category_theory.limits.shapes.terminal

@[simp]

theorem category_theory.limits.as_empty_cone_π_app {C : Type u} [category_theory.category C] (X : C) (X_1 : category_theory.discrete pempty) :

(category_theory.limits.as_empty_cone X).π.app X_1 = category_theory.limits.as_empty_cone._aux_1 X X_1

source

@[simp]

theorem category_theory.limits.as_empty_cone_X {C : Type u} [category_theory.category C] (X : C) :

(category_theory.limits.as_empty_cone X).X = X

source

def category_theory.limits.as_empty_cone {C : Type u} [category_theory.category C] (X : C) :

category_theory.limits.cone (category_theory.functor.empty C)

Construct a cone for the empty diagram given an object.

Equations

category_theory.limits.as_empty_cone X = {X := X, π := {app := category_theory.limits.as_empty_cone._aux_1 X, naturality' := _}}

source

@[simp]

theorem category_theory.limits.as_empty_cocone_X {C : Type u} [category_theory.category C] (X : C) :

(category_theory.limits.as_empty_cocone X).X = X

source

def category_theory.limits.as_empty_cocone {C : Type u} [category_theory.category C] (X : C) :

category_theory.limits.cocone (category_theory.functor.empty C)

Construct a cocone for the empty diagram given an object.

Equations

category_theory.limits.as_empty_cocone X = {X := X, ι := {app := category_theory.limits.as_empty_cocone._aux_1 X, naturality' := _}}

source

@[simp]

theorem category_theory.limits.as_empty_cocone_ι_app {C : Type u} [category_theory.category C] (X : C) (X_1 : category_theory.discrete pempty) :

(category_theory.limits.as_empty_cocone X).ι.app X_1 = category_theory.limits.as_empty_cocone._aux_1 X X_1

source

def category_theory.limits.is_terminal {C : Type u} [category_theory.category C] (X : C) :

Type (max u v)

X is terminal if the cone it induces on the empty diagram is limiting.

source

def category_theory.limits.is_initial {C : Type u} [category_theory.category C] (X : C) :

Type (max u v)

X is initial if the cocone it induces on the empty diagram is colimiting.

source

def category_theory.limits.is_terminal.of_unique {C : Type u} [category_theory.category C] (Y : C) [h : Π (X : C), unique (X ⟶ Y)] :

category_theory.limits.is_terminal Y

An object Y is terminal if for every X there is a unique morphism X ⟶ Y.

Equations

category_theory.limits.is_terminal.of_unique Y = {lift := λ (s : category_theory.limits.cone (category_theory.functor.empty C)), default (s.X ⟶ Y), fac' := _, uniq' := _}

source

def category_theory.limits.is_terminal.of_iso {C : Type u} [category_theory.category C] {Y Z : C} (hY : category_theory.limits.is_terminal Y) (i : Y ≅ Z) :

category_theory.limits.is_terminal Z

Transport a term of type is_terminal across an isomorphism.

Equations

hY.of_iso i = category_theory.limits.is_limit.of_iso_limit hY {hom := {hom := i.hom, w' := _}, inv := {hom := i.symm.hom, w' := _}, hom_inv_id' := _, inv_hom_id' := _}

source

def category_theory.limits.is_initial.of_unique {C : Type u} [category_theory.category C] (X : C) [h : Π (Y : C), unique (X ⟶ Y)] :

category_theory.limits.is_initial X

An object X is initial if for every Y there is a unique morphism X ⟶ Y.

Equations

category_theory.limits.is_initial.of_unique X = {desc := λ (s : category_theory.limits.cocone (category_theory.functor.empty C)), default (X ⟶ s.X), fac' := _, uniq' := _}

source

def category_theory.limits.is_initial.of_iso {C : Type u} [category_theory.category C] {X Y : C} (hX : category_theory.limits.is_initial X) (i : X ≅ Y) :

category_theory.limits.is_initial Y

Transport a term of type is_initial across an isomorphism.

Equations

hX.of_iso i = category_theory.limits.is_colimit.of_iso_colimit hX {hom := {hom := i.hom, w' := _}, inv := {hom := i.symm.hom, w' := _}, hom_inv_id' := _, inv_hom_id' := _}

source

def category_theory.limits.is_terminal.from {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_terminal X) (Y : C) :

Y ⟶ X

Give the morphism to a terminal object from any other.

Equations

t.from Y = t.lift (category_theory.limits.as_empty_cone Y)

source

theorem category_theory.limits.is_terminal.hom_ext {C : Type u} [category_theory.category C] {X Y : C} (t : category_theory.limits.is_terminal X) (f g : Y ⟶ X) :

f = g

Any two morphisms to a terminal object are equal.

source

@[simp]

theorem category_theory.limits.is_terminal.comp_from {C : Type u} [category_theory.category C] {Z : C} (t : category_theory.limits.is_terminal Z) {X Y : C} (f : X ⟶ Y) :

f ≫ t.from Y = t.from X

source

@[simp]

theorem category_theory.limits.is_terminal.from_self {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_terminal X) :

t.from X = 𝟙 X

source

def category_theory.limits.is_initial.to {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_initial X) (Y : C) :

X ⟶ Y

Give the morphism from an initial object to any other.

