# Categorical (co)products #

This file defines (co)products as special cases of (co)limits.

A product is the categorical generalization of the object Π i, f i where f : ι → C. It is a limit cone over the diagram formed by f, implemented by converting f into a functor Discrete ι ⥤ C.

A coproduct is the dual concept.

## Main definitions #

• a Fan is a cone over a discrete category
• Fan.mk constructs a fan from an indexed collection of maps
• a Pi is a limit (Discrete.functor f)

Each of these has a dual.

## Implementation notes #

As with the other special shapes in the limits library, all the definitions here are given as abbreviations of the general statements for limits, so all the simp lemmas and theorems about general limits can be used.

@[reducible, inline]
abbrev CategoryTheory.Limits.Fan {β : Type w} {C : Type u} (f : βC) :
Type (max (max w u) v)

A fan over f : β → C consists of a collection of maps from an object P to every f b.

Equations
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.Cofan {β : Type w} {C : Type u} (f : βC) :
Type (max (max w u) v)

A cofan over f : β → C consists of a collection of maps from every f b to an object P.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Fan.mk_π_app {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → P f b) (X : ) :
.app X = p X.as
@[simp]
theorem CategoryTheory.Limits.Fan.mk_pt {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → P f b) :
.pt = P
def CategoryTheory.Limits.Fan.mk {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → P f b) :

A fan over f : β → C consists of a collection of maps from an object P to every f b.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Cofan.mk_pt {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → f b P) :
.pt = P
@[simp]
theorem CategoryTheory.Limits.Cofan.mk_ι_app {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → f b P) (X : ) :
.app X = p X.as
def CategoryTheory.Limits.Cofan.mk {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → f b P) :

A cofan over f : β → C consists of a collection of maps from every f b to an object P.

Equations
Instances For
def CategoryTheory.Limits.Fan.proj {β : Type w} {C : Type u} {f : βC} (p : ) (j : β) :
p.pt f j

Get the jth "projection" in the fan. (Note that the initial letter of proj matches the greek letter in Cone.π.)

Equations
• p.proj j = p.app { as := j }
Instances For
def CategoryTheory.Limits.Cofan.inj {β : Type w} {C : Type u} {f : βC} (p : ) (j : β) :
f j p.pt

Get the jth "injection" in the cofan. (Note that the initial letter of inj matches the greek letter in Cocone.ι.)

Equations
• p.inj j = p.app { as := j }
Instances For
@[simp]
theorem CategoryTheory.Limits.fan_mk_proj {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → P f b) (j : β) :
.proj j = p j
@[simp]
theorem CategoryTheory.Limits.cofan_mk_inj {β : Type w} {C : Type u} {f : βC} (P : C) (p : (b : β) → f b P) (j : β) :
.inj j = p j
@[reducible, inline]
abbrev CategoryTheory.Limits.HasProduct {β : Type w} {C : Type u} (f : βC) :

An abbreviation for HasLimit (Discrete.functor f).

Equations
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.HasCoproduct {β : Type w} {C : Type u} (f : βC) :

An abbreviation for HasColimit (Discrete.functor f).

Equations
Instances For
theorem CategoryTheory.Limits.hasCoproduct_of_equiv_of_iso {β : Type w} {α : Type w₂} {C : Type u} (f : αC) (g : βC) (e : β α) (iso : (j : β) → g j f (e j)) :
theorem CategoryTheory.Limits.hasProduct_of_equiv_of_iso {β : Type w} {α : Type w₂} {C : Type u} (f : αC) (g : βC) (e : β α) (iso : (j : β) → g j f (e j)) :
@[simp]
theorem CategoryTheory.Limits.mkFanLimit_lift {β : Type w} {C : Type u} {f : βC} (t : ) (lift : (s : ) → s.pt t.pt) (fac : autoParam (∀ (s : ) (j : β), CategoryTheory.CategoryStruct.comp (lift s) (t.proj j) = s.proj j) _auto✝) (uniq : autoParam (∀ (s : ) (m : s.pt t.pt), (∀ (j : β), CategoryTheory.CategoryStruct.comp m (t.proj j) = s.proj j)m = lift s) _auto✝) (s : ) :
(CategoryTheory.Limits.mkFanLimit t lift fac uniq).lift s = lift s
def CategoryTheory.Limits.mkFanLimit {β : Type w} {C : Type u} {f : βC} (t : ) (lift : (s : ) → s.pt t.pt) (fac : autoParam (∀ (s : ) (j : β), CategoryTheory.CategoryStruct.comp (lift s) (t.proj j) = s.proj j) _auto✝) (uniq : autoParam (∀ (s : ) (m : s.pt t.pt), (∀ (j : β), CategoryTheory.CategoryStruct.comp m (t.proj j) = s.proj j)m = lift s) _auto✝) :

