTheory.Limits.biproduct.fromSubtype_toSubtype_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] {Z : C} (h : ⨁ Subtype.restrict p f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) h) = h

@[simp]

theorem CategoryTheory.Limits.biproduct.fromSubtype_toSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.toSubtype f p) = CategoryTheory.CategoryStruct.id (⨁ Subtype.restrict p f)

@[simp]

theorem CategoryTheory.Limits.biproduct.toSubtype_fromSubtype_assoc {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] {Z : C} (h : ⨁ f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map fun j => if p j then CategoryTheory.CategoryStruct.id (f j) else 0) h

@[simp]

theorem CategoryTheory.Limits.biproduct.toSubtype_fromSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) (CategoryTheory.Limits.biproduct.fromSubtype f p) = CategoryTheory.Limits.biproduct.map fun j => if p j then CategoryTheory.CategoryStruct.id (f j) else 0

def CategoryTheory.Limits.biproduct.isLimitFromSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι (CategoryTheory.Limits.biproduct.fromSubtype f fun j => j ≠ i) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f fun j => ¬j = i) (CategoryTheory.Limits.biproduct.π f i) = 0))

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.

instance CategoryTheory.Limits.instHasKernelBiproductπ {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.biproduct.π f i)

@[simp]

theorem CategoryTheory.Limits.kernelBiproductπIso_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : (CategoryTheory.Limits.kernelBiproductπIso f i).inv = CategoryTheory.Limits.limit.lift (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.π f i) 0) (CategoryTheory.Limits.KernelFork.ofι (CategoryTheory.Limits.biproduct.fromSubtype f fun j => ¬j = i) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f fun j => ¬j = i) (CategoryTheory.Limits.biproduct.π f i) = 0))

@[simp]

theorem CategoryTheory.Limits.kernelBiproductπIso_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : (CategoryTheory.Limits.kernelBiproductπIso f i).hom = CategoryTheory.Limits.IsLimit.lift (CategoryTheory.Limits.biproduct.isLimitFromSubtype f i) (CategoryTheory.Limits.limit.cone (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.π f i) 0))

def CategoryTheory.Limits.kernelBiproductπIso {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : CategoryTheory.Limits.kernel (CategoryTheory.Limits.biproduct.π f i) ≅ ⨁ Subtype.restrict (fun j => j ≠ i) f

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

def CategoryTheory.Limits.biproduct.isColimitToSubtype {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ (CategoryTheory.Limits.biproduct.toSubtype f fun j => j ≠ i) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f i) (CategoryTheory.Limits.biproduct.toSubtype f fun j => ¬j = i) = 0))

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

instance CategoryTheory.Limits.instHasCokernelBiproductι {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : CategoryTheory.Limits.HasCokernel (CategoryTheory.Limits.biproduct.ι f i)

@[simp]

theorem CategoryTheory.Limits.cokernelBiproductιIso_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : (CategoryTheory.Limits.cokernelBiproductιIso f i).hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.ι f i) 0) (CategoryTheory.Limits.CokernelCofork.ofπ (CategoryTheory.Limits.biproduct.toSubtype f fun j => ¬j = i) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f i) (CategoryTheory.Limits.biproduct.toSubtype f fun j => ¬j = i) = 0))

@[simp]

theorem CategoryTheory.Limits.cokernelBiproductιIso_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : (CategoryTheory.Limits.cokernelBiproductιIso f i).inv = CategoryTheory.Limits.IsColimit.desc (CategoryTheory.Limits.biproduct.isColimitToSubtype f i) (CategoryTheory.Limits.colimit.cocone (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.ι f i) 0))

def CategoryTheory.Limits.cokernelBiproductιIso {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : CategoryTheory.Limits.cokernel (CategoryTheory.Limits.biproduct.ι f i) ≅ ⨁ Subtype.restrict (fun j => j ≠ i) f

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

@[simp]

theorem CategoryTheory.Limits.kernelForkBiproductToSubtype_cone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.kernelForkBiproductToSubtype f p).cone = CategoryTheory.Limits.KernelFork.ofι (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (CategoryTheory.Limits.biproduct.toSubtype f p) = 0)