Equations

t.to Y = t.desc (category_theory.limits.as_empty_cocone Y)

source

theorem category_theory.limits.is_initial.hom_ext {C : Type u} [category_theory.category C] {X Y : C} (t : category_theory.limits.is_initial X) (f g : X ⟶ Y) :

f = g

Any two morphisms from an initial object are equal.

source

@[simp]

theorem category_theory.limits.is_initial.to_comp {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_initial X) {Y Z : C} (f : Y ⟶ Z) :

t.to Y ≫ f = t.to Z

source

@[simp]

theorem category_theory.limits.is_initial.to_self {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_initial X) :

t.to X = 𝟙 X

source

theorem category_theory.limits.is_terminal.mono_from {C : Type u} [category_theory.category C] {X Y : C} (t : category_theory.limits.is_terminal X) (f : X ⟶ Y) :

category_theory.mono f

Any morphism from a terminal object is mono.

source

theorem category_theory.limits.is_initial.epi_to {C : Type u} [category_theory.category C] {X Y : C} (t : category_theory.limits.is_initial X) (f : Y ⟶ X) :

category_theory.epi f

Any morphism to an initial object is epi.

source

def category_theory.limits.has_terminal (C : Type u) [category_theory.category C] :

Prop

A category has a terminal object if it has a limit over the empty diagram. Use has_terminal_of_unique to construct instances.

source

def category_theory.limits.has_initial (C : Type u) [category_theory.category C] :

Prop

A category has an initial object if it has a colimit over the empty diagram. Use has_initial_of_unique to construct instances.

source

def category_theory.limits.terminal (C : Type u) [category_theory.category C] [category_theory.limits.has_terminal C] :

C

An arbitrary choice of terminal object, if one exists. You can use the notation ⊤_ C. This object is characterized by having a unique morphism from any object.

source

def category_theory.limits.initial (C : Type u) [category_theory.category C] [category_theory.limits.has_initial C] :

C

An arbitrary choice of initial object, if one exists. You can use the notation ⊥_ C. This object is characterized by having a unique morphism to any object.

source

theorem category_theory.limits.has_terminal_of_unique {C : Type u} [category_theory.category C] (X : C) [h : Π (Y : C), unique (Y ⟶ X)] :

category_theory.limits.has_terminal C

We can more explicitly show that a category has a terminal object by specifying the object, and showing there is a unique morphism to it from any other object.

source

theorem category_theory.limits.has_initial_of_unique {C : Type u} [category_theory.category C] (X : C) [h : Π (Y : C), unique (X ⟶ Y)] :

category_theory.limits.has_initial C

We can more explicitly show that a category has an initial object by specifying the object, and showing there is a unique morphism from it to any other object.

source

def category_theory.limits.terminal.from {C : Type u} [category_theory.category C] [category_theory.limits.has_terminal C] (P : C) :

P ⟶ ⊤_C

The map from an object to the terminal object.

source

def category_theory.limits.initial.to {C : Type u} [category_theory.category C] [category_theory.limits.has_initial C] (P : C) :

⊥_C ⟶ P

The map to an object from the initial object.

source

@[instance]

def category_theory.limits.unique_to_terminal {C : Type u} [category_theory.category C] [category_theory.limits.has_terminal C] (P : C) :

unique (P ⟶ ⊤_C)

Equations

category_theory.limits.unique_to_terminal P = {to_inhabited := {default := category_theory.limits.terminal.from P}, uniq := _}

source

@[instance]

def category_theory.limits.unique_from_initial {C : Type u} [category_theory.category C] [category_theory.limits.has_initial C] (P : C) :

unique (⊥_C ⟶ P)

Equations

category_theory.limits.unique_from_initial P = {to_inhabited := {default := category_theory.limits.initial.to P}, uniq := _}

source

@[simp]

theorem category_theory.limits.terminal.comp_from {C : Type u} [category_theory.category C] [category_theory.limits.has_terminal C] {P Q : C} (f : P ⟶ Q) :

f ≫ category_theory.limits.terminal.from Q = category_theory.limits.terminal.from P

source

@[simp]

theorem category_theory.limits.initial.to_comp {C : Type u} [category_theory.category C] [category_theory.limits.has_initial C] {P Q : C} (f : P ⟶ Q) :

category_theory.limits.initial.to P ≫ f = category_theory.limits.initial.to Q

source

def category_theory.limits.terminal_is_terminal {C : Type u} [category_theory.category C] [category_theory.limits.has_terminal C] :

category_theory.limits.is_terminal (⊤_C)

A terminal object is terminal.