Make a fan f into a limit fan by providing lift, fac, and uniq -- just a convenience lemma to avoid having to go through Discrete

Equations
Instances For
def CategoryTheory.Limits.Fan.IsLimit.desc {β : Type w} {C : Type u} {F : βC} {c : } (hc : ) {A : C} (f : (i : β) → A F i) :
A c.pt

Constructor for morphisms to the point of a limit fan.

Equations
• = hc.lift
Instances For
@[simp]
theorem CategoryTheory.Limits.Fan.IsLimit.fac_assoc {β : Type w} {C : Type u} {F : βC} {c : } (hc : ) {A : C} (f : (i : β) → A F i) (i : β) {Z : C} (h : F i Z) :
=
@[simp]
theorem CategoryTheory.Limits.Fan.IsLimit.fac {β : Type w} {C : Type u} {F : βC} {c : } (hc : ) {A : C} (f : (i : β) → A F i) (i : β) :
= f i
theorem CategoryTheory.Limits.Fan.IsLimit.hom_ext {C : Type u} {I : Type u_1} {F : IC} {c : } (hc : ) {A : C} (f : A c.pt) (g : A c.pt) (h : ∀ (i : I), CategoryTheory.CategoryStruct.comp f (c.proj i) = CategoryTheory.CategoryStruct.comp g (c.proj i)) :
f = g
@[simp]
theorem CategoryTheory.Limits.mkCofanColimit_desc {β : Type w} {C : Type u} {f : βC} (s : ) (desc : (t : ) → s.pt t.pt) (fac : autoParam (∀ (t : ) (j : β), CategoryTheory.CategoryStruct.comp (s.inj j) (desc t) = t.inj j) _auto✝) (uniq : autoParam (∀ (t : ) (m : s.pt t.pt), (∀ (j : β), CategoryTheory.CategoryStruct.comp (s.inj j) m = t.inj j)m = desc t) _auto✝) (t : ) :
(CategoryTheory.Limits.mkCofanColimit s desc fac uniq).desc t = desc t
def CategoryTheory.Limits.mkCofanColimit {β : Type w} {C : Type u} {f : βC} (s : ) (desc : (t : ) → s.pt t.pt) (fac : autoParam (∀ (t : ) (j : β), CategoryTheory.CategoryStruct.comp (s.inj j) (desc t) = t.inj j) _auto✝) (uniq : autoParam (∀ (t : ) (m : s.pt t.pt), (∀ (j : β), CategoryTheory.CategoryStruct.comp (s.inj j) m = t.inj j)m = desc t) _auto✝) :

Make a cofan f into a colimit cofan by providing desc, fac, and uniq -- just a convenience lemma to avoid having to go through Discrete

Equations
Instances For
def CategoryTheory.Limits.Cofan.IsColimit.desc {β : Type w} {C : Type u} {F : βC} {c : } (hc : ) {A : C} (f : (i : β) → F i A) :
c.pt A

Constructor for morphisms from the point of a colimit cofan.

Equations
• = hc.desc
Instances For
@[simp]
theorem CategoryTheory.Limits.Cofan.IsColimit.fac_assoc {β : Type w} {C : Type u} {F : βC} {c : } (hc : ) {A : C} (f : (i : β) → F i A) (i : β) {Z : C} (h : A Z) :
@[simp]
theorem CategoryTheory.Limits.Cofan.IsColimit.fac {β : Type w} {C : Type u} {F : βC} {c : } (hc : ) {A : C} (f : (i : β) → F i A) (i : β) :
= f i
theorem CategoryTheory.Limits.Cofan.IsColimit.hom_ext {C : Type u} {I : Type u_1} {F : IC} {c : } (hc : ) {A : C} (f : c.pt A) (g : c.pt A) (h : ∀ (i : I), CategoryTheory.CategoryStruct.comp (c.inj i) f = CategoryTheory.CategoryStruct.comp (c.inj i) g) :
f = g
@[reducible, inline]

An abbreviation for HasLimitsOfShape (Discrete f).

Equations
Instances For
@[reducible, inline]

An abbreviation for HasColimitsOfShape (Discrete f).