@[simp]

theorem CategoryTheory.Limits.kernelForkBiproductToSubtype_isLimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.kernelForkBiproductToSubtype f p).isLimit = CategoryTheory.Limits.KernelFork.IsLimit.ofι (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (CategoryTheory.Limits.biproduct.toSubtype f p) = 0) (fun {W} g x => CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.toSubtype f pᶜ)) (_ : ∀ {W' : C} (g' : W' ⟶ ⨁ f) (w : CategoryTheory.CategoryStruct.comp g' (CategoryTheory.Limits.biproduct.toSubtype f p) = 0), CategoryTheory.CategoryStruct.comp ((fun {W} g x => CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.toSubtype f pᶜ)) W' g' w) (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) = g') (_ : ∀ {W' : C} (g' : W' ⟶ ⨁ f) (eq' : CategoryTheory.CategoryStruct.comp g' (CategoryTheory.Limits.biproduct.toSubtype f p) = 0) (m : W' ⟶ ⨁ Subtype.restrict pᶜ f), CategoryTheory.CategoryStruct.comp m (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) = g' → m = (fun {W} g x => CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.toSubtype f pᶜ)) W' g' eq')

def CategoryTheory.Limits.kernelForkBiproductToSubtype {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : CategoryTheory.Limits.LimitCone (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.toSubtype f p) 0)

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

instance CategoryTheory.Limits.instHasKernelBiproductHas_biproductHasBiproductsOfShape_finiteOf_fintypeSubtypeRestrictFintypePropDecidableToSubtype {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.biproduct.toSubtype f p)

@[simp]

theorem CategoryTheory.Limits.kernelBiproductToSubtypeIso_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.kernelBiproductToSubtypeIso f p).hom = CategoryTheory.Limits.IsLimit.lift (CategoryTheory.Limits.KernelFork.IsLimit.ofι (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (CategoryTheory.Limits.biproduct.toSubtype f p) = 0) (fun {W} g x => CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.toSubtype f pᶜ)) (_ : ∀ {W' : C} (g' : W' ⟶ ⨁ f) (w : CategoryTheory.CategoryStruct.comp g' (CategoryTheory.Limits.biproduct.toSubtype f p) = 0), CategoryTheory.CategoryStruct.comp ((fun {W} g x => CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.toSubtype f pᶜ)) W' g' w) (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) = g') (_ : ∀ {W' : C} (g' : W' ⟶ ⨁ f) (eq' : CategoryTheory.CategoryStruct.comp g' (CategoryTheory.Limits.biproduct.toSubtype f p) = 0) (m : W' ⟶ ⨁ Subtype.restrict pᶜ f), CategoryTheory.CategoryStruct.comp m (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) = g' → m = (fun {W} g x => CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.biproduct.toSubtype f pᶜ)) W' g' eq')) (CategoryTheory.Limits.limit.cone (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.toSubtype f p) 0))

@[simp]

theorem CategoryTheory.Limits.kernelBiproductToSubtypeIso_inv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.kernelBiproductToSubtypeIso f p).inv = CategoryTheory.Limits.limit.lift (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.toSubtype f p) 0) (CategoryTheory.Limits.KernelFork.ofι (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) (CategoryTheory.Limits.biproduct.toSubtype f p) = 0))

def CategoryTheory.Limits.kernelBiproductToSubtypeIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : CategoryTheory.Limits.kernel (CategoryTheory.Limits.biproduct.toSubtype f p) ≅ ⨁ Subtype.restrict pᶜ f

The kernel of biproduct.toSubtype f p is ⨁ Subtype.restrict pᶜ f.

