mathlib documentation

category_theory.limits.shapes.terminal

Initial and terminal objects in a category. #

References #

@[simp]
theorem category_theory.limits.as_empty_cone_π_app {C : Type u₁} [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

Construct a cone for the empty diagram given an object.

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Construct a cocone for the empty diagram given an object.

Equations
@[simp]
theorem category_theory.limits.as_empty_cocone_ι_app {C : Type u₁} [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
@[reducible]
def category_theory.limits.is_terminal {C : Type u₁} [category_theory.category C] (X : C) :
Type (max u₁ v₁)

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

@[reducible]
def category_theory.limits.is_initial {C : Type u₁} [category_theory.category C] (X : C) :
Type (max u₁ v₁)

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

def category_theory.limits.is_terminal_equiv_unique {C : Type u₁} [category_theory.category C] (F : category_theory.discrete pempty C) (Y : C) :
category_theory.limits.is_limit {X := Y, π := {app := category_theory.limits.is_terminal_equiv_unique._aux_1 F Y, naturality' := _}} Π (X : C), unique (X Y)

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

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An object Y is terminal if for every X there is a unique morphism X ⟶ Y (as an instance).

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Transport a term of type is_terminal across an isomorphism.

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def category_theory.limits.is_initial_equiv_unique {C : Type u₁} [category_theory.category C] (F : category_theory.discrete pempty C) (X : C) :
category_theory.limits.is_colimit {X := X, ι := {app := category_theory.limits.is_initial_equiv_unique._aux_1 F X, naturality' := _}} Π (Y : C), unique (X Y)

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

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An object X is initial if for every Y there is a unique morphism X ⟶ Y (as an instance).

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Transport a term of type is_initial across an isomorphism.

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Give the morphism to a terminal object from any other.

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Any two morphisms to a terminal object are equal.

@[simp]
theorem category_theory.limits.is_terminal.comp_from {C : Type u₁} [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

Give the morphism from an initial object to any other.

Equations

Any two morphisms from an initial object are equal.

@[simp]
theorem category_theory.limits.is_initial.to_comp {C : Type u₁} [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

Any morphism from a terminal object is split mono.

Any morphism to an initial object is split epi.

Any morphism from a terminal object is mono.

Any morphism to an initial object is epi.

If T and T' are terminal, they are isomorphic.

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If I and I' are initial, they are isomorphic.

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@[reducible]

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

@[reducible]

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

Being terminal is independent of the empty diagram, its universe, and the cone over it, as long as the cone points are isomorphic.

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Being initial is independent of the empty diagram, its universe, and the cocone over it, as long as the cocone points are isomorphic.

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@[reducible]

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.

@[reducible]

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.

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.

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.

@[reducible]

The map from an object to the terminal object.

@[reducible]

The map to an object from the initial object.

@[simp]

The (unique) isomorphism between the chosen initial object and any other initial object.

Equations
@[simp]

The (unique) isomorphism between the chosen terminal object and any other terminal object.

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@[protected, instance]

Any morphism from a terminal object is split mono.

@[protected, instance]

Any morphism to an initial object is split epi.

@[class]

A category is a initial_mono_class if the canonical morphism of an initial object is a monomorphism. In practice, this is most useful when given an arbitrary morphism out of the chosen initial object, see initial.mono_from. Given a terminal object, this is equivalent to the assumption that the unique morphism from initial to terminal is a monomorphism, which is the second of Freyd's axioms for an AT category.

TODO: This is a condition satisfied by categories with zero objects and morphisms.

Instances of this typeclass

To show a category is a initial_mono_class it suffices to give an initial object such that every morphism out of it is a monomorphism.

To show a category is a initial_mono_class it suffices to show every morphism out of the initial object is a monomorphism.

To show a category is a initial_mono_class it suffices to show the unique morphism from an initial object to a terminal object is a monomorphism.

To show a category is a initial_mono_class it suffices to show the unique morphism from the initial object to a terminal object is a monomorphism.

The comparison morphism from the image of a terminal object to the terminal object in the target category. This is an isomorphism iff G preserves terminal objects, see category_theory.limits.preserves_terminal.of_iso_comparison.

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Instances for category_theory.limits.terminal_comparison

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

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Instances for category_theory.limits.initial_comparison

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.

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

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From a functor F : J ⥤ C, given a terminal object of J, construct a cone for J, provided that the morphisms in the diagram are isomorphisms. In limit_of_diagram_terminal we show it is a limit cone.

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From a functor F : J ⥤ C, given a terminal object of J and that the morphisms in the diagram are isomorphisms, show the cone cone_of_diagram_terminal is a limit.

Equations
@[reducible]

For a functor F : J ⥤ C, if J has a terminal object and all the morphisms in the diagram are isomorphisms, then the image of the terminal object is isomorphic to the limit of F.

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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.

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

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From a functor F : J ⥤ C, given an initial object of J, construct a cocone for J, provided that the morphisms in the diagram are isomorphisms. In colimit_of_diagram_initial we show it is a colimit cocone.

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From a functor F : J ⥤ C, given an initial object of J and that the morphisms in the diagram are isomorphisms, show the cone cocone_of_diagram_initial is a colimit.

Equations
@[reducible]

For a functor F : J ⥤ C, if J has an initial object and all the morphisms in the diagram are isomorphisms, then the image of the initial object is isomorphic to the colimit of F.

Equations

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

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