Equations
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.piObj {β : Type w} {C : Type u} (f : βC) :
C

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

Equations
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.sigmaObj {β : Type w} {C : Type u} (f : βC) :
C

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

Equations
Instances For

notation for categorical products. We need to avoid conflict with Finset.prod.

Equations
Instances For

notation for categorical coproducts

Equations
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.Pi.π {β : Type w} {C : Type u} (f : βC) (b : β) :
∏ᶜ f f b

The b-th projection from the pi object over f has the form ∏ᶜ f ⟶ f b.

Equations
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.Sigma.ι {β : Type w} {C : Type u} (f : βC) (b : β) :
f b f

The b-th inclusion into the sigma object over f has the form f b ⟶ ∐ f.

Equations
Instances For
theorem CategoryTheory.Limits.Pi.hom_ext_iff {β : Type w} {C : Type u} {f : βC} {X : C} {g₁ : X ∏ᶜ f} {g₂ : X ∏ᶜ f} :
g₁ = g₂ ∀ (b : β),
theorem CategoryTheory.Limits.Pi.hom_ext {β : Type w} {C : Type u} {f : βC} {X : C} (g₁ : X ∏ᶜ f) (g₂ : X ∏ᶜ f) (h : ∀ (b : β), ) :
g₁ = g₂
theorem CategoryTheory.Limits.Sigma.hom_ext_iff {β : Type w} {C : Type u} {f : βC} {X : C} {g₁ : f X} {g₂ : f X} :
g₁ = g₂ ∀ (b : β),
theorem CategoryTheory.Limits.Sigma.hom_ext {β : Type w} {C : Type u} {f : βC} {X : C} (g₁ : f X) (g₂ : f X) (h : ∀ (b : β), ) :
g₁ = g₂
def CategoryTheory.Limits.productIsProduct {β : Type w} {C : Type u} (f : βC) :

The fan constructed of the projections from the product is limiting.

Equations
• One or more equations did not get rendered due to their size.
Instances For
def CategoryTheory.Limits.coproductIsCoproduct {β : Type w} {C : Type u} (f : βC) :

The cofan constructed of the inclusions from the coproduct is colimiting.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.π_comp_eqToHom_assoc {C : Type u} {J : Type u_1} (f : JC) {j : J} {j' : J} (w : j = j') {Z : C} (h : f j' Z) :
@[simp]
theorem CategoryTheory.Limits.Pi.π_comp_eqToHom {C : Type u} {J : Type u_1} (f : JC) {j : J} {j' : J} (w : j = j') :
@[simp]
theorem CategoryTheory.Limits.Sigma.eqToHom_comp_ι_assoc {C : Type u} {J : Type u_1} (f : JC) {j : J} {j' : J} (w : j = j') {Z : C} (h : f Z) :
@[simp]
theorem CategoryTheory.Limits.Sigma.eqToHom_comp_ι {C : Type u} {J : Type u_1} (f : JC) {j : J} {j' : J} (w : j = j') :
@[reducible, inline]
abbrev CategoryTheory.Limits.Pi.lift {β : Type w} {C : Type u} {f : βC} {P : C} (p : (b : β) → P f b) :

A collection of morphisms P ⟶ f b induces a morphism P ⟶ ∏ᶜ f.

Equations
Instances For
theorem CategoryTheory.Limits.Pi.lift_π {C : Type u} {β : Type w} {f : βC} {P : C} (p : (b : β) → P f b) (b : β) :
@[reducible, inline]
abbrev CategoryTheory.Limits.Sigma.desc {β : Type w} {C : Type u} {f : βC} {P : C} (p : (b : β) → f b P) :
f P

A collection of morphisms f b ⟶ P induces a morphism ∐ f ⟶ P.