@[simp]

theorem CategoryTheory.Limits.cokernelCoforkBiproductFromSubtype_isColimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.cokernelCoforkBiproductFromSubtype f p).isColimit = CategoryTheory.Limits.CokernelCofork.IsColimit.ofπ (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) = 0) (fun {W} g x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) g) (_ : ∀ {W : C} (g' : ⨁ f ⟶ W) (w : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) g' = 0), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) ((fun {W} g x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) g) W g' w) = g') (_ : ∀ {Z' : C} (g' : ⨁ f ⟶ Z') (eq' : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) g' = 0) (m : ⨁ Subtype.restrict pᶜ f ⟶ Z'), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) m = g' → m = (fun {W} g x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) g) Z' g' eq')

@[simp]

theorem CategoryTheory.Limits.cokernelCoforkBiproductFromSubtype_cocone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.cokernelCoforkBiproductFromSubtype f p).cocone = CategoryTheory.Limits.CokernelCofork.ofπ (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) = 0)

def CategoryTheory.Limits.cokernelCoforkBiproductFromSubtype {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : CategoryTheory.Limits.ColimitCocone (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.fromSubtype f p) 0)

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

instance CategoryTheory.Limits.instHasCokernelBiproductSubtypeRestrictHas_biproductHasBiproductsOfShape_finiteOf_fintypeFintypePropDecidableFromSubtype {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : CategoryTheory.Limits.HasCokernel (CategoryTheory.Limits.biproduct.fromSubtype f p)

@[simp]

theorem CategoryTheory.Limits.cokernelBiproductFromSubtypeIso_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.cokernelBiproductFromSubtypeIso f p).hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.fromSubtype f p) 0) (CategoryTheory.Limits.CokernelCofork.ofπ (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) = 0))

@[simp]

theorem CategoryTheory.Limits.cokernelBiproductFromSubtypeIso_inv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : (CategoryTheory.Limits.cokernelBiproductFromSubtypeIso f p).inv = CategoryTheory.Limits.IsColimit.desc (CategoryTheory.Limits.CokernelCofork.IsColimit.ofπ (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) = 0) (fun {W} g x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) g) (_ : ∀ {W : C} (g' : ⨁ f ⟶ W) (w : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) g' = 0), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) ((fun {W} g x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) g) W g' w) = g') (_ : ∀ {Z' : C} (g' : ⨁ f ⟶ Z') (eq' : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) g' = 0) (m : ⨁ Subtype.restrict pᶜ f ⟶ Z'), CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) m = g' → m = (fun {W} g x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f pᶜ) g) Z' g' eq')) (CategoryTheory.Limits.colimit.cocone (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.fromSubtype f p) 0))

def CategoryTheory.Limits.cokernelBiproductFromSubtypeIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Fintype K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : Set K) : CategoryTheory.Limits.cokernel (CategoryTheory.Limits.biproduct.fromSubtype f p) ≅ ⨁ Subtype.restrict pᶜ f

The cokernel of biproduct.fromSubtype f p is ⨁ Subtype.restrict pᶜ f.

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

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

@[simp]

theorem CategoryTheory.Limits.biproduct.matrix_π_assoc {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (k : K) {Z : C} (h : g k ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix m) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π g k) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.desc fun j => m j k) h

@[simp]

theorem CategoryTheory.Limits.biproduct.matrix_π {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (k : K) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix m) (CategoryTheory.Limits.biproduct.π g k) = CategoryTheory.Limits.biproduct.desc fun j => m j k

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_matrix_assoc {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (j : J) {Z : C} (h : (⨁ fun k => g k) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix m) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift fun k => m j k) h

@[simp]

theorem CategoryTheory.Limits.biproduct.ι_matrix {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.matrix m) = CategoryTheory.Limits.biproduct.lift fun k => m j k

def CategoryTheory.Limits.biproduct.components {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : ⨁ f ⟶ ⨁ g) (j : J) (k : K) : f j ⟶ g k

Extract the matrix components from a morphism of biproducts.

@[simp]

theorem CategoryTheory.Limits.biproduct.matrix_components {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (j : J) (k : K) : CategoryTheory.Limits.biproduct.components (CategoryTheory.Limits.biproduct.matrix m) j k = m j k

@[simp]

theorem CategoryTheory.Limits.biproduct.components_matrix {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : ⨁ f ⟶ ⨁ g) : (CategoryTheory.Limits.biproduct.matrix fun j k => CategoryTheory.Limits.biproduct.components m j k) = m

