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Mathlib.CategoryTheory.Limits.Shapes.IsTerminal

Initial and terminal objects in a category. #

In this file we define the predicates IsTerminal and IsInitial as well as the class InitialMonoClass.

The classes HasTerminal and HasInitial and the associated notations for terminal and inital objects are defined in Terminal.lean.

References #

Construct a cone for the empty diagram given an object.

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Construct a cocone for the empty diagram given an object.

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@[reducible, inline]
abbrev CategoryTheory.Limits.IsTerminal {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) :
Type (max (max 0 u₁) v₁)

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

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@[reducible, inline]
abbrev CategoryTheory.Limits.IsInitial {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) :
Type (max (max 0 u₁) v₁)

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

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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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def CategoryTheory.Limits.IsTerminal.ofUniqueHom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y : C} (h : (X : C) → X Y) (uniq : ∀ (X : C) (m : X Y), m = h X) :

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

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If α is a preorder with top, then is a terminal object.

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Transport a term of type IsTerminal across an isomorphism.

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If X and Y are isomorphic, then X is terminal iff Y is.

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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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def CategoryTheory.Limits.IsInitial.ofUniqueHom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} (h : (Y : C) → X Y) (uniq : ∀ (Y : C) (m : X Y), m = h Y) :

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

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If α is a preorder with bot, then is an initial object.

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Transport a term of type is_initial across an isomorphism.

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If X and Y are isomorphic, then X is initial iff Y is.

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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.

Give the morphism from an initial object to any other.

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theorem CategoryTheory.Limits.IsInitial.hom_ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (t : CategoryTheory.Limits.IsInitial X) (f : X Y) (g : X Y) :
f = g

Any two morphisms from an initial object are equal.

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.

@[simp]
theorem CategoryTheory.Limits.IsTerminal.uniqueUpToIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : C} {T' : C} (hT : CategoryTheory.Limits.IsTerminal T) (hT' : CategoryTheory.Limits.IsTerminal T') :
(hT.uniqueUpToIso hT').hom = hT'.from T
@[simp]
theorem CategoryTheory.Limits.IsTerminal.uniqueUpToIso_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : C} {T' : C} (hT : CategoryTheory.Limits.IsTerminal T) (hT' : CategoryTheory.Limits.IsTerminal T') :
(hT.uniqueUpToIso hT').inv = hT.from T'

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

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  • hT.uniqueUpToIso hT' = { hom := hT'.from T, inv := hT.from T', hom_inv_id := , inv_hom_id := }
@[simp]
theorem CategoryTheory.Limits.IsInitial.uniqueUpToIso_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {I : C} {I' : C} (hI : CategoryTheory.Limits.IsInitial I) (hI' : CategoryTheory.Limits.IsInitial I') :
(hI.uniqueUpToIso hI').inv = hI'.to I
@[simp]
theorem CategoryTheory.Limits.IsInitial.uniqueUpToIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {I : C} {I' : C} (hI : CategoryTheory.Limits.IsInitial I) (hI' : CategoryTheory.Limits.IsInitial I') :
(hI.uniqueUpToIso hI').hom = hI.to I'

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

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  • hI.uniqueUpToIso hI' = { hom := hI.to I', inv := hI'.to I, hom_inv_id := , inv_hom_id := }

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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Replacing an empty cone in IsLimit by another with the same cone point is an equivalence.

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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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Replacing an empty cocone in IsColimit by another with the same cocone point is an equivalence.

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An initial object is terminal in the opposite category.

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

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A terminal object is initial in the opposite category.

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

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A category is an InitialMonoClass 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

    The map from the (any as stated) initial object to any other object is a monomorphism

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

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

    From a functor F : J ⥤ C, given an initial object of J, construct a cone for J. In limitOfDiagramInitial 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 coneOfDiagramInitial 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 limitOfDiagramTerminal 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 coneOfDiagramTerminal is a limit.

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    From a functor F : J ⥤ C, given a terminal object of J, construct a cocone for J. In colimitOfDiagramTerminal 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 coconeOfDiagramTerminal 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 colimitOfDiagramInitial 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 coconeOfDiagramInitial is a colimit.

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    Any morphism between terminal objects is an isomorphism.

    Any morphism between initial objects is an isomorphism.