Equations
Instances For
theorem CategoryTheory.Limits.Sigma.ι_desc {C : Type u} {β : Type w} {f : βC} {P : C} (p : (b : β) → f b P) (b : β) :
instance CategoryTheory.Limits.instIsIsoDescι {β : Type w} {C : Type u} {f : βC} :
Equations
• =
@[simp]
theorem CategoryTheory.Limits.Cofan.ext_hom_hom {β : Type w} {C : Type u} {f : βC} {c₁ : } {c₂ : } (e : c₁.pt c₂.pt) (w : autoParam (∀ (b : β), CategoryTheory.CategoryStruct.comp (c₁.inj b) e.hom = c₂.inj b) _auto✝) :
.hom.hom = e.hom
@[simp]
theorem CategoryTheory.Limits.Cofan.ext_inv_hom {β : Type w} {C : Type u} {f : βC} {c₁ : } {c₂ : } (e : c₁.pt c₂.pt) (w : autoParam (∀ (b : β), CategoryTheory.CategoryStruct.comp (c₁.inj b) e.hom = c₂.inj b) _auto✝) :
.inv.hom = e.inv
def CategoryTheory.Limits.Cofan.ext {β : Type w} {C : Type u} {f : βC} {c₁ : } {c₂ : } (e : c₁.pt c₂.pt) (w : autoParam (∀ (b : β), CategoryTheory.CategoryStruct.comp (c₁.inj b) e.hom = c₂.inj b) _auto✝) :
c₁ c₂

A version of Cocones.ext for Cofans.

Equations
Instances For
def CategoryTheory.Limits.Cofan.isColimitOfIsIsoSigmaDesc {β : Type w} {C : Type u} {f : βC} (c : ) [hc : ] :

A cofan c on f such that the induced map ∐ f ⟶ c.pt is an iso, is a coproduct.

Equations
• One or more equations did not get rendered due to their size.
Instances For
theorem CategoryTheory.Limits.Cofan.isColimit_iff_isIso_sigmaDesc {β : Type w} {C : Type u} {f : βC} (c : ) :
def CategoryTheory.Limits.Cofan.isColimitTrans {α : Type w₂} {C : Type u} {X : αC} (c : ) (hc : ) {β : αType u_1} {Y : (a : α) → β aC} (π : (a : α) → (b : β a) → Y a b X a) (hs : (a : α) → ) :
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Cofan.mk c.pt fun (x : (a : α) × β a) => match x with | a, b => CategoryTheory.CategoryStruct.comp (π a b) (c.inj a))

A coproduct of coproducts is a coproduct

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]
abbrev CategoryTheory.Limits.Pi.map {β : Type w} {C : Type u} {f : βC} {g : βC} (p : (b : β) → f b g b) :

Construct a morphism between categorical products (indexed by the same type) from a family of morphisms between the factors.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.map_id {α : Type w₂} {C : Type u} {f : αC} :
(CategoryTheory.Limits.Pi.map fun (a : α) => ) =
theorem CategoryTheory.Limits.Pi.map_comp_map {α : Type w₂} {C : Type u} {f : αC} {g : αC} {h : αC} (q : (a : α) → f a g a) (q' : (a : α) → g a h a) :
instance CategoryTheory.Limits.Pi.map_mono {β : Type w} {C : Type u} {f : βC} {g : βC} (p : (b : β) → f b g b) [∀ (i : β), CategoryTheory.Mono (p i)] :
Equations
• =
def CategoryTheory.Limits.Pi.map' {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} (p : βα) (q : (b : β) → f (p b) g b) :

Construct a morphism between categorical products from a family of morphisms between the factors.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.map'_comp_π_assoc {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} (p : βα) (q : (b : β) → f (p b) g b) (b : β) {Z : C} (h : g b Z) :
@[simp]
theorem CategoryTheory.Limits.Pi.map'_comp_π {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} (p : βα) (q : (b : β) → f (p b) g b) (b : β) :
theorem CategoryTheory.Limits.Pi.map'_id_id {α : Type w₂} {C : Type u} {f : αC} :
(CategoryTheory.Limits.Pi.map' id fun (a : α) => ) =
@[simp]
theorem CategoryTheory.Limits.Pi.map'_id {α : Type w₂} {C : Type u} {f : αC} {g : αC} (p : (b : α) → f b g b) :
theorem CategoryTheory.Limits.Pi.map'_comp_map' {β : Type w} {α : Type w₂} {γ : Type w₃} {C : Type u} {f : αC} {g : βC} {h : γC} (p : βα) (p' : γβ) (q : (b : β) → f (p b) g b) (q' : (c : γ) → g (p' c) h c) :
= CategoryTheory.Limits.Pi.map' (p p') fun (c : γ) => CategoryTheory.CategoryStruct.comp (q (p' c)) (q' c)
theorem CategoryTheory.Limits.Pi.map'_comp_map {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} {h : βC} (p : βα) (q : (b : β) → f (p b) g b) (q' : (b : β) → g b h b) :
theorem CategoryTheory.Limits.Pi.map_comp_map' {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : αC} {h : βC} (p : βα) (q : (a : α) → f a g a) (q' : (b : β) → g (p b) h b) :
theorem CategoryTheory.Limits.Pi.map'_eq {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} {p : βα} {p' : βα} {q : (b : β) → f (p b) g b} {q' : (b : β) → f (p' b) g b} (hp : p = p') (hq : ∀ (b : β), = q' b) :
@[reducible, inline]
abbrev CategoryTheory.Limits.Pi.mapIso {β : Type w} {C : Type u} {f : βC} {g : βC} (p : (b : β) → f b g b) :