@[simp]

theorem CategoryTheory.Limits.biproduct.matrixEquiv_apply {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : ⨁ f ⟶ ⨁ g) (j : J) (k : K) : ↑CategoryTheory.Limits.biproduct.matrixEquiv m j k = CategoryTheory.Limits.biproduct.components m j k

@[simp]

theorem CategoryTheory.Limits.biproduct.matrixEquiv_symm_apply {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) : ↑CategoryTheory.Limits.biproduct.matrixEquiv.symm m = CategoryTheory.Limits.biproduct.matrix m

def CategoryTheory.Limits.biproduct.matrixEquiv {J : Type} [Fintype J] {K : Type} [Fintype K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] {f : J → C} {g : K → C} : (⨁ f ⟶ ⨁ g) ≃ ((j : J) → (k : K) → f j ⟶ g k)

Morphisms between direct sums are matrices.

instance CategoryTheory.Limits.biproduct.ι_mono {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (b : J) : CategoryTheory.IsSplitMono (CategoryTheory.Limits.biproduct.ι f b)

instance CategoryTheory.Limits.biproduct.π_epi {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (b : J) : CategoryTheory.IsSplitEpi (CategoryTheory.Limits.biproduct.π f b)

theorem CategoryTheory.Limits.biproduct.conePointUniqueUpToIso_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {b : CategoryTheory.Limits.Bicone f} (hb : CategoryTheory.Limits.Bicone.IsBilimit b) : (CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso hb.isLimit (CategoryTheory.Limits.biproduct.isLimit f)).hom = CategoryTheory.Limits.biproduct.lift b.π

Auxiliary lemma for biproduct.uniqueUpToIso.

theorem CategoryTheory.Limits.biproduct.conePointUniqueUpToIso_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {b : CategoryTheory.Limits.Bicone f} (hb : CategoryTheory.Limits.Bicone.IsBilimit b) : (CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso hb.isLimit (CategoryTheory.Limits.biproduct.isLimit f)).inv = CategoryTheory.Limits.biproduct.desc b.ι

Auxiliary lemma for biproduct.uniqueUpToIso.

@[simp]

theorem CategoryTheory.Limits.biproduct.uniqueUpToIso_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {b : CategoryTheory.Limits.Bicone f} (hb : CategoryTheory.Limits.Bicone.IsBilimit b) : (CategoryTheory.Limits.biproduct.uniqueUpToIso f hb).inv = CategoryTheory.Limits.biproduct.desc b.ι

@[simp]

theorem CategoryTheory.Limits.biproduct.uniqueUpToIso_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {b : CategoryTheory.Limits.Bicone f} (hb : CategoryTheory.Limits.Bicone.IsBilimit b) : (CategoryTheory.Limits.biproduct.uniqueUpToIso f hb).hom = CategoryTheory.Limits.biproduct.lift b.π

def CategoryTheory.Limits.biproduct.uniqueUpToIso {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] {b : CategoryTheory.Limits.Bicone f} (hb : CategoryTheory.Limits.Bicone.IsBilimit b) : b.pt ≅ ⨁ f

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.

instance CategoryTheory.Limits.hasZeroObject_of_hasFiniteBiproducts (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] : CategoryTheory.Limits.HasZeroObject C

A category with finite biproducts has a zero object.

@[simp]

theorem CategoryTheory.Limits.limitBiconeOfUnique_bicone_ι {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) (j : J) : CategoryTheory.Limits.Bicone.ι (CategoryTheory.Limits.limitBiconeOfUnique f).bicone j = CategoryTheory.eqToHom (_ : f j = f default)

@[simp]

theorem CategoryTheory.Limits.limitBiconeOfUnique_isBilimit_isLimit {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : (CategoryTheory.Limits.limitBiconeOfUnique f).isBilimit.isLimit = (CategoryTheory.Limits.limitConeOfUnique f).isLimit

@[simp]

theorem CategoryTheory.Limits.limitBiconeOfUnique_bicone_pt {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : (CategoryTheory.Limits.limitBiconeOfUnique f).bicone.pt = f default

@[simp]

theorem CategoryTheory.Limits.limitBiconeOfUnique_isBilimit_isColimit {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : (CategoryTheory.Limits.limitBiconeOfUnique f).isBilimit.isColimit = (CategoryTheory.Limits.colimitCoconeOfUnique f).isColimit