Equations

category_theory.limits.terminal_is_terminal = {lift := λ (s : category_theory.limits.cone (category_theory.functor.empty C)), category_theory.limits.terminal.from s.X, fac' := _, uniq' := _}

source

def category_theory.limits.initial_is_initial {C : Type u} [category_theory.category C] [category_theory.limits.has_initial C] :

category_theory.limits.is_initial (⊥_C)

An initial object is initial.

Equations

category_theory.limits.initial_is_initial = {desc := λ (s : category_theory.limits.cocone (category_theory.functor.empty C)), category_theory.limits.initial.to s.X, fac' := _, uniq' := _}

source

@[instance]

def category_theory.limits.terminal.mono_from {C : Type u} [category_theory.category C] {Y : C} [category_theory.limits.has_terminal C] (f : ⊤_C ⟶ Y) :

category_theory.mono f

Any morphism from a terminal object is mono.

source

@[instance]

def category_theory.limits.initial.epi_to {C : Type u} [category_theory.category C] {Y : C} [category_theory.limits.has_initial C] (f : Y ⟶ ⊥_C) :

category_theory.epi f

Any morphism to an initial object is epi.

source

def category_theory.limits.terminal_op_of_initial {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_initial X) :

category_theory.limits.is_terminal (opposite.op X)

An initial object is terminal in the opposite category.

Equations

category_theory.limits.terminal_op_of_initial t = {lift := λ (s : category_theory.limits.cone (category_theory.functor.empty Cᵒᵖ)), (t.to (opposite.unop s.X)).op, fac' := _, uniq' := _}

source

def category_theory.limits.terminal_unop_of_initial {C : Type u} [category_theory.category C] {X : Cᵒᵖ} (t : category_theory.limits.is_initial X) :

category_theory.limits.is_terminal (opposite.unop X)

An initial object in the opposite category is terminal in the original category.

Equations

category_theory.limits.terminal_unop_of_initial t = {lift := λ (s : category_theory.limits.cone (category_theory.functor.empty C)), (t.to (opposite.op s.X)).unop, fac' := _, uniq' := _}

source

def category_theory.limits.initial_op_of_terminal {C : Type u} [category_theory.category C] {X : C} (t : category_theory.limits.is_terminal X) :

category_theory.limits.is_initial (opposite.op X)

A terminal object is initial in the opposite category.

Equations

category_theory.limits.initial_op_of_terminal t = {desc := λ (s : category_theory.limits.cocone (category_theory.functor.empty Cᵒᵖ)), (t.from (opposite.unop s.X)).op, fac' := _, uniq' := _}

source

def category_theory.limits.initial_unop_of_terminal {C : Type u} [category_theory.category C] {X : Cᵒᵖ} (t : category_theory.limits.is_terminal X) :

category_theory.limits.is_initial (opposite.unop X)

A terminal object in the opposite category is initial in the original category.

Equations

category_theory.limits.initial_unop_of_terminal t = {desc := λ (s : category_theory.limits.cocone (category_theory.functor.empty C)), (t.from (opposite.op s.X)).unop, fac' := _, uniq' := _}

source

def category_theory.limits.cone_of_diagram_initial {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_initial X) (F : J ⥤ C) :

category_theory.limits.cone F

From a functor F : J ⥤ C, given an initial object of J, construct a cone for J. In limit_of_diagram_initial we show it is a limit cone.

Equations

category_theory.limits.cone_of_diagram_initial tX F = {X := F.obj X, π := {app := λ (j : J), F.map (tX.to j), naturality' := _}}

source

@[simp]

theorem category_theory.limits.cone_of_diagram_initial_π_app {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_initial X) (F : J ⥤ C) (j : J) :

(category_theory.limits.cone_of_diagram_initial tX F).π.app j = F.map (tX.to j)

source

@[simp]

theorem category_theory.limits.cone_of_diagram_initial_X {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_initial X) (F : J ⥤ C) :

(category_theory.limits.cone_of_diagram_initial tX F).X = F.obj X

source

def category_theory.limits.limit_of_diagram_initial {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_initial X) (F : J ⥤ C) :

category_theory.limits.is_limit (category_theory.limits.cone_of_diagram_initial tX F)

From a functor F : J ⥤ C, given an initial object of J, show the cone cone_of_diagram_initial is a limit.

Equations

category_theory.limits.limit_of_diagram_initial tX F = {lift := λ (s : category_theory.limits.cone F), s.π.app X, fac' := _, uniq' := _}

source

def category_theory.limits.limit_of_initial {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] (F : J ⥤ C) [category_theory.limits.has_initial J] [category_theory.limits.has_limit F] :

category_theory.limits.limit F ≅ F.obj (⊥_J)

For a functor F : J ⥤ C, if J has an initial object then the image of it is isomorphic to the limit of F.