Construct an isomorphism between categorical products (indexed by the same type) from a family of isomorphisms between the factors.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.cone_pt {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] :
= ∏ᶜ fun (j : α) => X.obj { as := j }
@[simp]
theorem CategoryTheory.Limits.Pi.cone_π {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] :
= CategoryTheory.Discrete.natTrans fun (x : ) => CategoryTheory.Limits.Pi.π (fun (j : α) => X.obj { as := j }) x.as
def CategoryTheory.Limits.Pi.cone {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] :

A limit cone for X : Discrete α ⥤ C that is given by ∏ᶜ (fun j => X.obj (Discrete.mk j)).

Equations
• One or more equations did not get rendered due to their size.
Instances For
def CategoryTheory.Limits.productIsProduct' {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] :

The cone Pi.cone X is a limit cone.

Equations
Instances For
def CategoryTheory.Limits.Pi.isoLimit {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] :
(∏ᶜ fun (j : α) => X.obj { as := j })

The isomorphism ∏ᶜ (fun j => X.obj (Discrete.mk j)) ≅ limit X.

Equations
• = .conePointUniqueUpToIso
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.isoLimit_inv_π_assoc {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] (j : α) {Z : C} (h : X.obj { as := j } Z) :
@[simp]
theorem CategoryTheory.Limits.Pi.isoLimit_inv_π {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] (j : α) :
@[simp]
theorem CategoryTheory.Limits.Pi.isoLimit_hom_π_assoc {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] (j : α) {Z : C} (h : X.obj { as := j } Z) :
= CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Pi.π (fun (j : α) => X.obj { as := j }) j) h
@[simp]
theorem CategoryTheory.Limits.Pi.isoLimit_hom_π {α : Type w₂} {C : Type u} (X : ) [CategoryTheory.Limits.HasProduct fun (j : α) => X.obj { as := j }] (j : α) :
@[reducible, inline]
abbrev CategoryTheory.Limits.Sigma.map {β : Type w} {C : Type u} {f : βC} {g : βC} (p : (b : β) → f b g b) :
f g

Construct a morphism between categorical coproducts (indexed by the same type) from a family of morphisms between the factors.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Sigma.map_id {α : Type w₂} {C : Type u} {f : αC} :
theorem CategoryTheory.Limits.Sigma.map_comp_map {α : Type w₂} {C : Type u} {f : αC} {g : αC} {h : αC} (q : (a : α) → f a g a) (q' : (a : α) → g a h a) :
instance CategoryTheory.Limits.Sigma.map_epi {β : Type w} {C : Type u} {f : βC} {g : βC} (p : (b : β) → f b g b) [∀ (i : β), CategoryTheory.Epi (p i)] :
Equations
• =
def CategoryTheory.Limits.Sigma.map' {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} (p : αβ) (q : (a : α) → f a g (p a)) :
f g

Construct a morphism between categorical coproducts from a family of morphisms between the factors.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Sigma.ι_comp_map'_assoc {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} (p : αβ) (q : (a : α) → f a g (p a)) (a : α) {Z : C} (h : g Z) :
@[simp]
theorem CategoryTheory.Limits.Sigma.ι_comp_map' {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} (p : αβ) (q : (a : α) → f a g (p a)) (a : α) :
theorem CategoryTheory.Limits.Sigma.map'_id_id {α : Type w₂} {C : Type u} {f : αC} :
(CategoryTheory.Limits.Sigma.map' id fun (a : α) => ) =
@[simp]
theorem CategoryTheory.Limits.Sigma.map'_id {α : Type w₂} {C : Type u} {f : αC} {g : αC} (p : (b : α) → f b g b) :
theorem CategoryTheory.Limits.Sigma.map'_comp_map' {β : Type w} {α : Type w₂} {γ : Type w₃} {C : Type u} {f : αC} {g : βC} {h : γC} (p : αβ) (p' : βγ) (q : (a : α) → f a g (p a)) (q' : (b : β) → g b h (p' b)) :
theorem CategoryTheory.Limits.Sigma.map'_comp_map {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} {h : βC} (p : αβ) (q : (a : α) → f a g (p a)) (q' : (b : β) → g b h b) :
theorem CategoryTheory.Limits.Sigma.map_comp_map' {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : αC} {h : βC} (p : αβ) (q : (a : α) → f a g a) (q' : (a : α) → g a h (p a)) :
theorem CategoryTheory.Limits.Sigma.map'_eq {β : Type w} {α : Type w₂} {C : Type u} {f : αC} {g : βC} {p : αβ} {p' : αβ} {q : (a : α) → f a g (p a)} {q' : (a : α) → f a g (p' a)} (hp : p = p') (hq : ∀ (a : α), = q' a) :
@[reducible, inline]
abbrev CategoryTheory.Limits.Sigma.mapIso {β : Type w} {C : Type u} {f : βC} {g : βC} (p : (b : β) → f b g b) :
f g