@[simp]

theorem CategoryTheory.Limits.limitBiconeOfUnique_bicone_π {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) (j : J) : CategoryTheory.Limits.Bicone.π (CategoryTheory.Limits.limitBiconeOfUnique f).bicone j = CategoryTheory.eqToHom (_ : f default = f j)

def CategoryTheory.Limits.limitBiconeOfUnique {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : CategoryTheory.Limits.LimitBicone f

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

instance CategoryTheory.Limits.hasBiproduct_unique {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : CategoryTheory.Limits.HasBiproduct f

@[simp]

theorem CategoryTheory.Limits.biproductUniqueIso_inv {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : (CategoryTheory.Limits.biproductUniqueIso f).inv = CategoryTheory.Limits.biproduct.lift (CategoryTheory.Limits.limitBiconeOfUnique f).bicone.π

@[simp]

theorem CategoryTheory.Limits.biproductUniqueIso_hom {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : (CategoryTheory.Limits.biproductUniqueIso f).hom = CategoryTheory.Limits.biproduct.desc (CategoryTheory.Limits.limitBiconeOfUnique f).bicone.ι

def CategoryTheory.Limits.biproductUniqueIso {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [Unique J] (f : J → C) : ⨁ f ≅ f default

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

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

• 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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• fst : s.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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• snd : s.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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• inl : P ⟶ s.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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• inr : Q ⟶ s.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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• inl_fst : CategoryTheory.CategoryStruct.comp s.inl s.fst = CategoryTheory.CategoryStruct.id 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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• inl_snd : CategoryTheory.CategoryStruct.comp s.inl s.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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• inr_fst : CategoryTheory.CategoryStruct.comp s.inr s.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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

• inr_snd : CategoryTheory.CategoryStruct.comp s.inr s.snd = CategoryTheory.CategoryStruct.id 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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 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 inl ≫ fst = 𝟙 P, inl ≫ snd = 0, inr ≫ fst = 0, and inr ≫ snd = 𝟙 Q

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.inl_fst_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (self : CategoryTheory.Limits.BinaryBicone P Q) {Z : C} (h : P ⟶ Z) : CategoryTheory.CategoryStruct.comp self.inl (CategoryTheory.CategoryStruct.comp self.fst h) = h

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.inr_fst_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (self : CategoryTheory.Limits.BinaryBicone P Q) {Z : C} (h : P ⟶ Z) : CategoryTheory.CategoryStruct.comp self.inr (CategoryTheory.CategoryStruct.comp self.fst h) = CategoryTheory.CategoryStruct.comp 0 h

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.inr_snd_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (self : CategoryTheory.Limits.BinaryBicone P Q) {Z : C} (h : Q ⟶ Z) : CategoryTheory.CategoryStruct.comp self.inr (CategoryTheory.CategoryStruct.comp self.snd h) = h

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.inl_snd_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (self : CategoryTheory.Limits.BinaryBicone P Q) {Z : C} (h : Q ⟶ Z) : CategoryTheory.CategoryStruct.comp self.inl (CategoryTheory.CategoryStruct.comp self.snd h) = CategoryTheory.CategoryStruct.comp 0 h

def CategoryTheory.Limits.BinaryBicone.toCone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.Limits.Cone (CategoryTheory.Limits.pair P Q)

Extract the cone from a binary bicone.

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toCone_pt {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : (CategoryTheory.Limits.BinaryBicone.toCone c).pt = c.pt

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toCone_π_app_left {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : (CategoryTheory.Limits.BinaryBicone.toCone c).π.app { as := CategoryTheory.Limits.WalkingPair.left } = c.fst

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toCone_π_app_right {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : (CategoryTheory.Limits.BinaryBicone.toCone c).π.app { as := CategoryTheory.Limits.WalkingPair.right } = c.snd

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.binary_fan_fst_toCone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.Limits.BinaryFan.fst (CategoryTheory.Limits.BinaryBicone.toCone c) = c.fst

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.binary_fan_snd_toCone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.Limits.BinaryFan.snd (CategoryTheory.Limits.BinaryBicone.toCone c) = c.snd

def CategoryTheory.Limits.BinaryBicone.toCocone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.Limits.Cocone (CategoryTheory.Limits.pair P Q)

Extract the cocone from a binary bicone.