Equations

category_theory.limits.limit_of_initial F = (category_theory.limits.limit.is_limit F).cone_point_unique_up_to_iso (category_theory.limits.limit_of_diagram_initial category_theory.limits.initial_is_initial F)

source

@[simp]

theorem category_theory.limits.cocone_of_diagram_terminal_ι_app {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_terminal X) (F : J ⥤ C) (j : J) :

(category_theory.limits.cocone_of_diagram_terminal tX F).ι.app j = F.map (tX.from j)

source

def category_theory.limits.cocone_of_diagram_terminal {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_terminal X) (F : J ⥤ C) :

category_theory.limits.cocone F

From a functor F : J ⥤ C, given a terminal object of J, construct a cocone for J. In colimit_of_diagram_terminal we show it is a colimit cocone.

Equations

category_theory.limits.cocone_of_diagram_terminal tX F = {X := F.obj X, ι := {app := λ (j : J), F.map (tX.from j), naturality' := _}}

source

@[simp]

theorem category_theory.limits.cocone_of_diagram_terminal_X {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_terminal X) (F : J ⥤ C) :

(category_theory.limits.cocone_of_diagram_terminal tX F).X = F.obj X

source

def category_theory.limits.colimit_of_diagram_terminal {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {X : J} (tX : category_theory.limits.is_terminal X) (F : J ⥤ C) :

category_theory.limits.is_colimit (category_theory.limits.cocone_of_diagram_terminal tX F)

From a functor F : J ⥤ C, given a terminal object of J, show the cocone cocone_of_diagram_terminal is a colimit.

Equations

category_theory.limits.colimit_of_diagram_terminal tX F = {desc := λ (s : category_theory.limits.cocone F), s.ι.app X, fac' := _, uniq' := _}

source

def category_theory.limits.colimit_of_terminal {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] (F : J ⥤ C) [category_theory.limits.has_terminal J] [category_theory.limits.has_colimit F] :

category_theory.limits.colimit F ≅ F.obj (⊤_J)

For a functor F : J ⥤ C, if J has a terminal object then the image of it is isomorphic to the colimit of F.

Equations

category_theory.limits.colimit_of_terminal F = (category_theory.limits.colimit.is_colimit F).cocone_point_unique_up_to_iso (category_theory.limits.colimit_of_diagram_terminal category_theory.limits.terminal_is_terminal F)

source

def category_theory.limits.terminal_comparison {C : Type u} [category_theory.category C] {D : Type u₂} [category_theory.category D] (G : C ⥤ D) [category_theory.limits.has_terminal C] [category_theory.limits.has_terminal D] :

G.obj (⊤_C) ⟶ ⊤_D

The comparison morphism from the image of a terminal object to the terminal object in the target category.

Equations

category_theory.limits.terminal_comparison G = category_theory.limits.terminal.from (G.obj (⊤_C))

source

def category_theory.limits.initial_comparison {C : Type u} [category_theory.category C] {D : Type u₂} [category_theory.category D] (G : C ⥤ D) [category_theory.limits.has_initial C] [category_theory.limits.has_initial D] :

⊥_D ⟶ G.obj (⊥_C)

The comparison morphism from the initial object in the target category to the image of the initial object.

Equations

category_theory.limits.initial_comparison G = category_theory.limits.initial.to (G.obj (⊥_C))

source

theorem category_theory.limits.is_iso_π_of_is_initial {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {j : J} (I : category_theory.limits.is_initial j) (F : J ⥤ C) [category_theory.limits.has_limit F] :

category_theory.is_iso (category_theory.limits.limit.π F j)

If j is initial in the index category, then the map limit.π F j is an isomorphism.

source

@[instance]

def category_theory.limits.is_iso_π_initial {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] [category_theory.limits.has_initial J] (F : J ⥤ C) [category_theory.limits.has_limit F] :

category_theory.is_iso (category_theory.limits.limit.π F (⊥_J))

source

theorem category_theory.limits.is_iso_ι_of_is_terminal {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] {j : J} (I : category_theory.limits.is_terminal j) (F : J ⥤ C) [category_theory.limits.has_colimit F] :

category_theory.is_iso (category_theory.limits.colimit.ι F j)

If j is terminal in the index category, then the map colimit.ι F j is an isomorphism.

source

@[instance]

def category_theory.limits.is_iso_ι_terminal {C : Type u} [category_theory.category C] {J : Type v} [category_theory.small_category J] [category_theory.limits.has_terminal J] (F : J ⥤ C) [category_theory.limits.has_colimit F] :

category_theory.is_iso (category_theory.limits.colimit.ι F (⊤_J))

mathlib documentation

Initial and terminal objects in a category. #

References #