Construct an isomorphism between categorical coproducts (indexed by the same type) from a family of isomorphisms between the factors.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.whiskerEquiv_hom {C : Type u} {J : Type u_1} {K : Type u_2} {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) :
.hom = CategoryTheory.Limits.Pi.map' e.symm fun (k : K) => CategoryTheory.CategoryStruct.comp (w (e.symm k)).inv
@[simp]
theorem CategoryTheory.Limits.Pi.whiskerEquiv_inv {C : Type u} {J : Type u_1} {K : Type u_2} {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) :
.inv = CategoryTheory.Limits.Pi.map' e fun (j : J) => (w j).hom
def CategoryTheory.Limits.Pi.whiskerEquiv {C : Type u} {J : Type u_1} {K : Type u_2} {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) :

Two products which differ by an equivalence in the indexing type, and up to isomorphism in the factors, are isomorphic.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem CategoryTheory.Limits.Sigma.whiskerEquiv_hom {C : Type u} {J : Type u_1} {K : Type u_2} {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) :
.hom = CategoryTheory.Limits.Sigma.map' e fun (j : J) => (w j).inv
@[simp]
theorem CategoryTheory.Limits.Sigma.whiskerEquiv_inv {C : Type u} {J : Type u_1} {K : Type u_2} {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) :
.inv = CategoryTheory.Limits.Sigma.map' e.symm fun (k : K) => CategoryTheory.CategoryStruct.comp (w (e.symm k)).hom
def CategoryTheory.Limits.Sigma.whiskerEquiv {C : Type u} {J : Type u_1} {K : Type u_2} {f : JC} {g : KC} (e : J K) (w : (j : J) → g (e j) f j) :
f g

Two coproducts which differ by an equivalence in the indexing type, and up to isomorphism in the factors, are isomorphic.

Equations
• One or more equations did not get rendered due to their size.
Instances For
instance CategoryTheory.Limits.instHasProductSigmaFstSndOfPiObj {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasProduct fun (i : ι) => ∏ᶜ g i] :
CategoryTheory.Limits.HasProduct fun (p : (i : ι) × f i) => g p.fst p.snd
Equations
• =
@[simp]
theorem CategoryTheory.Limits.piPiIso_inv {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasProduct fun (i : ι) => ∏ᶜ g i] :
.inv = CategoryTheory.Limits.Pi.lift fun (i : ι) => CategoryTheory.Limits.Pi.lift fun (x : f i) => CategoryTheory.Limits.Pi.π (fun (p : (i : ι) × f i) => g p.fst p.snd) i, x
@[simp]
theorem CategoryTheory.Limits.piPiIso_hom {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasProduct fun (i : ι) => ∏ᶜ g i] :
.hom = CategoryTheory.Limits.Pi.lift fun (x : (i : ι) × f i) => match x with | i, x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Pi.π (fun (i : ι) => ∏ᶜ g i) i) (CategoryTheory.Limits.Pi.π (g i) x)
def CategoryTheory.Limits.piPiIso {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasProduct fun (i : ι) => ∏ᶜ g i] :
(∏ᶜ fun (i : ι) => ∏ᶜ g i) ∏ᶜ fun (p : (i : ι) × f i) => g p.fst p.snd

An iterated product is a product over a sigma type.