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toCocone_pt {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : (CategoryTheory.Limits.BinaryBicone.toCocone c).pt = c.pt

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toCocone_ι_app_left {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : (CategoryTheory.Limits.BinaryBicone.toCocone c).ι.app { as := CategoryTheory.Limits.WalkingPair.left } = c.inl

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toCocone_ι_app_right {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : (CategoryTheory.Limits.BinaryBicone.toCocone c).ι.app { as := CategoryTheory.Limits.WalkingPair.right } = c.inr

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.binary_cofan_inl_toCocone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.Limits.BinaryCofan.inl (CategoryTheory.Limits.BinaryBicone.toCocone c) = c.inl

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.binary_cofan_inr_toCocone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.Limits.BinaryCofan.inr (CategoryTheory.Limits.BinaryBicone.toCocone c) = c.inr

instance CategoryTheory.Limits.BinaryBicone.instIsSplitMonoPtInl {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.IsSplitMono c.inl

instance CategoryTheory.Limits.BinaryBicone.instIsSplitMonoPtInr {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.IsSplitMono c.inr

instance CategoryTheory.Limits.BinaryBicone.instIsSplitEpiPtFst {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.IsSplitEpi c.fst

instance CategoryTheory.Limits.BinaryBicone.instIsSplitEpiPtSnd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (c : CategoryTheory.Limits.BinaryBicone P Q) : CategoryTheory.IsSplitEpi c.snd

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toBicone_pt {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) : (CategoryTheory.Limits.BinaryBicone.toBicone b).pt = b.pt

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toBicone_ι {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) (j : CategoryTheory.Limits.WalkingPair) : CategoryTheory.Limits.Bicone.ι (CategoryTheory.Limits.BinaryBicone.toBicone b) j = CategoryTheory.Limits.WalkingPair.casesOn j b.inl b.inr

@[simp]

theorem CategoryTheory.Limits.BinaryBicone.toBicone_π {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) (j : CategoryTheory.Limits.WalkingPair) : CategoryTheory.Limits.Bicone.π (CategoryTheory.Limits.BinaryBicone.toBicone b) j = CategoryTheory.Limits.WalkingPair.casesOn j b.fst b.snd

def CategoryTheory.Limits.BinaryBicone.toBicone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)

Convert a BinaryBicone into a Bicone over a pair.

def CategoryTheory.Limits.BinaryBicone.toBiconeIsLimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.Bicone.toCone (CategoryTheory.Limits.BinaryBicone.toBicone b)) ≃ CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.BinaryBicone.toCone b)

A binary bicone is a limit cone if and only if the corresponding bicone is a limit cone.

def CategoryTheory.Limits.BinaryBicone.toBiconeIsColimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Bicone.toCocone (CategoryTheory.Limits.BinaryBicone.toBicone b)) ≃ CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryBicone.toCocone b)

A binary bicone is a colimit cocone if and only if the corresponding bicone is a colimit cocone.

@[simp]

theorem CategoryTheory.Limits.Bicone.toBinaryBicone_inl {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : (CategoryTheory.Limits.Bicone.toBinaryBicone b).inl = CategoryTheory.Limits.Bicone.ι b CategoryTheory.Limits.WalkingPair.left

@[simp]

theorem CategoryTheory.Limits.Bicone.toBinaryBicone_inr {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : (CategoryTheory.Limits.Bicone.toBinaryBicone b).inr = CategoryTheory.Limits.Bicone.ι b CategoryTheory.Limits.WalkingPair.right

@[simp]

theorem CategoryTheory.Limits.Bicone.toBinaryBicone_fst {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : (CategoryTheory.Limits.Bicone.toBinaryBicone b).fst = CategoryTheory.Limits.Bicone.π b CategoryTheory.Limits.WalkingPair.left

@[simp]

theorem CategoryTheory.Limits.Bicone.toBinaryBicone_pt {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : (CategoryTheory.Limits.Bicone.toBinaryBicone b).pt = b.pt

