open import Cat.Instances.Shape.Terminal
open import Cat.Functor.Adjoint.Hom
open import Cat.Functor.Adjoint
open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Terminal where

module _ {o h} (C : Precategory o h) where
  open Cat.Reasoning C

Terminal objects🔗

An object $⊤$ of a category $C$ is said to be terminal if it admits a unique map from any other object:

  is-terminal : Ob → Type _
  is-terminal ob = ∀ x → is-contr (Hom x ob)

  record Terminal : Type (o ⊔ h) where
    field
      top : Ob
      has⊤ : is-terminal top

We refer to the centre of contraction as !. Since it inhabits a contractible type, it is unique.

    ! : ∀ {x} → Hom x top
    ! = has⊤ _ .centre

    !-unique : ∀ {x} (h : Hom x top) → ! ≡ h
    !-unique = has⊤ _ .paths

    !-unique₂ : ∀ {x} (f g : Hom x top) → f ≡ g
    !-unique₂ = is-contr→is-prop (has⊤ _)

  open Terminal

Uniqueness🔗

If a category has two terminal objects $t_{1}$ and $t_{2},$ then there is a unique isomorphism $t_{1} ≅ t_{2} .$ We first establish the isomorphism: Since $t_{1}$ (resp. $t_{2})$ is terminal, there is a unique map $!_{1} : t_{1} \to t_{2}$ (resp. $!_{2} : t_{2} \to t_{1}).$ To show these maps are inverses, we must show that $!_{1} \circ!_{2}$ is $id;$ But these morphisms inhabit a contractible space, namely the space of maps into $t_{2},$ so they are equal.

  !-invertible : (t1 t2 : Terminal) → is-invertible (! t1 {top t2})
  !-invertible t1 t2 = make-invertible (! t2) (!-unique₂ t1 _ _) (!-unique₂ t2 _ _)

  ⊤-unique : (t1 t2 : Terminal) → top t1 ≅ top t2
  ⊤-unique t1 t2 = invertible→iso (! t2) (!-invertible t2 t1)

Hence, if $C$ is additionally a category, it has a propositional space of terminal objects:

  ⊤-is-prop : is-category C → is-prop Terminal
  ⊤-is-prop ccat x1 x2 i .top =
    ccat .to-path (⊤-unique x1 x2) i

  ⊤-is-prop ccat x1 x2 i .has⊤ ob =
    is-prop→pathp
      (λ i → is-contr-is-prop {A = Hom _
        (ccat .to-path (⊤-unique x1 x2) i)})
      (x1 .has⊤ ob) (x2 .has⊤ ob) i

  is-terminal-iso : ∀ {A B} → A ≅ B → is-terminal A → is-terminal B
  is-terminal-iso isom term x = contr (isom .to ∘ term x .centre) λ h →
    isom .to ∘ term x .centre ≡⟨ ap (isom .to ∘_) (term x .paths _) ⟩
    isom .to ∘ isom .from ∘ h ≡⟨ cancell (isom .invl) ⟩
    h                         ∎

In terms of right adjoints🔗

We prove that the inclusion functor of an object $x$ of $C$ is right adjoint to the unique functor $C \to ⊤$ if and only if $x$ is terminal.

  module _ (x : Ob) (term : is-terminal x) where
    is-terminal→inclusion-is-right-adjoint : !F ⊣ !Const {C = C} x
    is-terminal→inclusion-is-right-adjoint =
      hom-iso→adjoints (e _ .fst) (e _ .snd)
        λ _ _ _ → term _ .paths _
      where
        e : ∀ y → ⊤ ≃ Hom y x
        e y = is-contr→≃ (hlevel 0) (term y)

  module _ (x : Ob) (adj : !F ⊣ !Const {C = C} x) where
    inclusion-is-right-adjoint→is-terminal : is-terminal x
    inclusion-is-right-adjoint→is-terminal y = Equiv→is-hlevel 0
      (Σ-contr-snd (λ _ → hlevel 0) e⁻¹)
      (R-adjunct-is-equiv adj .is-eqv _)

module _ {o h} {C : Precategory o h} where
  open Cat.Reasoning C
  private unquoteDecl eqv = declare-record-iso eqv (quote Terminal)

  instance
    Extensional-Terminal
      : ∀ {ℓr}
      → ⦃ sa : Extensional Ob ℓr ⦄
      → Extensional (Terminal C) ℓr
    Extensional-Terminal ⦃ sa ⦄ =
      embedding→extensional
        (Iso→Embedding eqv ∙emb (fst , Subset-proj-embedding (λ _ → hlevel 1)))
        sa