Equations
• One or more equations did not get rendered due to their size.
Instances For
instance CategoryTheory.Limits.instHasCoproductSigmaFstSndOfSigmaObj {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasCoproduct fun (i : ι) => g i] :
CategoryTheory.Limits.HasCoproduct fun (p : (i : ι) × f i) => g p.fst p.snd
Equations
• =
@[simp]
theorem CategoryTheory.Limits.sigmaSigmaIso_hom {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasCoproduct fun (i : ι) => g i] :
.hom = CategoryTheory.Limits.Sigma.desc fun (i : ι) => CategoryTheory.Limits.Sigma.desc fun (x : f i) => CategoryTheory.Limits.Sigma.ι (fun (p : (i : ι) × f i) => g p.fst p.snd) i, x
@[simp]
theorem CategoryTheory.Limits.sigmaSigmaIso_inv {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasCoproduct fun (i : ι) => g i] :
.inv = CategoryTheory.Limits.Sigma.desc fun (x : (i : ι) × f i) => match x with | i, x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Sigma.ι (g i) x) (CategoryTheory.Limits.Sigma.ι (fun (i : ι) => g i) i)
def CategoryTheory.Limits.sigmaSigmaIso {C : Type u} {ι : Type u_1} (f : ιType u_2) (g : (i : ι) → f iC) [∀ (i : ι), ] [CategoryTheory.Limits.HasCoproduct fun (i : ι) => g i] :
( fun (i : ι) => g i) fun (p : (i : ι) × f i) => g p.fst p.snd

An iterated coproduct is a coproduct over a sigma type.

Equations
• One or more equations did not get rendered due to their size.
Instances For
def CategoryTheory.Limits.piComparison {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasProduct fun (b : β) => G.obj (f b)] :
G.obj (∏ᶜ f) ∏ᶜ fun (b : β) => G.obj (f b)

The comparison morphism for the product of f. This is an iso iff G preserves the product of f, see PreservesProduct.ofIsoComparison.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.piComparison_comp_π_assoc {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasProduct fun (b : β) => G.obj (f b)] (b : β) {Z : D} (h : G.obj (f b) Z) :
@[simp]
theorem CategoryTheory.Limits.piComparison_comp_π {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasProduct fun (b : β) => G.obj (f b)] (b : β) :
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Pi.π (fun (b : β) => G.obj (f b)) b) = G.map
@[simp]
theorem CategoryTheory.Limits.map_lift_piComparison_assoc {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasProduct fun (b : β) => G.obj (f b)] (P : C) (g : (j : β) → P f j) {Z : D} (h : (∏ᶜ fun (b : β) => G.obj (f b)) Z) :
@[simp]
theorem CategoryTheory.Limits.map_lift_piComparison {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasProduct fun (b : β) => G.obj (f b)] (P : C) (g : (j : β) → P f j) :
= CategoryTheory.Limits.Pi.lift fun (j : β) => G.map (g j)
def CategoryTheory.Limits.sigmaComparison {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasCoproduct fun (b : β) => G.obj (f b)] :
( fun (b : β) => G.obj (f b)) G.obj ( f)

The comparison morphism for the coproduct of f. This is an iso iff G preserves the coproduct of f, see PreservesCoproduct.ofIsoComparison.

Equations
Instances For
@[simp]
theorem CategoryTheory.Limits.ι_comp_sigmaComparison_assoc {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasCoproduct fun (b : β) => G.obj (f b)] (b : β) {Z : D} (h : G.obj ( f) Z) :
@[simp]
theorem CategoryTheory.Limits.ι_comp_sigmaComparison {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasCoproduct fun (b : β) => G.obj (f b)] (b : β) :
@[simp]
theorem CategoryTheory.Limits.sigmaComparison_map_desc_assoc {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasCoproduct fun (b : β) => G.obj (f b)] (P : C) (g : (j : β) → f j P) {Z : D} (h : G.obj P Z) :
@[simp]
theorem CategoryTheory.Limits.sigmaComparison_map_desc {β : Type w} {C : Type u} {D : Type u₂} [] (G : ) (f : βC) [CategoryTheory.Limits.HasCoproduct fun (b : β) => G.obj (f b)] (P : C) (g : (j : β) → f j P) :
= CategoryTheory.Limits.Sigma.desc fun (j : β) => G.map (g j)
@[reducible, inline]

An abbreviation for Π J, HasLimitsOfShape (Discrete J) C

Equations
Instances For
@[reducible, inline]

An abbreviation for Π J, HasColimitsOfShape (Discrete J) C

Equations
Instances For
theorem CategoryTheory.Limits.hasProducts_of_limit_fans {C : Type u} (lf : {J : Type w} → (f : JC) → ) (lf_isLimit : {J : Type w} → (f : JC) → ) :

(Co)products over a type with a unique term.