@[simp]

theorem CategoryTheory.Limits.Bicone.toBinaryBicone_snd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : (CategoryTheory.Limits.Bicone.toBinaryBicone b).snd = CategoryTheory.Limits.Bicone.π b CategoryTheory.Limits.WalkingPair.right

def CategoryTheory.Limits.Bicone.toBinaryBicone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : CategoryTheory.Limits.BinaryBicone X Y

Convert a Bicone over a function on WalkingPair to a BinaryBicone.

def CategoryTheory.Limits.Bicone.toBinaryBiconeIsLimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.BinaryBicone.toCone (CategoryTheory.Limits.Bicone.toBinaryBicone b)) ≃ CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.Bicone.toCone b)

A bicone over a pair is a limit cone if and only if the corresponding binary bicone is a limit cone.

def CategoryTheory.Limits.Bicone.toBinaryBiconeIsColimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryBicone.toCocone (CategoryTheory.Limits.Bicone.toBinaryBicone b)) ≃ CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Bicone.toCocone b)

A bicone over a pair is a colimit cocone if and only if the corresponding binary bicone is a colimit cocone.

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

• isLimit : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.BinaryBicone.toCone b)

  Structure witnessing that a binary bicone is a limit cone and a limit cocone.

• isColimit : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryBicone.toCocone b)

  Structure witnessing that a binary bicone is a limit cone and a limit cocone.

Structure witnessing that a binary bicone is a limit cone and a limit cocone.

def CategoryTheory.Limits.BinaryBicone.toBiconeIsBilimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.BinaryBicone X Y) : CategoryTheory.Limits.Bicone.IsBilimit (CategoryTheory.Limits.BinaryBicone.toBicone b) ≃ CategoryTheory.Limits.BinaryBicone.IsBilimit b

A binary bicone is a bilimit bicone if and only if the corresponding bicone is a bilimit.

def CategoryTheory.Limits.Bicone.toBinaryBiconeIsBilimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)) : CategoryTheory.Limits.BinaryBicone.IsBilimit (CategoryTheory.Limits.Bicone.toBinaryBicone b) ≃ CategoryTheory.Limits.Bicone.IsBilimit b

A bicone over a pair is a bilimit bicone if and only if the corresponding binary bicone is a bilimit.

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

• bicone : CategoryTheory.Limits.BinaryBicone P Q

  A bicone over P Q : C, which is both a limit cone and a colimit cocone.

• isBilimit : CategoryTheory.Limits.BinaryBicone.IsBilimit s.bicone

  A bicone over P Q : C, which is both a limit cone and a colimit cocone.

A bicone over P Q : C, which is both a limit cone and a colimit cocone.

class CategoryTheory.Limits.HasBinaryBiproduct {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P : C) (Q : C) : Prop

• mk' :: (
• exists_binary_biproduct : Nonempty (CategoryTheory.Limits.BinaryBiproductData P Q)

  HasBinaryBiproduct P Q expresses the mere existence of a bicone which is simultaneously a limit and a colimit of the diagram pair P Q.

• )

HasBinaryBiproduct P Q expresses the mere existence of a bicone which is simultaneously a limit and a colimit of the diagram pair P Q.

theorem CategoryTheory.Limits.HasBinaryBiproduct.mk {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} (d : CategoryTheory.Limits.BinaryBiproductData P Q) : CategoryTheory.Limits.HasBinaryBiproduct P Q

def CategoryTheory.Limits.getBinaryBiproductData {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P : C) (Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.BinaryBiproductData P Q

Use the axiom of choice to extract explicit BinaryBiproductData F from HasBinaryBiproduct F.

def CategoryTheory.Limits.BinaryBiproduct.bicone {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P : C) (Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.BinaryBicone P Q

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

def CategoryTheory.Limits.BinaryBiproduct.isBilimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P : C) (Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.BinaryBicone.IsBilimit (CategoryTheory.Limits.BinaryBiproduct.bicone P Q)

BinaryBiproduct.bicone P Q is a limit bicone.

def CategoryTheory.Limits.BinaryBiproduct.isLimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P : C) (Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.BinaryBicone.toCone (CategoryTheory.Limits.BinaryBiproduct.bicone P Q))

BinaryBiproduct.bicone P Q is a limit cone.

def CategoryTheory.Limits.BinaryBiproduct.isColimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P : C) (Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryBicone.toCocone (CategoryTheory.Limits.BinaryBiproduct.bicone P Q))

BinaryBiproduct.bicone P Q is a colimit cocone.

class CategoryTheory.Limits.HasBinaryBiproducts (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] : Prop

• has_binary_biproduct : ∀ (P Q : C), CategoryTheory.Limits.HasBinaryBiproduct P Q

HasBinaryBiproducts C represents the existence of a bicone which is simultaneously a limit and a colimit of the diagram pair P Q, for every P Q : C.

theorem CategoryTheory.Limits.hasBinaryBiproducts_of_finite_biproducts (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteBiproducts C] : CategoryTheory.Limits.HasBinaryBiproducts C

A category with finite biproducts has binary biproducts.

This is not an instance as typically in concrete categories there will be an alternative construction with nicer definitional properties.

instance CategoryTheory.Limits.HasBinaryBiproduct.hasLimit_pair {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.pair P Q)

instance CategoryTheory.Limits.HasBinaryBiproduct.hasColimit_pair {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P : C} {Q : C} [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.pair P Q)

instance CategoryTheory.Limits.hasBinaryProducts_of_hasBinaryBiproducts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBinaryBiproducts C] : CategoryTheory.Limits.HasBinaryProducts C

instance CategoryTheory.Limits.hasBinaryCoproducts_of_hasBinaryBiproducts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBinaryBiproducts C] : CategoryTheory.Limits.HasBinaryCoproducts C

def CategoryTheory.Limits.biprodIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) (Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] : X ⨯ Y ≅ X ⨿ Y

The isomorphism between the specified binary product and the specified binary coproduct for a pair for a binary biproduct.

@[inline, reducible]

abbrev CategoryTheory.Limits.biprod {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) (Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] : C

An arbitrary choice of biproduct of a pair of objects.

def CategoryTheory.Limits.«term_⊞_» : Lean.TrailingParserDescr

An arbitrary choice of biproduct of a pair of objects.

@[inline, reducible]

abbrev CategoryTheory.Limits.biprod.fst {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : X ⊞ Y ⟶ X

The projection onto the first summand of a binary biproduct.

@[inline, reducible]

abbrev CategoryTheory.Limits.biprod.snd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : X ⊞ Y ⟶ Y

The projection onto the second summand of a binary biproduct.

@[inline, reducible]

abbrev CategoryTheory.Limits.biprod.inl {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : X ⟶ X ⊞ Y

The inclusion into the first summand of a binary biproduct.

@[inline, reducible]

abbrev CategoryTheory.Limits.biprod.inr {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : Y ⟶ X ⊞ Y

The inclusion into the second summand of a binary biproduct.

@[simp]

theorem CategoryTheory.Limits.BinaryBiproduct.bicone_fst {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : (CategoryTheory.Limits.BinaryBiproduct.bicone X Y).fst = CategoryTheory.Limits.biprod.fst

@[simp]

theorem CategoryTheory.Limits.BinaryBiproduct.bicone_snd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : (CategoryTheory.Limits.BinaryBiproduct.bicone X Y).snd = CategoryTheory.Limits.biprod.snd

@[simp]

theorem CategoryTheory.Limits.BinaryBiproduct.bicone_inl {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : (CategoryTheory.Limits.BinaryBiproduct.bicone X Y).inl = CategoryTheory.Limits.biprod.inl

@[simp]

theorem CategoryTheory.Limits.BinaryBiproduct.bicone_inr {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : (CategoryTheory.Limits.BinaryBiproduct.bicone X Y).inr = CategoryTheory.Limits.biprod.inr

theorem CategoryTheory.Limits.biprod.inl_fst_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] {Z : C} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.fst h) = h

theorem CategoryTheory.Limits.biprod.inl_fst {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.fst = CategoryTheory.CategoryStruct.id X