@[simp]
theorem CategoryTheory.Limits.limitConeOfUnique_cone_pt {β : Type w} {C : Type u} [] (f : βC) :
.pt = f default
@[simp]
theorem CategoryTheory.Limits.limitConeOfUnique_isLimit_lift {β : Type w} {C : Type u} [] (f : βC) :
.isLimit.lift s = s.app default
@[simp]
theorem CategoryTheory.Limits.limitConeOfUnique_cone_π {β : Type w} {C : Type u} [] (f : βC) :
= CategoryTheory.Discrete.natTrans fun (x : ) => match x with | { as := j } =>
def CategoryTheory.Limits.limitConeOfUnique {β : Type w} {C : Type u} [] (f : βC) :

The limit cone for the product over an index type with exactly one term.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[instance 100]
instance CategoryTheory.Limits.hasProduct_unique {β : Type w} {C : Type u} [] (f : βC) :
Equations
• =
@[simp]
theorem CategoryTheory.Limits.productUniqueIso_hom {β : Type w} {C : Type u} [] (f : βC) :
@[simp]
theorem CategoryTheory.Limits.productUniqueIso_inv {β : Type w} {C : Type u} [] (f : βC) :
def CategoryTheory.Limits.productUniqueIso {β : Type w} {C : Type u} [] (f : βC) :
∏ᶜ f f default

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

Equations
• = .conePointUniqueUpToIso .isLimit
Instances For
@[simp]
theorem CategoryTheory.Limits.colimitCoconeOfUnique_isColimit_desc {β : Type w} {C : Type u} [] (f : βC) :
.isColimit.desc s = s.app default
@[simp]
theorem CategoryTheory.Limits.colimitCoconeOfUnique_cocone_pt {β : Type w} {C : Type u} [] (f : βC) :
.cocone.pt = f default
@[simp]
theorem CategoryTheory.Limits.colimitCoconeOfUnique_cocone_ι {β : Type w} {C : Type u} [] (f : βC) :
.cocone = CategoryTheory.Discrete.natTrans fun (x : ) => match x with | { as := j } =>
def CategoryTheory.Limits.colimitCoconeOfUnique {β : Type w} {C : Type u} [] (f : βC) :

The colimit cocone for the coproduct over an index type with exactly one term.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[instance 100]
instance CategoryTheory.Limits.hasCoproduct_unique {β : Type w} {C : Type u} [] (f : βC) :
Equations
• =
@[simp]
theorem CategoryTheory.Limits.coproductUniqueIso_hom {β : Type w} {C : Type u} [] (f : βC) :
@[simp]
theorem CategoryTheory.Limits.coproductUniqueIso_inv {β : Type w} {C : Type u} [] (f : βC) :
def CategoryTheory.Limits.coproductUniqueIso {β : Type w} {C : Type u} [] (f : βC) :
f f default

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

Equations
• One or more equations did not get rendered due to their size.
Instances For
def CategoryTheory.Limits.Pi.reindex {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] :
∏ᶜ f ε ∏ᶜ f

Reindex a categorical product via an equivalence of the index types.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem CategoryTheory.Limits.Pi.reindex_hom_π_assoc {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) {Z : C} (h : f (ε b) Z) :
@[simp]
theorem CategoryTheory.Limits.Pi.reindex_hom_π {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) :
@[simp]
theorem CategoryTheory.Limits.Pi.reindex_inv_π_assoc {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) {Z : C} (h : (f ε) b Z) :
@[simp]
theorem CategoryTheory.Limits.Pi.reindex_inv_π {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) :
=
def CategoryTheory.Limits.Sigma.reindex {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] :
f ε f

Reindex a categorical coproduct via an equivalence of the index types.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem CategoryTheory.Limits.Sigma.ι_reindex_hom_assoc {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) {Z : C} (h : f Z) :
@[simp]
theorem CategoryTheory.Limits.Sigma.ι_reindex_hom {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) :
=
@[simp]
theorem CategoryTheory.Limits.Sigma.ι_reindex_inv_assoc {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) {Z : C} (h : f ε Z) :
@[simp]
theorem CategoryTheory.Limits.Sigma.ι_reindex_inv {β : Type w} {C : Type u} {γ : Type w'} (ε : β γ) (f : γC) [] (b : β) :